Advanced SAT Type Questions

Question 1

Congratulations! You’ve studied your way through the core concept decks and arrived here.

What is the purpose of this deck?

Answer 1

The purpose of this deck is to improve your critical thinking skills by showing you a logical solution path and guiding you through a detailed analysis of various SAT problems using the core concepts reviewed in other decks.

There are 4 to 7 problems per topic, in order of increasing difficulty.

Question 2

Are the problems in this deck difficult?

Do they resemble the ones you would see on the test?

Answer 2

The problems in this deck range in difficulty from medium to advanced.

They definitely resemble the actual questions you might see on the test.

Question 3

What do you think you might need to solve the problems in this deck?

Pen and paper?

A calculator?

Answer 3

The questions in this deck are not straightforward. Solving them may require you to work with pen and paper. Hey, Einstein wrote his ideas on napkins and the cuffs of his shirt. Why can’t you? Oh, you, guys don’t wear dress shirts…. LOL

If you can avoid using your calculator, please do so.

Question 4

Does this deck include shortcuts and strategies that every student is looking for in order to beat the test?

Answer 4

Yes, this deck covers key strategies, time saving techniques and shortcuts to help you answer questions more efficiently and effectively.

The two most efficient and effective strategies are, however, to use your brain and practice, practice, practice.

Question 5

It’s tempting to reveal the answer if the problem looks difficult or confusing. Resist the temptation.

What approach should you take to working through problems in this deck?

Answer 5

We suggest taking this approach to maximize the benefits of this deck:

• Give yourself sufficient time to solve each problem step-by-step
• Make sure you understand the explanation in the answer
• The first time you study this deck, rate all the cards “perfect” regardless of whether or not you solved the problem
• If many questions present difficulties for you, review the core concepts again
• In a week, come back to this deck and work through the questions again but now rate them fairly

Question 6

You may want to take a diagnostic test. Depending on your score you could take different approaches to problems.

Are you an advanced student or an intermediate student?

Answer 6

If you are an advanced student; i.e scoring above 650 on the SAT Math section, please, feel free to ignore the steps and solve the problem any way you want. What matters is that you get right answers on a consistent basis.

Students with scores from 450 to 650, please do yourself a favor and stick to the recommended solution path.

Question 7

In this card, we demonstrate the critical thinking process in action.

The cost of a coat is $200. At what retail price should this item be put out on the floor so that the store can offer a 25% discount and still make a 20% profit on the cost?

• (a) 290
• (b) 300
• (c) 320
• (d) 900
• (e) 150

Answer 7

* State the purpose. Find the percent increase/decrease

• Define the question. Find the retail price of the coat
• Extract key info. Cost of the coat is $200. The retail price has to allow for a 25% discount and a 20% profit
• Make logical conclusions. Profit is calculated from the cost while the discount is taken from the retail price
• Identify formulas and strategy.

Increased/Decreased Value =

(100% +/- % increase) x Original Value

• Apply all knowledge.

To make a 20% profit on $200, the store needs to get $240 from the customer.

200 * 1.2 = 240

To account for a 25% off sale, the retail price R (100%) minus 25% has to equal $240.

100% of R - 25% of R = 75% of R = 240.

0. 75R = 240 ⇒ R = $320 - choice (c)
   * Verify the answer against the question. Verify if it makes sense

Question 8

The cost of a coat is $200. At what retail price should this item be put out on the floor so that the store can offer a 25% discount and still make a 20% profit on the cost?

• (a) 290
• (b) 300
• (c) 320
• (d) 900
• (e) 150

Is your answer (c)320? If not, you made a mistake somewhere in your thought process.

Where did your logic fail?

Answer 8

If you didn’t get the correct answer, it’s easy to predict your potential mistake:

• If your answer was (a) 290, you added 20% and 25% together and increased the cost by 45%
• If your answer was (b) 300, you increased $200 by 20%, then added 25% to that number
• (d) 900 and (e) 150 choices are way out of the ballpark

Obviously, both ways are incorrect because the problem clearly states that 20% should be added to the cost of the coat but the 25% discount should be taken off the retail price. Realize that:

cost + 20% = retail - 25%

Question 9

To develop the ability to use many different pieces of information to come up with a solution process to a math problem is to think critically in math.

A good critical thinker first picks apart the question at hand and uses the pieces of the information to devise a process to solve it.

How should you analyze an SAT problem in front of you?

Answer 9

Use this step-by-step analysis for any type of SAT problem. Train your brain to effectively extract key information, then use it to create a solution plan.

• State the purpose of the problem
• Define the question
• Extract key information
• Make logical conclusions from the information given
• Identify math formulas and strategies that may apply
• Apply all your knowledge to arrive at a logical answer
• Make sure your answer is the answer to the question

Question 10

Understanding the purpose of a math problem is the first step in the critical thinking process.

What helps you determine the purpose of an SAT problem?

Answer 10

To determine the purpose of the problem, determine what type of problem it is.

Test makers love to test your ability to solve these types of arithmetic questions, commonly seen on the test:

• percents
• average
• ratios/proportions
• sets/sequences
• divisibility
• number properties
• probability

Question 11

In the next few cards we will review some helpful strategies that may make the problem solving process easier and improve your timing on the test.

What are some strategies that we recommend using in order to get optimal results on the math section of the SAT?

Answer 11

Make these strategies part of your preparation routine for the SAT math section:

• Know core concepts and learn to apply them
• Pick Numbers
• Read actively
• Look for shortcuts
• Avoid traps
• Work back from the answers
• Guess if absolutely necessary

Question 12

Nothing in the world will help you to score high on the SAT if you don’t have the proper core knowledge!

Knowledge of definitions, rules, and formulas should be cemented in your brain and subject to immediate recall. It will help you gain math confidence and solve problems faster.

How do you build a good “math” foundation?

Answer 12

This App is designed to help you study core concepts! Your goal is not to just memorize the facts in math but to understand them.

Study the decks repeatedly and answer practice questions before moving on to the advanced material deck.

Don’t skip tutorials and topic reviews in your other SAT materials.

Question 13

The sum of two positive integers a and b is even. The sum of 2a, b and c is odd. Which statement below must be true?

• (a) a is even
• (b) a is odd
• (c) the product of a and b is even
• (d) if b is even, then c is odd
• (e) if c is even, then the product of a and b is even

If a, b, and c are too abstract, maybe substituting them with numbers would help?

Answer 13

As a matter of fact, this problem lends itself perfectly to picking easy, small numbers to stand for variables. Make sure they abide by the restrictions set in the problem and are positive integers.

a + b = even and 2a + b + c = odd

Realize that an even sum could be the result of either adding two even numbers or adding two odd numbers. So, test odd/odd and even/even pairs.

• a* = 4, b = 2 ⇒ 4 + 2 = 6
• a* = 3, b = 3⇒ 3 + 3 = 6
• a* and b could be either odd or even, but 2a (8 or 6) will always be even.

even 2a + b + c = odd

6 + 3 + 2 = 11 or 8 + 2 + 1 = 11

⇒ b + c must be odd

So, either b is even when c is odd or vice versa. Therefore, choice (d) must be true.

Question 14

How do you know what problems lend themselves to using the “pick numbers” strategy?

Answer 14

It would be helpful to use the “pick numbers” strategy when you see:

• variables in the body of the question or in the answer choices
• problems pertaining to number properties asking for what “must be”, “could be” or “cannot be”
• problems asking to find a fraction or a percent of an unknown whole

Question 15

When you decide to use numbers instead of variables, how do you know which numbers to pick?

Answer 15

• Numbers have to be small and easy to work with

Don’t pick 368 or 5, 246. Pick 1 or 2 or 3. Or 10 or 100.

• Numbers have to fit the problem. Abide by the restrictions

If the problem calls for a positive odd number, don’t pick 4 or a negative number. If it says that a > b, picking b = 5 and a = 3 is wrong!

*** Note: once you have picked your numbers in a word problem, re-read it with numbers instead of variables. It will be easier to process the information.

Question 16

The “pick numbers” strategy sometimes involves testing various options. When should you do that?

Answer 16

Make sure you test all possible choices when not much is specified in a problem.

• Example:* If only “positive” is specified, make sure to pick both a whole number and a fraction to test.
• Example*: If a problems calls for an integer, make sure you test both positive and negative numbers and zero.

Question 17

The Blue Sky limousine service charges $3 for the first mile plus $1.50 for each additional mile traveled. Yellow Cab charges an initial fee of $5 dollars plus $0.50 for each half-mile traveled. Suppose you take a ride for 10 miles with Blue Sky Limo. How many miles can you travel with Yellow Cab if you spend the same amount of money as you did with Blue Sky?

Too many words…. What can help you to process the information better?

Answer 17

For any word problem, you should use active reading skills to sort out the information.

Active Reading involves reading aloud, re-telling, highlighting and drawing diagrams of the problem. Chances are that if you do all of it or some of it, you won’t say “Ooohh, i didn’t notice that” or “That’s what it is??”.

The key here is to realize that the 1st mile with Blue Sky Limo is already accounted for. With Yellow Cab, notice that their charges are per half-mile. If you miss those, you’ll fall into traps that test makers prepared for you.

3 + 1.50 * 9 = 5 + 0.5 * 2x ⇒ x = 11.5

Question 18

While it’s a good habit to always read actively, it’s simply necessary to do when solving word problems on the SAT.

What does “read actively” mean?

Answer 18

Active reading involves keeping your mind working at all times while trying to anticipate where the information is leading as you read it. Don’t let the words just wash over you. Solving word problems requires strong reading comprehension skills.

Ok, let’s learn a few helpful techniques.

Question 19

What are some simple techniques you should practice to improve your active reading skills?

