Numbers: Word Problems

Question 1

What is a word problem?

Answer 1

A word problem is any mathematics exercise expressed in a hypothetical situation described in words.

Question 2

There are seven houses; in each house there are seven cats; each cat kills seven mice; each mouse has eaten seven grains of barley; each grain would have produced hekat (means volume unit). What is the sum of all the enumerated things?

Answer 2

19, 607

This word problem goes back to Ancient Egypt! To solve this problem, you need to add 7, 49, 343, 2401 and 16807.

Question 3

A ship sails the ocean. It left Boston with a cargo of wool. It grosses 200 tons. It is bound for Le Havre. The mainmast is broken, the cabin boy is on deck, there are 12 passengers aboard, the wind is blowing East-North-East, the clock points to a quarter past three in the afternoon. It’s the month of May. How old is the captain?

Answer 3

Gustave Flaubert (a famous 19th century French writer) wrote this nonsensical problem to make fun of confusing nature of some word problems. So, you are not alone; he must have hated them too!

On the SAT, word problems do make sense even though the wordiness might throw you off. Learn strategies in this App and do practice drills to improve your results.

Question 4

GENERAL: Word problems are challenging. In this deck, we will show you a the steps which should make solving them easier.

1. Read actively.
2. Identify the type of word problem in front of you.
3. Draw a simple diagram.
4. List information on the diagram and define the unknown(s).
5. Identify the question.
6. Write an equation.
7. Solve the equation.
8. Check if the answer makes sense.

We will review these steps in details on separate cards and reinforce them through a variety of sample problems in this deck.

What are you going to do?

Answer 4

We wrote this card to introduce the word problem solving process. You will now learn it in individual steps.

Feel free to rate this card a “5” so you will see it very infrequently going forward. Don’t worry: there are plenty of other individual cards that you will rate a “1” or a “2” that you will want to repeat!

Question 5

What are some of the things you might do when solving word problems but really should not?

Answer 5

Like we said before, word problems are challenging. To make them easier, here are a few suggestiong of what NOT to do:

Question 6

Previously, we suggested you follow a certain pattern when solving word problems. Can you put these steps in a proper order and identify wrong steps?

• Read first sentence and start solving the problem.
• Draw a simple diagram.
• Write an equation.
• Identify the type of word problem in front of you.
• List information on the diagram and define the unknown(s).
• Identify what is being asked.

Answer 6

1. Read first sentence and start solving the problem. *** NEVER! Read the problem entirely.
2. Identify the type of word problem in front of you.
3. Draw a simple diagram.
4. List information on the diagram and define the unknown(s).
5. Identify the question.
6. Write an equation.

Question 7

The first step to word problems on the SAT is to read actively.

What does it mean?

Answer 7

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 8

What types of word problems are common on the SAT test?

Answer 8

The most common types of word problems are distance problems, age problems, percentage problems, and number problems. You may also see mixture and work word problems on the test.

Question 9

Active reading is the first critical thing you need to get good at.

Which of the following do you think would make excellent active reading techniques:

• Read aloud for practice
• Test yourself as to whether you can re-tell the story after reading it once or twice
• drinking heavily
• tapping with each new data point or
• underlining key words and phrases that might go into an equation
• drinking more heavily

Answer 9

Active reading is:

• bbbb
• dddd
• ddddd
• rrrr

It is not:

• vvvv
• vvvv

You should practice this to learn it. . . ready?

Question 10

What are the simple techniques you should practice to train yourself to read actively?

Answer 10

• Read the problem aloud when you practice.
• Every word problem is a story. You should be able to re-tell the story after reading it once or twice. Don’t worry if you don’t remember the exact numbers. Concentrate on key facts.

*** Never start trying to solve anything until you have read the problem entirely!!

Question 11

What are the simple techniques you should practice to train yourself to read actively?

Answer 11

• As you read the problem, tap every time you think you found a key detail.
• When you are reading a word problem on paper, it helps active reading process if you highlight and/or underline key information.

Question 12

Active Reading: What information could you underline and/or highlight as important here?

Mary is three times as old as her sister Carry, who is five years older than their cousin Jessy. If in 15 years Mary will be twice as old as Jessy will be then, how old is Carry?

Answer 12

Active Reading is important on word problems. One method of improving your active reading in the SAT is to underline key words and phrases.

Mary is three times as old as her sister Carry, who is five years older than their cousin Jessy. If in 15 years Mary will be twice as old as Jessy will be then, how old is Carry?

*** We have underlined main characters in this problem - Mary, Carry and Jessy - and bolded key information pertaining to their ages.

Question 13

What information would you underline and/or highlight in this word problem?

Anne is twice as old as Mike, but three years ago, she was two years older than Mike is now. How old is Mike now?

Answer 13

Anne is twice as old as Mike, but three years ago, she was two years older than Mike is now. How old is Mike now?

*** We have underlined main characters in this problem - Anne and Mike - and bolded key information pertaining to their ages. Pay attention to words indicating time… “ago” and “now”.

Question 14

So, you’ve read the problem… Next, you have to identify what type of problem it is. Why is it so important?

Answer 14

SAT test makers love distance, age, percentage, and numbers word problems. You may also see mixture, rate and work problems.

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

Example:

You need to know this formula D = S x T in order to solve a distance word problem.

Question 15

What type of word problem is this….?

If a factory produces 3,000 plastic forms in H hours, how many plastic forms can it produce in M minutes?