Answer 19

• Read the problem aloud when you practice. Accentuate key information with your voice
• Every word problem is a story. Practice re-telling it after reading it once or twice
• Most important: highlight and/or underline key information
• Create a simple diagram with the information given

Question 20

The product of three consecutive integers is given by the formula: x³ + 3x² + 2x, where x is the first integer. Find the product of 372, 373, and 374.

On the test, you will have five answer choices to pick from.

Let’s try to anticipate what you should look for in the correct answer.

Answer 20

Certainly don’t plug in 372 into the formula. Also, don’t multiply 372 x 373 x 374 even if you have a calculator.

You look for the asnwer choice ending in 4. Why? Multiplying just the unit digits of three numbers results in 24.

What if there is more than one answer that ends in 4? Pick the one that is divisible by 6. Why? The product must be divisible by 3 since it’s the product of three consecutive integers. The two factors are even, so the product is divisible by 2.

Question 21

Look for shortcuts when you see questions that look like they can take a long time to solve.

What helps you find shortcuts?

Answer 21

1. It’s hard to find a shortcut unless you know how to solve the problem the “long” way.
2. Shortcuts don’t always exist so don’t look for a shortcut in every problem.
3. Your ability to find a shortcut is a result of being comfortable with math problems which is achieved by hours of practice.

Question 22

What problems and/or math operations make it necessary to look for shortcuts?

Answer 22

Look for shortcuts when you see:

• Large or confusing fractions or fractional expressions. Always try to reduce first
• Large numbers. Try to factor them into simple, small numbers for easy calculations
• Problems that require long arithmetic calculations

Question 23

a + b + 2c = 200, where a, b, and c are positive integers.

If 2a = 5b and 3a = 10c, find the value of c?

(a) 10
(b) 15
(c) 20
(d) 25
(e) 30

Answer 23

You can solve this problem algebraically by expressing b and c in terms of a, solving for a, then finding c. A faster way for some of you may be to plug the answer choices back into the question.

Let’s pick (c) 20 and plug into the equation 3a = 10c. If c = 20, then 3a = 200. Clearly, this choice doesn’t work since a, in this case, is not an integer.

Since a has to be an integer, 10c must be divisible by 3.

Choice (b) 15 and choice (e) 30 meet this requirement. 15 is too small so the correct answer is (e) 30.

Question 24

When does it make sense to work back from the answer choices?

Answer 24

Backsolve when you have no idea how to solve the problem mathematically or it looks like solving it might take a long time.

Use this strategy when the answer choices are simple, small numbers.

While for some of you plugging in answer choices saves time, we recommend to use backsolving as a “fall back” strategy.

Question 25

If you decide to backsolve, which answer choice do you plug in first?

Answer 25

Start with the middle answer choice “C”.

In some cases work backwards from choice “E”. For example, when you are asked to find the largest possible value.

Question 26

The makers of the SAT like to take advantage of your uncertainty in math.

They can anticipate where and what kind of mistakes you might make so they intentionally put the most common wrong answers as answer choices, creating so called “traps”.

Is there a way to recognize and avoid “traps”?

Answer 26

To avoid “traps”:

• improve your core knowledge
• read slowly
• solve problems to the end

Don’t stop half-way and try not to semi-guess. Arrive at your own answer, then compare it to the choices available.

Question 27

k, 3k, 6k, 9k, 12k….. where k is not zero. Which of these five numbers is the smallest?

We present this problem to illustrate what common traps you should learn to avoid on the SAT math section.

Answer 27

You cannot determine which of the five numbers in the problem is the smallest with the information given.

If k is a positive integer, then the smallest number is k. But not all numbers are positive integers!

k can be negative. In this case, k is the greatest number, not the smallest.

Question 28

What should you be careful about when you pick numbers to substitute for variables?

Answer 28

When you pick numbers to substitute for variables, make sure to test all possible choices.

Example:

If the problem states that n > 0, n can be a whole number as well as a fraction.

If it says that n is an integer, don’t forget to test negative numbers.

Question 29

Consecutively numbered raffle tickets were given to all the kids in your school. The first ticket was number 29. What was the number of the last ticket if 254 students (one ticket per student) received raffle tickets?

Answer 29

of tickets distributed =

Last ticket # - First ticket # + 1

• x* - 29 + 1 = 254 ⇒ x = 254 + 29 -1 ⇒
• x* = 282

If your answer was 283, you added 29 and 254 but didn’t subtract 1. You’ve fallen into a trap the test makers prepared for you.

*** Change the numbers to small numbers (like 2 and 10) and count.

Question 30

What is one common mistake students make when they count consecutive integers?

*** Note: that mistake may be set as a trap answer by the test makers.

Answer 30

To count consecutive integers in a range, subtract the smaller integer from the larger and then add 1 to the difference.

Example:

How many consecutive integers are between 5 and 10 inclusively? Let’s count. 5, 6, 7, 8, 9, and 10. There are six integers between 5 and 10.

10 - 5 + 1 = 6

Question 31

Suppose you solved a problem correctly but still got a quarter of a point deduction. How could this happen?

You probably fell into a trap….

Answer 31

Make sure you answer the question that is being asked.

Yes, you could have solved the problem correctly but didn’t read the instructions carefully.

The last step in the successful solution process is to evaluate the answer.

Question 32

The ratio of wins to losses of games played this year by your favorite team is ⁷/₁₃.

1. Does it mean that the team played 20 games?
2. If you subtract from or add 5 to both parts of the ratio above, will it be the same ratio?
3. If you multiply or divide both parts of the ratio by 5, will it be the same ratio?

Answer 32

1. The team didnt necessarily play 20 games this year. It may have played 20, 40, … or any multiple of 20 games this year.
2. Subtracting from or adding numbers to the ratio will change the ratio. A 7 to 13 ratio doesn’t equal a 2 to 8 ratio.
3. Multiplying or dividing both parts of the ratio by the same number will not change the ratio. A 7 to 13 ratio equals a 35 to 65 ratio.

Question 33

The ratio of blondes to red heads in your school is 9 : 2, and the ratio of blondes to brunettes is 3 : 5. What is the ratio of brunettes to red heads?

• (a) 5 : 2
• (b) 2 : 5
• (c) 8 : 11
• (d) 15 : 2
• (e) 2 : 15

Hint: beware of traps…

Answer 33

The common part (blonde students) of two ratios is not the same number. The numbers in these ratios are not proportional because the ratios don’t refer to the same whole.

• Blondes* : Reds = 9 : 2
• Blondes* : Brunettes = 3 : 5

Re-state the ratios so that the common part is represented by the same number. The least common multiple of 3 and 9 is 9. Re-state 2nd ratio.

Blondes : Brunettes = 9 : 15.

Only now you can compare the ratios. But… make sure you are answering the question! You are asked for the ratio of brunettes to red heads, so it’s 15 : 2.

Question 34

What are the most common misconceptions about ratios that could be set as trap answers on the SAT?

Answer 34

Students are often confused about these simple concepts.

• Ratios may not represent actual quantities.
• You cannot add or subtract from a ratio or part of the ratio.
• When common parts of two ratios don’t refer to the same whole, the numbers in the ratios are not proportional.

*** Remember, ratios are a lot like fractions. The same rules apply.

Question 35

You read n pages of a book at an average rate of 10 pages a day. On vacation, there was more time to read so you read the remaining n pages at an average rate of 30 pages a day. What was your average rate of reading for the entire book?

• (a) 14
• (b) 15
• (c) 17
• (d) 19
• (e) 20

Hint: try to avoid the trap.

Answer 35

Avg Rate = ^{Total # of pages} / _{Total time}

We know that the book has 2n pages. We don’t know the time it took you to read each part but we can write equations to find it:

time ₁ = ⁿ/₁₀ (n pages at 10ppd rate)

time ₂ = ⁿ/₃₀ (n pages at 30 ppd rate)

To make it easier, pick a number for n that is a multiple of both 10 and 30. Or solve it algebraically.

Total time = time ₁ + time ₂ = ⁿ/₁₀ + ⁿ/₃₀ = ^{<em>3n + n</em>}/₃₀ = ^<em>4n</em>/₃₀

Avg Rate = 2n ÷ ⁴ⁿ/₃₀ = 2n * ³⁰/_4n = 15

The correct answer is choice (b) 15.

Question 36

You read n pages of a book at an average rate of 10 pages/day. On vacation, there was more time to read so you read the remaining n pages at an average rate of 30 pages/day. What was your average rate of reading for the entire book?

• (a) 14
• (b) 15
• (c) 17
• (d) 19
• (e) 20

Have you picked (e) 20 answer choice? Clearly, it’s wrong. Why?

Answer 36

Taking the average of 10 and 30 to find the average rate of reading is wrong! It doesn’t take into account that it took you more time to read the first part of the book because your average rate of reading was slower.

You can pick numbers to solve this problem instead of writing equations.

The LCM of 10 and 30 is 30. Let’s set n as 30 pages. There are 60 pages in the entire book. It took you 3 days (30 ÷ 10) to read the 1st part and 1 day (30 ÷ 30) to read the 2nd part. So, you read 60 pages in 4 days. Your average rate of reading for the entire book is 15 pages/day.

Question 37

What is the most common misconception about finding the average rate that could be set as a trap answer on the SAT?

Answer 37

Many students forget that the average rate of A per B is the total of A divided by the total of B.

For example, the average rate of speed of the whole trip is not the average of the rates of speed on different legs of the trip.

Avg Rate of Speed = ^{Total Distance}/_{Total Time}

Question 38

The population of town A increased by 10% in 5 years. In the next 5 years, the population increased again by 20%. What is the percent increase in the population in 10 years compared to the original population?

• (a) 30%
• (b) 32%
• (c) 38%
• (d) 40%
• (e) 42%

Answer 38

You can solve the problem algebraically but let’s practice the “pick numbers” strategy.

Suppose the original population of town A was 100. In 5 years, it became 110 (100 * 1.1). Over the next 5 years it increased upto 132 (110 * 1.2). The change in population is 32, the original population is 100. Therefore, the population increased by 32%, choice (b).

If you picked choice (a) 30%, you’ve added 20% to 10% and got caught in a trap.

Question 39

You invested $1,000 in stocks. After the first year the value of your portfolio increased by 25%. After the second year, it shrunk by 25%.

1. Does the value of the increase equal the value of the decrease?
2. Is the value of your portfolio after two years the same as the original value? Less? More?

Answer 39

• Even without calculating exact amounts you can determine that the increase is smaller than the decrease. The reason is that the 25% increase is calculated off of 1,000 while the 25% decrease is taken off of a greater amount.
• The value of your portfolio after two years is smaller than the original value.

1st year: 1,000 x 1.25 = 1,250

2nd year: 1,250 x 0.75 = 937.50

Question 40

On the SAT, this problem would be presented with multiple choice answers where some may be designed to be traps.

You invested $1,000 in stocks. After the first year the value of your portfolio increased by 25%. After the second year, it shrunk by 25%. What percent of the original value is the value of your portfolio after two years?

• (a) 150%
• (b) 125.6%
• (c) 100%
• (d) 93.8%
• (e) 75%

Answer 40

1st year: 1,000 x 1.25 = 1,250

2nd year: 1,250 x 0.75 = 937.5

^937.5/₁₀₀₀ * 100% = 93.75%, choice (e).

Answer choices (a) and (c) are traps.

If you picked (a), you didn’t read the problem carefully and thought that both years the value increased by 25%. Then, you added up percent increases.

If you picked (c), you mistakenly assumed that the amount of the increase equals the amount of the decrease.

Question 41

What are some common misconceptions about figuring out percent changes that may be set as trap answers on the SAT?

Answer 41

• Multiple percent changes are not additive. A 20% decrease and an additional 60% decrease off the original price do not equate to an 80% decrease. Percentages can be added or subtracted only when they are calculated from the same amount.
• The same percent increases or decreases don’t necessarily equal the same amounts of decrease of increase.

Question 42

You are having a mental block and can’t solve a problem.

When should you guess?

Answer 42

Guessing is smart as long as you are using “educated” guessing.

• Educated guesses are guesses that are made based on a familiarity with the math concepts in the question being asked
• Advanced students should make educated guesses whenever they can eliminate one answer choice
• Intermediate students should make educated guesses only when they can eliminate two answer choices
• Always guess on grid-ins because points are not lost for wrong answers!

Question 43

How should you manage the limited amount of time you have to get through all questions in a section?

Answer 43

Math questions are arranged in order of difficulty - easy to hard. On the average, you can spend 60 to 90 seconds per question. But, with practice, you will be able to solve easier questions in the beginning of each section faster, leaving more time for harder ones.

• In a section that gives you 25 minutes for 20 questions, try to get those first 10 easy questions solved in about 7 minutes.
• Try to have 2 mins per question towards the end of the section, on the last 3 or 4 problems.

Question 44

What are some of the things you can do to make sure you don’t run out of time?

Answer 44

It’s very important that you manage your time wisely during the test.

• Don’t get stuck on a question. If you see that it’s taking you too long, move on. If time allows, you can come back to it
• All questions are worth the same points. Make sure you answer all easy questions (usually 1 through 10) in each section
• Keep track of time. To get the feel of how much time you are spending per question, take a lot of practice tests as if they were the real tests

Question 45

Remainder problems are very common on the SAT. Most of them are medium difficulty.

What is our strategic advice on how to approach remainder problems on the SAT?

Answer 45

To solve remainder problems, you can use this formula to figure out any missing element.

N = Q x D + R

• N - Number in question (or Dividend)
• Q - quotient
• D - divisor
• R - remainder

Example:

When N is divided by 5, the remainder is 4. What are the first three values of N?

N = Q x 5 + 4

The first three values of N are 9 (when Q = 1), 14 (when Q = 2), and 19 (when Q = 3).

Question 46

When a number is divided by 5, the remainder is 3. What is the remainder when double that number is divided by 5?

How do you solve this problem using the recommended step-by-step approach and our strategic advice?

Answer 46

• The purpose of the problem is to divide certain numbers to find a remainder
• The question is to find a remainder of a certain number when divided by 5
• Extract key info: when a divisor of a certain number is 5, the remainder equals 3
• Identify formula/strategy:

N = Q x D + R

(N - number in question, Q - quotient, D - divisor, R - remainder)

• Apply all knowledge to solve:

If N = Q x 5 + 3 ⇒

2N = 2 (5Q + 3) = 10Q + 6

10Q is divisible by 5. When 6 is divided by 5, the remainder is 1.

• Make sure your answer makes sense and is the answer to the question.

Question 47

When a certain number N is divided by 6, the remainder is 4. Which of the following will have no remainder when divided by 6?

• (a) N + 1
• (b) N + 8
• (c) N + 3
• (d) N + 5
• (e) N + 4

Go through all the steps of the critical thinking process on your own. The back of the card gives you a condensed version.

Answer 47

Once you’ve analyzed the information, set up the formula N = Q x D + R and plug in what you know.

(N - number in question, Q - quotient, D - divisor, R - remainder).

N = Q x 6 + 4. Possible choices for N are 10 (Q = 1), 16 (Q = 2), 22 (Q = 3), etc. Examining answer choices, you will see that adding 8 to those numbers will leave no remainder when divided by 6. The correct answer is (b) N + 8.

Another way to solve it is to use logic. A zero remainder when divided by 6 means that we are looking for multiples of 6; i.e. 6, 12, 18, … What number should you add to 4 to make it one of those multiples? 2, 8, 16, … N + 8 is choice (b).

Question 48

When each student gets 6 pens, there are 4 pens left in the box. If two of the students are absent, then each student in the class can be given 8 pens. How many pens were in the box before distribution?

The back of the card gives you a short version of the process, but if you are an intermediate student, you shouldn’t skip any steps of critical thinking process.

Answer 48

The essence of this divisibility problem is that if a certain number is divided by 6, the remainder is 4. When that number is reduced by 2 and divided by 8, there is no remainder.

Let’s set up the remainder formula.

N = Q x D + R

Realize that Q is, in this case, the number of students, D is the number of pens given to each.

• N* (# of pens) = S (students) x 6 + 4
• N* (# of pens) = S - 2 (students) x 8

6S + 4 = (S - 2) x 8 ⇒ 6S + 4 = 8S - 16

Solving for S, you get that there are 10 students in the class originally. When 2 students are absent, there are 8 students left in the class. 8 x 8 = 64 - # of pens before distribution.

Question 49

When 18 is divided by the positive integer X, the remainder is 3. How many positive integers X of different value can you think of that make this statement true?

How do you analyze information using the steps of the critical thinking process?

Answer 49

Think of purpose, define the question, underline key info, set up the remainder formula, and plug in what you know:

N = Q x D + R

In this case, N = 18, R = 3

18 = Q x D + 3 ⇒ Q x D = 15

There are three pairs of integers whose product is 15. (1; 15), (3; 5), and (5; 3). So, there are three positive integers that make the statement true.

Logically, 18 - 3 = 15. So, we need to find positive integers that are factors of 15. They are 1, 3, 5, and 15 but only 3, 5, and 15 work for this problem. 18 divided by 1 doesn’t leave a remainder of 3.

Question 50

If n is divisible by 3 and m is divisible by 5, which expression below must be divisible by 15? Pick one, two or all three from the answers below.

• i. n * m
• ii. 5 (m + n)
• iii. 5n + 3m

Run thought the steps of the critical thinking process in your head. Which strategy would be best to use here?

Answer 50

You can pick small numbers to substitue for variables. Pick n = 6 and m = 10, plug into the answer choices and determine which ones are divisible by 15. The correct answer is choice i and iii.

You can solve this problem logically or algebraically as well. The number is divisible by 15 if it’s divisible by both 3 and 5.

n is a multiple of 3 and can be expressed as 3x. m is a multiple of 5, so you can write it as 5y. n = 3x; m = 5y.

• n * m = 3x * 5y = 15xy - yes
• 5 (m + n) = 5 (3x + 5y) = 15x + 25y - not always
• 5n + 3m = 5* 3x + 3* 5y - yes

Question 51

A certain integer N is divisible by 5 with no remainder. 38 < N < 88. When N is divided by 4 and by 6, the remainder is 2. What is that integer?

How do you solve this problem using the recommended step-by-step approach?

Answer 51

• Purpose of the problem: Intersection of sets, division operation
• Question: Find N that meets all criteria
• _Key info:_38 < N < 88

When D = 5, R = 0. When D = 4, R = 2. When D = 6, R = 2

• Formulas/Strategy: Find the intersection of 3 sets using just logic.
• Apply all knowledge and solve:

List all multiples of 5 between 38 and 88.

{40, 45, 50, 55, 60, 65, 70, 75, 80, 85}.

The number that is divisible by both 4 and 6 is a multiple of 12. List all multiples of 12 in a target range.

{48, 60, 72, 84}

But our N leaves a remainder of 2 when divided by 4 or 6 so N has to be 2 greater that listed multiple of 12. The only number that’s in both sets is 50.

• Make sure you answered the question. Yes, N = 50

Question 52

Problems pertaining to the average value may look different, but they all have one thing in common. To solve them, you need to know the formula for finding the average (arithmetic mean) of a set of numbers and be able to manipulate it to fit the problem.

How do you find the average of a set of numbers?

Answer 52

Average (A) =

^{Sum of the terms (S)}/_{Number of terms (N)}

and therefore,

Sum = Avg x N

In the next few cards, we will show you how to use these basic formulas to solve different types of average problems.

Question 53

What do you find when you divide the sum of a set of numbers by the average (arithmetic mean) of those numbers?

Answer 53

When you divide the sum of numbers in a set by the average of those numbers, you find how many of those numbers are in the set.

Question 54

The median of five positive integers is 36. The difference between the largest and the smallest integer is 4. What is the average (arithmetic mean) of the numbers?

• (a) 34
• (b) 34.5
• (c) 36
• (d) 36.5
• (e) 37

What are the steps of your critical thinking process?

Answer 54

• The purpose and the question is to find the average of five positive integers number
• Extract key info: The median of five positive integers is 36. The difference between the smallest and the largest is 4
• Make logical conclusions: These five integers must be consecutive since the difference between the smallest and the largest integers is 4
• Identify formulas and strategy: In any set of evenly spaced numbers, the median number is also the average. The answer is 36, choice (c)
• Is it the answer to the question? Yes.

Question 55

The average price of five CDs you bought is $12. If another CD is added to your shopping cart, the average price increases to $13. How expensive is the disc that is added to the original purchase?

Approach the problem using the recommended method to problem solving.

Answer 55

• The purpose is to find the missing term in a set using the average of the set.
• The question is to find the price of the added CD.
• Extract key info: the average price of 5 CDs is $12, the average price of 6 CDs is $13.
• Make logical conclusions: if you know the average and the number of terms, you can find the sum.
• Identify math formulas and strategies:

Sum (S) ÷ Number of terms (N) = Avg

⇒S = Avg x N

• Apply all your knowledge to solve:

Total cost of 5 CDs: 5 x 12 = 60

Total cost of 6 CDs: 6 x 13 = 78

78 - 60 = $18 (price of the added CD)

• Make sure the answer answers the question: yes.

Question 56

The average (arithmetic mean) of a, b, c, and d is f. Express a + b in terms of c, d and f.

• (a) (f + c + d) ÷ 4
• (b) 4f + c - d
• (c) 4f - c - d
• (d) 4(f + d + c)
• (e) d - c - 4f

Use the recommended method to analyze and solve the problem.

Answer 56

• The purpose is to use the average to find the sum of terms
• The question is to express a + b in terms c, d, and f
• Extract key info: The average of 4 terms a, b, c and d is f
• Identify math formulas and strategies: You can use simple, small numbers to replace variables. Or solve it algebraically

Sum = Avg * Number of Terms

• Apply all your knowledge to solve:

(a + b + c + d) ÷ 4 = f ⇒

• a + b + c + d* = 4f ⇒
• a + b* = 4f - c - d …. choice (c)
• Is it the answer to the question? Yes

Question 57

If the average (arithmetic mean) of x and y is 30, and the average of x and z is 45, what is the difference of z and y?

How should you approach a problem like that?

Answer 57

Analyze the purpose, the question and the key info on your own.

• Identify math formulas and strategies:

Sum = Avg * Number of Terms

• Logical conclusion: given the average of two numbers you can find their sum.
• Apply all knowledge to solve:

x + y = 60 x + z = 90

Subtract 1st equation from the 2nd equation.

• z - y* = 30
• Make sure you answered what was asked. Yes

Question 58

The average (arithmetic mean) weight of X people in one group is 140 lbs. The average weight of Y people in another group is 128 lbs. When the weights of all people in both groups are combined, the average weight becomes 130 lbs. What is the ratio of X to Y?

Due to space limitations, not every card will give you all the steps of the critical thinking process. You may refer to previous cards if you don’t see a solution path.

Answer 58

You should analyze the purpose, the question, key information and formulas and strategies that may apply here.

Sum = Avg x Number of Terms

• Key info you are given:

group 1: X ppl, 140 lbs ave

group 2: Y ppl, 128 lbs ave

combined: X + Y ppl, 130 lbs ave
* Use the formula above to find the sum of weights in each group of people:

Sum of weights group 1 = X * 140

Sum of weights group 2 = Y * 128

Combined groups’ weight = (X + Y) * 130

Set up an equation:

X * 140 + Y * 128 = (X + Y) * 130 ⇒

140X + 128Y = 130X + 130Y

10X = 2Y ⇒ X : Y = 2 : 10

Question 59

Each of 7 people wrote a positive integer. The average (arithmetic mean) of these numbers is 9. What is the greatest possible integer that could have been written?

How should you approach this type of problem?

Answer 59

Again, you have to mathematically connect the average, the number of terms, and the sum of terms. The formula that ties all three together is

Sum = Avg x Number of Terms

The key information here is that the average of seven positive numbers is 9. You need to find the greatest of them.

Sum = 7 x 9 = 63

One term is the greatest possible when other terms in a set are the smallest possible. The smallest positive integer is 1. So, 6 out of 7 people could have written 1. 63 - 6 x 1 = 57

The greatest possible integer is 57.

Question 60

The average (arithmetic mean) of two numbers is w. If one more number p is added, what is the average of the three numbers, in terms of w and p?

What should your general approach be to problem solving on the SAT? In this particular problem, what core knowledge should you have?

Answer 60

Your general approach should be the step-by-step logical process taught in this deck. A must know core knowledge is the average formula. The best strategy is to set up the formula and solve the problem algebraically.

Sum = Avg x Number of Terms

Let’s call the first two numbers a and b. Their average is w.

So, (a + b) ÷ 2 = w ⇒ a + b = 2w

One more number p is added to the set. The sum of three numbers is 2w + p. The average of three numbers is:

Avg = (2w + p) ÷ 3

Question 61

The sum of six consecutive odd integers is 60. What is the median of the set of those six numbers and 60?

What core knowledge do you need to solve this problem?

Answer 61

By definition of the average, if the sum of six integers is 60, their average is 10. The average of a set of evenly spaced numbers is also the median of the set, so the median is 10 as well.

In a set with even number of terms, the median is the average of the two middle values, 9 and 11. {5, 7, 9, 11, 13, 15}

Or solving algebraically… n + (n + 2) + (n + 4) + (n + 6) + (n + 8) + (n + 10) = 60

6n + 30 = 60 ⇒ 6n = 30 ⇒ n = 5

If we add 60, we are adding the 7th term. The median becomes 11.

{5, 7, 9, 11, 13, 15, 60}

Question 62

Which of the following operations will never affect the median of a set of numbers?

(a) Increasing the smallest number
(b) Multiplying each number by 5
(c) Dividing each number by 2
(d) Increasing the largest number
(e) Decreasing the largest number

What approach and strategy should you use to solve this problem?

Answer 62

In this problem, concetrate your efforts on analyzing answer choices. Picking numbers works as a strategy and there are no restrictions to the numbers you pick.

For example, the set is {2, 6, 8, 14, 15}.

The median is the middle value of the set. Review each answer choice.

Increasing the smallest number (a) or decreasing the largest number (e) may or may not change the median. Choices (b) and (c) will definitely change the median. The only operation that will never change the median value of a set is increasing the largest number, choice (d).

Question 63

To answer questions pertaining to sets of numbers, you need to know the definition of a union of sets and an intersection of sets.

What is a union of sets?

What is an intersection of sets?

What is an empty set?

Answer 63

A union of sets is all elements in those sets without repeated elements.

An intersection of sets is only common elements that are in all sets.

An empty set is a set with no elements.

Question 64

If one set contains negative integers and another set contains the absolute value of each number if the first set, how many numbers are in the intersection of the two sets?

Answer 64

The set that contains absolute values has all positive integers. Since the first set contains all negative integers and the second set contains all positive integers, their intersection is an empty set.

You have to know the definitions of an intersection of sets and an empty set. The intersection of two sets is a set that includes only common elements in both sets. An empty set has zero elements.

Question 65

Set X =

{all prime numbers between 5 and 20}

Set X U Y =

{all positive odd numbers less than 20}

What is the minimum number of elements in set Y?

• (a) 4
• (b) 5
• (c) 6
• (d) 3
• (e) 2

Answer 65

If you know the definition of a union of sets, you can easily solve this problem. The union of two sets is all elements in both sets without repeating.

Write out the elements of both sets:

• X* = {5, 7, 11, 13, 17, 19}
• X* U Y = {1, 3, 5, 7, 9, 11, 13, 15, 17, 19}

Set Y must be {1, 3, 9, 15}

The minimum number of elements in set Y is 4, choice (a).

Question 66

• Set A = {5, 9, 16, 12, 3}
• Set B = {4, …., 12, ….}
• Intersection of these two sets form another set {5, 16, 12}.

What are the missing elements of set B?

What is the recommeded strategy for this type of problem?

Answer 66

The strategy is to understand the concept of intersections of sets really well so you can work backwards.

The intersection of two sets consists of only elements that are in both sets.

The intersection set consists of 5, 12 and 16. Therefore, these numbers have to be in both set A and set B. 12 is in both sets but 5 and 16 are only in set A. By definition, 5 and 16 have to be in set B as well.

The missing elements of set B are 5 and 16.

Question 67

• The elemets of set A are divisible by 3 or by 4.
• The elements of set B are divisible by 4 or by 5.
• The elements of set C are divisible by 3 or by 5.

List the first five elements of an intersection set of the three sets above.

Answer 67

An intersection set consists of elements common to all three sets.

To figure out the first five elements of the intersection set you can list a certain number of elements of each of the three sets and find the common elements.

A much faster way is to realize that you need to find the least common multiple of 12, 20 and 15. The LCM of these numbers is 60 (3 x 4 x 5). Therefore, the intersection set is a set of multiples of 60.

The first five elements in the intersection of A, B, and C are {60, 120, 180, 240, 300}.

Question 68

• The elemets of set A are divisible by 3 or by 4.
• The elements of set B are divisible by 4 or by 5.
• The elements of set C are divisible by 3 or by 5.

Which combination gives you the set with the least number of elements?

• (a) union of all three sets
• (b) union of two sets intersecting with the 3rd set
• (c) intersection of all three sets

Answer 68

By definition, the union of A, B, and C is all elements of A, B, and C without repetition. The intersection of A, B, and C is the common elements of the three sets.

The union of A, B, and C is the set of all multiples of 12, 20, and 15. It gives you the most number of elements.

The intersection of A, B, and C is the set of multiples of 60 (the LCM of 12, 20, and 15). It gives you the least number of elements. Choice (c) is correct.

Question 69

Strategic advice:

When you see a sequence, look for a pattern. The pattern will determine the type of sequence.

• What type of sequences do you know?
• What is the difference between them?
• How do you find the nth term?

Answer 69

The sequences (or progressions) you need to know for the SAT are arithmetic and geometric.

The difference between them is that in an arithmetic sequence there is a constant difference between two consecutive terms; in a geometric sequence there is a constant ratio between two consecutive terms.

Arithmetic: a_n = a₁ + d * (n - 1)

Geometric: a_n = a₁ * r^{(n - 1)}

Question 70

An arithmetic sequence is given by 302, 298, 294, …. Find the 20th term of this sequence.

Remember the successful problem solving method that we recommend.

Answer 70

* State the purpose: Determine the pattern and terms of a sequence with simple arithmetic operations

• Define the question: Find the 20th term
• Extract key info: You know the first 3 terms of an arithmetic sequence
• Make logical conclusions from the info given: You can find the common difference
• Identify math formulas and strategies that may apply: Core knowledge such as the definition of an arithmetic sequence, the common difference formula and the formula for the nth term

An arithmetic sequence is defined by a common difference (d) between the terms. d = a₂ - a₁ = 298 - 302 = - 4

nth term formula: a_n = a₁ + d * (n - 1)

a₂₀ = 302 + (- 4) * (20 - 1) = 226

Question 71

Which is true for an arithmetic sequence with an even number of elements?

• (i) The mean equals the mode
• (ii) The mean equals the median
• (iii) The mean is the average of the smallest and the largest number in the sequence

What strategy are you going to use?

Answer 71

You can pick small numbers to make up an arithmetic sequence. There is a constant difference between any two consecutive terms.

{3, 9, 15, 21, 27, 33}

The mode is the number that appears most often. The median is the middle value of a set of numbers in order. The arithmetic mean is the average.

Let’s analyze the answer choices.

• (i) false since there is no mode
• (ii) true since the median is 18 ((15 + 21) ÷ 2). The mean is also 18 (108 ÷ 6)
• (iii) true (3 + 33) ÷ 2 = 18

*** In an arithmetic sequence, the terms are evenly spaced. Therefore, the median is also the average of the set. And the average can be figured out by taking the average of the smallest and the largest term.

Question 72

The first term of a geometric sequence is 1. The common ratio is - 2. How many of the first 21 terms are less than 21?

Answer 72

• The purpose: To write a geometric sequence using multiplication
• The question: Find how many terms of the first 21 are
• Extract key info: It’s a geometric sequence. The first term a₁ = 1. The common ratio r = - 2
• Make logical conclusions: Write the sequence with info given and realize that negative and positive numbers alternate
• Determine formulas and strategy: Core knowledge and logic

a₂ = a₁ * r , a₃ = a₂ * r, …..

1, -2, 4, -8, 16, -32, 64, …..

• Apply all you know and solve:

2nd, 4th, 6th, … terms (i.e. even numbered) are negative so they all are less than 21. There are 10 of them among 21 terms. The 1st, 3rd, and 5th terms of the sequence are less than 21.

• 13 terms of the first 21 are less than 21*
• Check the question again

Question 73

In a given geometric sequence, a₁ = 27 and a₃ = 3. What is one-half of a₅ if the average of the first three terms equals 13?

Answer 73

Analyze the purpose, the question, the key information, make a logical conclusion, determine the formulas and the strategy to use on your own.

If the average of the first three terms is 13, their sum is 39. 39 - 3 - 27 = 9.

a₁ = 27, a₂ = 9, a₃ = 3, …

Common ratio r = ¹/₃ ⇒ a₅ = ¹/₃.

Found the answer? No! You were asked to find one-half of a₅. ¹/₃ * ¹/₂ = ¹/₆

Question 74

11, 18, 25, 32, 39 …..

In the arithmetic sequence above, which term would equal 11 + (25 * 7 - 14)?

• (a) 10th
• (b) 13th
• (c) 23rd
• (d) 24th
• (e) 25th

How should you approach this problem?

Answer 74

Your strategy here is to follow the steps of the critical thinking process and have an excellent knowledge of sequences.

Since this is an arithmetic sequence, there is a common difference (d) between its consecutive terms. You should know the formula for the nth term.

• d = a₂ - a₁ = 18 - 11 = 7
• a_n = a₁ + d (n - 1)

Plug in…. a_n = 11 + 7(n - 1)

The trick here is to realize that the original expression in the problem should be distributed to look like the formula.

a_n = 11 + 7 (25 - 2) = 11 + 7 * 23

n - 1 = 23 ⇒ n = 24, choice (d).

Question 75

In a certain arithmetic sequence, the 2nd and the 6th terms are 3 and 11 respectively. What is the sum of the first twenty terms of that sequence?

*** If you thought of manually adding all twenty terms of the sequence, think again!

Answer 75

• According to the formula below, to find the sum, you need to know the 1st and the 20th terms

Sum_n = ¹/₂n (a₁ + a_n)

• The formula for the nth term in an arithmetic sequence requires you to find d (common difference between consecutive terms)

a_n = a₁ + d *(n - 1)

• a₆* = a₁ + 5 * d ; a₂ = a₁ + d
• a₁* = a₆ - 5 * d = a₂ - d
• a₆ - a₂* = 11 - 3 = 4d ⇒ d = 2
• Next, find the 1st and the 20th terms
• a₁* = 3 - 2 = 1
• a₂₀* = 1 + 2(20 - 1) = 39
• Now you know all you need to figure out the sum:
• S₂₀* = 1/2 * 20(1 + 39) = 400

Question 76

In the geometric sequence below, all the numbers are positive. What is the value of b?

a, 4, 16b, ab

Answer 76

• The purpose: To find the missing terms of a geometric sequence.
• The question: Find the value of b
• Key info: You know four terms of the geometric sequence: a, 4, 16b, ab
• Logical conclusion: In a geometric sequence, there is a common ratio between consecutive terms.
• Math formulas and strategy: Set up proportions and cross-multiply to solve

4 : a = 16b : 4 = ab : 16b

16 = 16ab and 16²b² = 4ab ⇒

ab = 1 and 64b = a ⇒

64b² = 1 ⇒ b = ¹/₈

• Does the answer make sense? Plug in a and b to double check

Question 77

What should your strategic approach to percent problems be?

Answer 77

Core knowledge is a must. If the following problems present difficulties, go back to the “Percent, Ratios and Proportions” deck.

All percent problems can be solved easily algebraically. In our view, setting up a proportion to find the unknown is the best way to approach more advanced questions.

You can also pick numbers to substitute for variables or when figuring out a part of the unknown whole.

Question 78

Class A is 75% of class B.

Class C is 50% of class B.

What is the ratio (in percentages) of the number of students in class A to the total number of students in all three classes?

Follow the recommended step-by-step approach.

Answer 78

* State the purpose: Divide numbers to find a ratio

• Define the question: Find the ratio of A to the total T in percentages
• Extract key info: A is 75% of B, C is 50% of B
• Make logical conclusions: A and C are compared to B so B is 100% in this case. Total number of students equals 75% + 100% + 50% = 225% of B
• Identify the strategy: You can solve the problem algebraically or pick a value for the number of students in class B
• Apply all knowledge:

^A/_T = ^{75% of B}/_{225% of B} = 0.33 or 33.3%

• Make sure your answer is the answer to the question. Yes.

Question 79

Increasing the price of an item by 30% is the same as increasing the price by $24. What would be the discounted price of that item if the original price was cut in half?

• (a) 40
• (b) 50
• (c) 60
• (d) 80
• (e) 90

Answer 79

You know that 30% of an original price is $24. Conclude that 100% is the original price of the item and set up a proportion. You can write the proportion 2 ways:

³⁰/₁₀₀ = ²⁴/_x or ²⁴/₃₀ = ^x/₁₀₀

Cross-multiply to solve for x. x = 80

Is choice (d) the correct answer? No. It’s a trap. You found the original price. The question asks for half of the original price. So, it’s 40, choice (a) .

Question 80

If the length of a rectangle is increased by 50% and the width of the same rectangle is decreased by 50%, how does it change the area of the rectangle?

• (a) the area doesn’t change
• (b) the area increases by 50%
• (c) the area decreases by 50%
• (d) the area increases by 25%
• (e) the area decreases by 25%

Use the critical thinking process steps to analyze the problem.

Answer 80

• The purpose: Find a percent decrease or increase
• The question: Find change in the area in %
• Extract key info: Length increases by 50% while width decreases by 50%
• Make logical conclusions: The new length is 150% of the original or 1.5L. The new width is 50% of the original or 0.5W
• Identify formulas and the strategy: Know the percent change formula. Pick simple numbers for length and width or solve it algebraically

Percent change =

^{Change in value}/_{Old value} x 100%

• Apply all knowledge and solve:

Let’s assume L = 10, W = 20 so the original area is 200. New L = 15, new W = 10 so the area is now 150.

200 - 150 = 50 (the amount of change)

50/200 x 100% = 25%

The area decreased by 25%, choice (e)

Question 81

If the length of a rectangle is increased by 50% and the width of the same rectangle is decreased by 50%, how does it change the area of the rectangle?

• (a) the area doesn’t change
• (b) the area increases by 50%
• (c) the area decreases by 50%
• (d) the area increases by 25%
• (e) the area decreases by 25%

Now solve the problem algebraically.

Answer 81

Percent change =

^{Change in value}/_{Old value} x 100%

The original area: L x W

The new area: 1.5L x 0.5W = 0.75 L x W

The difference between them is 0.25. Therefore, you can make the conclusion that the area decreased by 25%, choice (e).

Question 82

A pair of jeans costs n dollars. The store first puts it on sale for m% off, then discounts another p dollars. Which expression shows the final price of the item in dollars?

• (a) ¹/₁₀₀nm - p
• (b) p + (1 - ¹/₁₀₀m) n
• (c) ((1 - ¹/₁₀₀m) n) - p
• (d) ((1 - 100m) n) - p
• (e) 100nm - p

Answer 82

Analyze the problem the way we recommend to approach all SAT questions. You can solve it algebraically or pick numbers for n, m and p.

Decreased value =

(100 - % decrease) * original value

Let n be $50, m - 10% and p - $20. Now, figure out the cost after two reductions (50 - 10% of 50) - 20 =

(50 - ¹⁰/₁₀₀* 50) - 20 …… or

(n - ^m/₁₀₀* n) - p = ((1 - ¹/₁₀₀m) n) - p

Realize that you needed to distribute to make the expression look like one of the answer choices. Choice (c) is the correct answer.

Question 83

John bought a lap top at a discount of 25%. He installed some software and sold it to Mark for 25% more than he paid. Mark paid $200 for upgrades, increased his cost by 8% and sold it to Jim. If Jim paid $1,377 for the lap top, what was its original retail price?

What is the recommended strategy for this type of question?

Answer 83

This is a long-winded word problem. We suggest that you use the active reading strategy for these types of questions. When working with paper SAT materials, highlight and/or underline key information. Re-tell the story described in the problem. Diagram the information and clearly label it. If you are an intermediate student, do not go straight to setting up an equation.

• x* - original price ; 0.75x - John’s cost
  0. 75x * 1.25 - Mark’s cost
  (1. 25 * 0.75x + 200) * 1.08 - Jim’s cost

Now, let’s solve for x.

(1. 25 * 0.75x + 200) * 1.08 = 1,377
0. 94 x * 1.08 + 216 = 1,377 ⇒ x = 1,139

Question 84

What are the main subjects of Number Theory and Operations being tested on the SAT?

Answer 84

The problems on Numbers Theory and Operations primarily test your knowledge of properties of the integers such as prime and composite numbers, odd and even numbers, positive and negative numbers. You should know how to find the LCM, the GCF and prime factorization of two or more integers.

Review “Classification and Definitions” and “Factors and Multiples” decks to be certain you know basic terms, concepts and rules.

Question 85

How should you approach problems testing your understanding of Number Theory and Operations?

Answer 85

• If the problem and the answer choices contain variables, you can substitute them with small numbers
• If you can’t solve it with straightforward math, and the answers have numerical values, try working backwards from them

Question 86

Number k is a prime number. Number A is a composite number (less than k) which has two unique prime number factors n and m.

How many unique prime factors does the square of the product of k and A have?

Answer 86

We need to find the number of unique prime factors in the square of k * A.

You can easily solve it algebraically. If you decide to pick numbers for variables, make sure they fit the problem.

(k * A)² = (k * m * n)²

The product of k and A has three unique prime factors. Squaring the product will change the number of factors, however, it will not change the number of unique prime factors.

The square of the product of k and A has three unique prime number factors.

Question 87

Number X is a product of three unique prime numbers greater than 2. How many positive factors including 1 and X does X have?

• (a) 5
• (b) 6
• (c) 7
• (d) 8
• (e) 9

What strategy will you pick to solve this problem?

Answer 87

Whether you solve it algebraically or pick numbers, beware of the answers that are too obvious.

Let’s find X by figuring out the product of three unique prime numbers greater than 2. X = 3 x 5 x 7 = 105. The obvious factors of 105 are 1, 3, 5, 7, and 105. So are there 5 factors? No. (a) choice is a trap.

What about 15 (3 x 5), 21 (3 x 7), and 35 (5 x 7)? 105 is also divisible by 15, 21 and 35, i.e. the products of any two of the three factors.

105 has 8 positive factors, choice (d).

Algebraic method: X = k * m * n, where k, m, and n are prime factors greater than 2. Factors of X are k, m, n, k * m, k * n, m * n, k * m * n. And 1 - a factor of every number.

Question 88

• a, b* and c are positive integers. Which of the following lists all the possible ways for the sum of these three integers to be an even number?
  (i) one of the numbers is even.
  (ii) two of the numbers are even.
  (iii) three of the numbers are even.
• (i) only
• (ii) only
• (i) and (iii)
• (i) and (ii)
• (i), (ii) and (iii)

Answer 88

You can replace variables with numbers but remember you are looking for all possible ways to make the sum an even number. Let’s test the triples below:

(3, 4, 5) , (4, 5, 6) , (2, 4, 6) , (1, 3, 5)

Only the combinations with either one even number or three even numbers result in an even sum. The correct answer is (i) and (iii).

Another way to solve this problem is to express odd as even + 1. We need to test (i) and (ii) since (iii) obviously works.

Even + Odd + Odd = Even + Even + 1 + Even + 1 = 3 Even + 2 is definitely even.

Even + Even + Odd = Even + Even + Even + 1 is odd.

Question 89

The traffic lights at three different intersections change after 12, 36 and 60 seconds respectively. All three traffic lights changed at 9 a.m.

What is the next time they will change simultaneously?

Answer 89

This problem is pretty straightforward if you know what you really need to find in order to find the time the traffic lights will all change again.

You are looking for the Least Common Multiple (LCM) of 12, 36 and 60. Remember how to find the LCM by using prime factorization?

• 12 = 2 x 2 x 3
• 36 = 2 x 2 x 3 x 3
• 60 = 2 x 2 x 3 x 5
• LCM = 2 x 2 x 3 x 3 x 5 = 180

180 seconds is 3 minutes. So, the next time all three lights change simultaneously is 9:03 a.m.

Question 90

P is a prime number. (P - 1) divided by 4 or by 6 is prime. (P + 1) divided by 2 is also a prime number. Which of the following could be P?

• (a) 13
• (b) 19
• (c) 23
• (d) 29
• (e) 37

What strategy is the easiest to use here?

Answer 90

The backsolving strategy will help you find the right answer in seconds.

Start with the middle choice (c) and work your way up or down. Choice (a) meets all the requirements of the problem.

Logically, if P - 1 is divisible by both 4 and 6, this number is a multiple of 12. With that in mind, examine the answers to find multiples of 12 + 1. (a), (c), and (e) work. However, only (a) 13 fits the problem.

Question 91

DC

+CD

ABA

Each of the four letters above represent a different digit from 0 to 9. All the numbers below can be the value of D except?

• (a) 3
• (b) 4
• (c) 5
• (d) 8
• (e) 9

Answer 91

This problem lends itself to using the picking numbers strategy once you’ve analyzed the information thoroughly.

A, B, C and D are different digits. Both DC and CD are two digit numbers; therefore, the hundredth digit in their sum ABA is 1. So, A = 1. Re-write the problem.

D C

+ C D

1 B 1

The sum of C and D has to equal 11. Single digit numbers that add up to 11 are (2;9) (3;8) (4;7) (5;6).

Remember, all the digits have to be different. So, B can not be 1 or be one of the digits already used for D or C. Try all the pairs. (2; 9) does not work. 2 is not part of the answer choices but 9 is. So, it’s (e).

Question 92

If n is an even integer, which of the following must be an odd integer?

• (a) n⁴
• (b) 4n
• (c) ⁿ/₄
• (d) 3(n - 1)
• (e) 4(n - 1)

What strategy do you want to use to solve this problem?

Answer 92

There is a variable in the answer choices so we can pick a small and admissible number to replace it. Let n equal 4 and plug it in to check every answer choice. Be careful with choice (c). When n = 4, the result is odd. When n = 8, it becomes even. Choice (d) is the correct answer.

Logically, an even number multiplied by any number is even. This eliminates (a), (b) and (e). An odd number multiplied by an odd number is always odd. That’s choice (d).

Question 93

• A part-to-part or a part-to-whole ratio?
• A direct or an indirect (inverse) relationship between numbers?

Why is it important to consider the questions above when solving ratio problems?

Answer 93

“Parts” of the ratio add up to the “whole”. Depending on the problem at hand you might need to convert a part-to-part ratio into a part-to-whole ratio or vice verse.

Well, if you don’t know how numbers or variables relate to each other, you might use the wrong formula and get a wrong answer.

Example:

If you drive twice as fast, you will go twice as far in a given time. Speed and distance are directly proportional.

If you drive twice as fast, you need half the time to go the distance. Speed and time are inversely proportional.

Question 94

How should you approach SAT problems testing your knowledge of ratios and proportions?

Answer 94

Determine whether:

• ratios are part-to-part or part-to-whole
• numbers (variables) relate directly or inversely
• you know actual quantities of items that correspond to parts of the ratio
• the total quantity of items

Set up a proportion with two or more ratios from the problem and solve by cross-multiplying.

Question 95

In a mixture of almonds and cashews, the ratio by weight of almonds to cashews is 7 to 4. How many pounds of cashews will there be in 16.5 pounds of this mixture?

Answer 95

Analyze this problem according to the recommended strategy.

The ratio of almonds to cashews is 7 to 4. The mixture weighs 16.5 pounds. The ratio is a part-to-part ratio but the actual quantity you are given is the total weight of the mixture. Therefore, you need to convert the ratio into part-to-whole.

“Parts” add up to “whole”. 7 + 4 = 11 so the ratio of cashews to total is 4 to 11. Let’s set up a proportion.

⁴/₁₁ = ^x/_16.5 ⇒ x = 6

The weight of cashews in the mixture is 6 lbs.

Question 96

The ratio of rap to pop to R&B songs on your iPod is 6 : 13 : 4. You erase a certain number of rap and pop songs making the ratio 4 : 17 : 7.

If you originally had 91 pop songs on your iPod, how many songs have you erased?

Answer 96

* State the purpose: Set up proportions, cross-multiply

• Define the question: Find the number of songs erased
• Extract key info: You had 91 pop songs

The original ratio R : P : R&B = 6 : 13 : 4

The new ratio R : P : R&B = 4 : 17 : 7

• Identify strategy: Determine the type of ratio (part-to-part) and set up proportions to find the original and the new number of songs in each genre

original R : P : R&B = 6 : 13 : 4

original R : P : R&B = x : 91 : y

⇒ 13 : 91 = 6 : x = 4 : y ⇒ x = 42, y = 28

The number of R&B songs remained the same. However, some songs were erased, changing the total and the numbers in the ratio.

new R : P : R&B = 4 : 17 : 7

new R : P : R&B = m : n : 28

⇒ 7 : 28 = 17 : n = 4 : m ⇒ n = 68, m = 16

Original total was 161. New total is 112. You erased 49 songs. The answer matches the question so you are done.

Question 97

Which of the following does not equal a : b = c : d?

1. b : a = d : c
2. c : a = d : b
3. a : c = b : d
4. b : d = c : a

Answer 97

Use the cross-multiplication method to re-write this proportion.

• a : b = c : d ⇒ ad = bc*
  (1) , (2), and (3) answer choices result in the same equal products.

The only answer choice that does not yield the original statement is choice (4).

dc = ba is not the same as ad = bc.

Question 98

The weight W of an object varies inversely as the square of the distance d from the center of the earth. Which expression can be used to figure out the weight of an astronaut in the spacecraft?

• (a) W = 100,056d²
• (b) W = ^100,056/_<em>d²</em>
• (c) W = ^{(<em>d</em> + 100,056)}/_d
• (d) W = d² - 100,056
• (e) W = d² + 100,056

Answer 98

• The purpose: Set up an inverse proportion
• The question: Which expression can be used to find the weight of an astronaut?
• Key info: the weight varies inversly as the square of the distance
• Formulas and strategies: Know the definition and the formula for the inverse proportion. y = ^k/_x
• Logical conclusion: d² has to be in the denominator of the correct expression. Choice (b) is the correct answer.

Question 99

The ratio of men to women in two dance classes is the same. If there are 10 men and 14 women in one class, what could be a possible number of people in the other class?

• (a) 12
• (b) 28
• (c) 34
• (d) 38
• (e) 42

Answer 99

Analyze the problem according to the recommended method and strategy.

M = 10, W = 14 in one class and therefore, the ratio of M to W in the first class is 5 to 7. The ratio is a part-to-part ratio. “Parts” add up to the “whole”. 5 + 7 = 12.

Since the ratios of dancers in two classes are equal, the number of people in the second class has to be a multiple of 12. Choice (a) is correct.

Question 100

What core knowledge is required to solve probability problems on the SAT?

Answer 100

A good understanding of the concept of the probability of an event happening is a must.

You need to know the difference between:

• simple (probability of one event) and compound (probability of two or more events)
• dependent and independent events
• simultaneous and mutually exclusive events

If these concepts are unfamiliar to you, review the mini-tutorials in the “Fractions and Probability” deck.

Question 101

The probability that a red bead is chosen at random from a bowl of beads is ³/₇. Which number below cannot represent the total number of beads in the bowl?

• (a) 63
• (b) 77
• (c) 81
• (d) 112
• (e) 126

Answer 101

The probability of randomly choosing a red bead from a bowl of beads is ³/₇. According to the definition of the probability, it means that there is a total of 7 beads in the bowl, of which 3 are red. Thus, the numerator can be any multiple of 3 and the denominator any multiple of 7.

Evaluate the answer choices to find a number that is not a multiple of 7. It’s choice (c) 81.

Question 102

There are x odd numbers and y even numbers in a list of numbers. You randomly pick a number from the list. The probability that the number is odd is ¹³/₁₉. What is the ratio of even to odd numbers in this list?

Answer 102

Use the recommended approach to problem solving to analyze the information. Do it on your own this time.

The probability of picking an odd number is ¹³/₁₉ which means that there are 13 odd numbers out of a total of 19 numbers on the list. It also means that there are 6 even numbers in the total. So, ⁶/₁₃ would be the ratio of even to odd numbers in this list.

Question 103

A full cycle of a dishwasher has three phases. During each cycle, the machine washes dishes for 50% of the time, rinses dishes 30% of the time and dries dishes the rest of the time. At a randomly chosen time, what is the probability that the dishwasher is not in a washing or drying phase?

Hint: Think of the probability of an event not happening.

Answer 103

The washing phase takes 50% of total time, rinsing takes 30% and drying takes the rest of the time (20%).

You should know that the probability of an event(s) not happening is

P not A = 100% - P (A)

100% - 50% - 20% = 30%

The probability that the dishwasher is not in a washing or drying phase is 30%.

Question 104

A drawer contains white, black and striped socks.

• The number of pairs of striped socks is 3 times the number of pairs of both white and black socks
• The probability of pulling a pair of white socks is 3 times the probability of pulling a pair of black socks

If there are 6 pairs of white socks in the drawer, what is the total number of socks in the drawer?

Answer 104

This is a typical word problem…long and confusing. To help you digest the information, we’ve broken it into 4 blocks and underlined key units. Test makers will not do that for you! Active reading is an important technique (reviewed in this deck) that keeps you focused.

Analyze key facts to make the following conclusions:

Total = S + B + W

S = 3B + 3W ⇒ T = 4B + 4W

P(W) = 3 P(B). It means that the number of white socks is 3 times the number of black socks. If W = 6, B = 2.

Total = 4 * 2 + 4 * 6 = 32

Question 105

A school survey found that 4 out of 6 students like pizza. If 3 students are chosen at random with “replacement”, what is the probability that all three students like pizza?

• (a) ²/₃
• (b) ¹/₅
• (c) ⁴/₂₇
• (d) ⁸/₉
• (e) ⁸/₂₇

Answer 105

• The purpose: Evaluate the probability of 3 events
• The question: Find the probability that all three students like pizza
• Key info: 4 out of 6 students like pizza. 3 students are chosen at random with “replacement”
• Logic conclusion: The probability that any one of three students chosen likes pizza is ⁴/₆ = ²/₃. The events are independent. The total number of possible outcomes is the same for each choice
• Formula and strategy: Know the meaning of the words with “replacement”. Know to multiply individual probabilities to find the probability that all three students like pizza

²/₃ x ²/₃ x ²/₃ = ⁸/₂₇- choice (e)

• Is it the answer to the question? Yes.

Question 106

There are 3 multicolored, 5 black and 6 white t-shirts in your closet. What is the probability that you will first pull a black t-shirt, then, without replacement, a white t-shirt?

Answer 106

• The purpose: To evaluate the probability of two events
• The question: Find the probability of pulling a black t-shirt and then a white t-shirt
• Key info: There are 3 multicolored, 5 black and 6 white t-shirts. You pull shirts without replacement
• Formulas and strategy: Know the definition of dependent events. Know to multiply individual probabilities if both events must happen
• Logical conclusions: There are a total of 14 t-shirts in the closet. The probability of pulling a black t-shirt is ⁵/₁₄. The probability of then pulling a white t-shirt without replacement is ⁶/₁₃
• Apply all knowledge: Multiply the two to find the probability of both events

⁵/₁₄ x ⁶/₁₃= ¹⁵/₉₁

Question 107

What should be your approach to word problems on the SAT?

Note: A word problem is any mathematics exercise expressed in a hypothetical situation described in words. It is basically a story.

Answer 107

You need to follow the recommended process. However, challenging, long word problems require you to do a few extra things to process the information easier.

Improve your active reading skills to make sure your brain registers every important detail!

Reminder: Active reading involves reading the problem aloud and re-telling it. It also involves highlighting or underlining key information as well as creating a simple diagram or mapping out the information.

Question 108

What type of word problems are common on the SAT?

Answer 108

SAT test makers love distance, age, percentage, and work word problems.

Identifying the type of problem will help trigger your brain to think of what formula or what approach to use to solve it.

Question 109

What are some of the things you should not do when approaching an SAT word problem?

Answer 109

When solving long, challenging word problems, you should not:

• Read the first sentence and start solving the problem
• Disregard our advice on active reading
• Write an equation without thinking the problem through

Question 110

This problem was created by Alex who is 8 years old and thinks he is a math whiz.

There are n socks in y drawers of Alex’s closet. How many socks can there be in z drawers?

Answer 110

That’s an easy problem. Some of you might prefer to work with small, simple numbers instead of variables.

n = 10, y = 2, z = 3

Re-read the problem using numbers instead of variables. There are 10 socks in 2 drawers of Alex’s closet. How many socks can there be in 3 drawers?

(10 ÷ 2) * 3 = 15.

Now, plug the variables back in.

(n ÷ y ) * z = ^nz/_y

Question 111

There are n socks in y drawers of your closet. You take c socks from each drawer. What is the percentage of socks left in each drawer to the total number of socks?

A real problem like this on the SAT would have five different expressions that look the same.

Answer 111

Follow the steps of the critical thinking process. Employ the active reading strategy. Pick small numbers to replace the variables if it helps.

n = 18, y = 3,* c = 2*

The problem makes more sense once you re-read it with numbers instead of variables.

(18 ÷ 3) - 2 = 4.

Write the expression to figure out what percent of 18 is 4…. ⁴/₁₈ x 100%

Now, plug the variables back in…

^{<span><span><em>(n </em></span></span><span>÷ <em>y)- c</em></span>}/_n * 100% ⇒

^{100(n - cy)}/_ny

This expression shows what percent of socks are left in each drawer out of the total number of socks.

Question 112

5 friends decided to split the cost of a friend’s gift at a price of X dollars. If Y additional people want to contribute to pay for the gift, the amount of money that each of 5 friends is initially supposed to pay will be reduced by:

• (a) ^{x + y}/₅
• (b) ^xy/_{5 + y}
• (c) ^{5(5 + y)}/_xy
• (d) ^xy/_{y - 5}
• (e) ^xy/_{25 + 5y}

Answer 112

Highlight key units of information and map out the problem. The algebraic solution is probably easier in this case but if you are more comfortable with numerals, replace the variables with manageable numbers.

• ^x/₅ - original $$ each is supposed to pay
• ^x /_{5 + y} - $$ each has to pay after more people decided to contribute
• ^x/₅ - ^x/_{5 + y} - the $$ difference

Don’t get discouraged if the expression you came up with is not among the answer choices. Think how to re-work the fractions above. Find the LCD of the two and simplify to get to the correct answer (e) ^xy/_{25 + 5y}.

Question 113

A driver, setting off from New York City, gets to Boston in 6 hours. For the first 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for a half hour. Realizing he is late, the driver increases his speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

What approach and what strategy are you going to use?

Answer 113

Use active reading tips to really read the problem. Re-tell and underline or highlight key facts. Create a simple diagram to better sort out the information as shown below.

Core knowledge you need here is the distance formula. Make sure the rates have the same unit measures as the other components.

Distance = Speed x Time

• Distance (D₁) from NYC to the rest area is 150 miles (2.5 x 60)
• The speed on the 2nd part of the trip is 66 mph.
• It takes the driver 3 hours (6 - 2.5 - 0.5) to cover the distance between the rest area and Boston. That distance (D₂) is 198 miles (3 x 66)

The total distance beween NYC and Boston is the sum of D₁ and D₂ = 198 + 150 = 348

Question 114

The Cooper family decided to drive to Florida for a vacation. They covered 480 miles in 8 hours, then stopped for 2 hours, then drove 324 miles in 6 hours. What was the car’s average speed for the total distance traveled?

What formulas and strategies should you consider applying to find the solution?

Answer 114

Employ the recommended step-by-step method as well as the active reading strategy. Practice both on your own.

Core knowledge: You have to know the average speed formula and realize what the average speed is not. The average speed is NOT the average of speeds on different parts of the trip.

Avg Speed = ^{Total Distance}/_{Total Time}

Total distance is 804 miles (480 + 324). It took Cooper family 16 hours (8 + 6 + 2) to cover that distance.

Divide total distance by total time to come up with the average speed of the entire trip. It equals 50.45 mph.

Question 115

Some of your classmates like soccer, while others like football or both sports. If 22 of your classmates like soccer, 21 like football, 15 like both sports, and 3 like neither sport, how many kids are there in your class?

Hint: What type of diagram would help solve it?

Answer 115

This problem involves drawing a Venn diagram, which is usually made up of two overlapping circles.

It would be a mistake to just add 22, 21, 15 and 3. You would be counting some people twice.

15 kids play both sports; that’s the intersection of two sets.

of kids who like only soccer = 22 - 15

of kids who like only football = 21 - 15

The total # of kids = only soccer + only football + both + neither

7 + 6 + 15 + 3 = 31

Question 116

There are 23 students in a school who play guitar. 20 students in that school play violin.15 students play piano. Every student plays at least one instrument, 8 students play two instruments but no student plays three instruments. How many students are in the school?

Answer 116

Extract key facts from the problem with the help of active reading method. Draw a 3-circle Venn diagram.

No student plays three instruments, so zero goes where the 3 sets overlap. 8 students play two instruments. It doesn’t matter how you break it down as long as the areas where 2 sets intersect contain a total of 8 people.

guitar only → 23 - 1 - 2 = 20 students

piano only →15 - 1 - 1 = 13 students

violin only → 20 - 1 - 2 = 17 students

Total = 20 + 13 + 17= 50

Or…. you can do it this way. The total # of students in the school is comprised of the # of students who play one instrument minus the # of students who play two instruments. You subtract that # because you don’t want to count them twice.

Total = 23 + 20 + 15 - 8 = 50

Question 117

You have three siblings. Kate is older than Chris, Michael is younger than Kate, and Chris is older than you. Which of the following is not possible?

• (a) You are the youngest member of the family.
• (b) Michael is older than you.
• (c) Michael is younger than Chris
• (d) You are older than Kate.
• (e) Michael is the youngest.

Have you read it three times, but you’re still not clear about who is older and who is younger?

Answer 117

The ages of all four siblings are interrelated but not in the obvious way. Let’s map out the problem using math “language”.

Kate is older than Chris: K > C

Michael is younger than Kate:

M < K ⇒ K > M

Chris is older than you: C > Y

Based on the information given, we can conclude that K > C > you.

Evaluate the answer choices. Obviously, if Kate is older than you, you cannot be older than Kate, choice (d).

Question 118

It takes one copy machine 4 hours to complete a job, another copy machine 3 hours to finish the same job. The 3rd copy machine takes twice as long as the 2nd. How many hours would it take all three machines, working together at their constant rates, to complete the job?

What process should you use to analyze the information?

Answer 118

• The purpose: Add the individual work rates to find the combined work rate
• The question: Find the time it takes all 3 machines working together to complete the job
• Key info: The 1st copy machine takes 4 hrs to complete the job, the 2nd takes 3 hrs, the 3rd takes twice as long as the 2nd
• Formula and strategy: Use the active reading method and the knowledge of the combined work formula.

^x/_{T together} = ^x/_Ta + ^x/_Tb + ^x/_Tc

• X* is the total job. T_a,b,c is the time that each machine by itself requires to complete the total job. T _together is the time all 3 machines require to complete the job working together
• _Apply all knowledge:__ _Since you don’t know what the job is, assume it’s 100% or 1.

¹/_{T together} = ¹/₄ + ¹/₃ + ¹/₆ = ⁹/₁₂

T _together = ¹²/₉ = 1 ¹/₃

• Have you answered the question? Does your answer make sense? Yes

Question 119

Two friends are running around a quarter-mile track. One runs at a constant speed of 6 mph for 90 minutes. The second person’s speed is 1.5 times faster but he only runs for 40 minutes.

Find the combined number of laps both friends ran and their respective average speed rates.

Answer 119

Use the active reading method to extract key information and diagram or map out the problem. Use your knowledge of core concepts such as the formulas for the average speed and the distance.

Speed x Time = Distance

Avg Speed = ^{Total Distance}/_{Total Time}

Since the unit rate of speed is in miles per hour and the time is given in minutes, convert the time into hours.

1st person: 6 x 1.5 = 9 miles

2nd person: 9 x ²/₃ = 6 miles

Each lap is 1/4 of a mile, therefore, two friends ran a total of 60 laps (9 + 6) x 4.

Together, both friends ran 15 miles in 2.17 hours.

Avg speed = ¹⁵/_2.17≈ 6.9 mph

Question 120

In 4 years, Alex’s grandfather will be 8 times as old as Alex was last year. When Alex’s age now is added to Alex’s grandfather’s age now, the total is 87.

How much younger is Alex than his grandfather?

*** Note: Don’t get lost in the words.

Answer 120

On your own, analyze the information using the step-by-step approach. Read actively and diagram (visual helps sort out the information). A = Alex, G = grandfather.

• (A - 1) * 8 = G + 4
• A + G = 87

Express G in terms of A in the 1st equation and plug into the 2nd equation.

8A - 8 = G + 4 ⇒ G = 8A - 12

A + 8A - 12 = 87

9A = 99 ⇒ A = 11 ⇒ G = 76

G - A = 76 - 11 = 65. Alex is 65 years younger than his grandfather.

Question 121

There are p number of choices of appetizers, (p + 2) choices of main courses, and (p - 2) choices of dessert. How many possible ways you can order your dinner if you order one of each?

• (a) 3p
• (b) p² - 4
• (c) p³ - 4
• (d) p³ - 4p
• (e) p³ + 4p

How would you approach this problem?

Answer 121

How many possible ways can more than one event occur? Use the Basic Counting Principle to determine it.

of possible ways = p * (p + 2) * (p - 2) =

p(p² - 4) = p³ - 4p

You can order your dinner p³ - 4p different ways.

Question 122

The original price of a pair of jeans is x dollars. The price of the first pair you buy is reduced by 20%. On each additional pair bought you receive an extra 10% discount. Which formula below represents the cost n pair of jeans, n >1?

• (a) 0.8x + 0.9xn
• (b) 0.8x + 0.7xn
• (c) 0.1x + 0.9xn
• (d) 0.1x + 0.7xn
• (e) 0.8x + 0.9x(n - 1)

Answer 122

• The purpose: Figure out and add the discounted prices of n pairs of jeans
• The question: Find the formula that expresses the total cost of n pairs
• Key info: The 1st pair gets a 20% discount off the original price x. On each additional pair you receive an extra 10% discount. Quantity n > 1.
• Conclusions: The price of the 1st pair is 80% of x or 0.8x. The price of each additional pair is 70% of x or 0.7x. The trick here is to realize that at 30% off you buy (n - 1) pairs.
• Formulas and strategies: Solve algebraically or pick numbers.

The cost of n pairs is 0.8x + 0.7x (n - 1). Distribute the expressions to get the final answer 0.1x + 0.7xn, choice (d).

Or pick $100 for x and 4 for n, re-read the problem and solve. $80 + $70 * 3 = $290. Plug the numbers into the answer choices. The correct expression will give you the same result.