(a) ^M/_50H
(b) 50MH
(c) ^MH/₅₀
(d) ^50M/_H
(e) ^50H/_M

Answer 15

It’s a production rate problem. It’s asking you to find out how many units per minute factory can produce. Remember the formula?

Production rate = ^Units/_Time

Units = 3,000. Time = H. ⇒ P = ^3,000/_H.

Ooops, there is no such answer among the choices. Can you guess where we made a mistake?

Question 16

Can you identify the type of the following word problem?

If John’s age is equal to the sum of Michael’s age and twice Rowan’s age, what is the average age of John and Michael?

Answer 16

This is an age word problem. That’s easy to determine. It contains the word “age”! LOL

There is no specific formula to remember. To successfully solve age word problems, your goal is to name things and pick variables to stand for the unknows. Name and label clearly so you know what exactly they stand for.

Question 17

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Question 18

How should you approach the following type of question?

Ellen is making twenty pancakes for six friends. The amount of required pancake mix is proportional to the number of pancakes that are being made. The recipe calls for 1 cup of mix and three-quarters of a cup of water to make 8 pieces. How many cups of water does Ellen need in order to make enough pancakes for her hungry friends?

Answer 18

This is a mixture word problem. On these kinds of questions, you should be:

1. Do this first
2. This second
3. this third

Solution to sample problem:

Question 19

You run 2 miles in 10 minutes, at a constant speed. How many feet do you run in 30 seconds?

*** 1 mile = 5,280 feet

Question 20

Laura makes 5% more than Jim. If this year, Jim gets a $1,500 raise, their salaries would become equal. How much money does Jim earn a year?

Answer 20

Diagram:

L: 5% > J

J: $1500 raise

L = J

J + 1500 = L

L = 1.05*J

30,000

Question 21

The sum of Mary’s age and Simon’s age is 12. The product of their ages is 32. What is the difference between their ages?

Answer 21

4

M + S = 12

M x S = 32

Question 22

In a 4-digit number, the sum of all digits is 12. The sum of my first and third digits equals its last digit. The third digit is 60% of the last digit. What is that 4-digit number?

Answer 22

2,235

The last digit is 5. That’s the only digit 60% of which would give you a whole number. The third digit is 3. The first digit is 2. Therefore, the second digit is 2.

Question 23

How would you diagram the following speed word problem?

You decided you needed exercise and walked from school to the mall at an average speed of 4 mph. Your friend left the school 45 minutes later but rode a bike to the mall. His speed was 8 mph. If the distance between the school and the mall is 6 miles, will your friend catch you on the way? if yes, in how many hours/minutes?

Answer 23

Once you you have actively read and identified the type of word problem, it’s time to diagram.

For this speed word problem, you might use the following diagram:

<vicky></vicky>

Question 24

Total of 4,950 pounds of tomatoes were harvested on two different farms. The first farm packed them in 20-pound boxes. The second farm packed tomatoes in 30-pound boxes. How many pounds of tomatoes did each farm collect if there were 10 more of heavier (30-pound) boxes?

Answer 24

You should be thinking:

1. Step 1
2. Step 2
3. Step 3

Solution:

Question 25

The distance around a track field is 600 meters. Lizzy runs around the field 6 times in three-quarters of an hour. What is her average speed in meters per minute?

Question 26

Alex, Chris and John save money to buy concert tickets. The amount of money that Alex and Chris have is $125. The amount of money that Chris and John have is $95. The amount of money that Alex and John have is $100. How much money do three of them have in total?

Question 27

Two electricians need to check 150 electrical units. One can do this job in 15 days, the other one can do it in 10 days. If they work together, how many days would be required to complete the job?

Question 28

One tank has three times as much water as the other one. The first tank was filled with additional 46 liters of water, while only 18 liters were added to the second tank. Both tanks now contain 184 liters of water. How many liters of water did each tank contain originally?

Question 29

Two boats leave the pier going in the opposite directions. The speed of the first boat is 25 mph. What will be the distance between them after 4 hours if the second boat was 48 miles from the pier after two hours?

Question 30

Two trains leave town A and town B respectively moving towards each other. The distance between A and B is 777 miles. The first train departed 3 hours ealier than the second train, going at a constant rate of speed of 75 mph. The trains meet in 4 hours after the departure of the second train. What is the speed of the second train?

Question 31

A certain company needs 1,200 brochures for a meeting. One printing shop can do the work in 3 days, the other shop will complete the work in 6 days. How many days would both shops need to finish the job working together?

Question 32

The ratio of gold beads to silver beads in a box was 7 : 4. After Sarah used 81 gold beads to make a necklace, the ratio of silver beads to gold beads became 10 : 13. How many beads of each color were in the box originally? How many beads of each color were left in the end?

Question 33

How would you map the following word problem?

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Answer 33

Draw a sort of number line to indicate a trip from point N (New York) to point B (Boston).

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Question 34

How would you map the following word problem?

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Answer 34

Draw a sort of number line to indicate a trip from point N (New York) to point B (Boston).

<placeholder></placeholder>

Question 35

Step 1: How to Read Actively for the following question:

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Answer 35

On this question, to Actively Read, you highlight the key info

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Question 36

How would you map the following word problem?

A driver, setting off from New York City, gets to Boston in 5.5 hours. First 2.5 hours he drives at an average speed of 60 mph, then stops at a rest area for one-half hour. Realizing he is late, the driver increases the speed by 10% for the second part of the trip. What is the distance between New York City and Boston?

Answer 36

Draw a sort of number line to indicate a trip from point N (New York) to point B (Boston).

<placeholder></placeholder>

Question 37

The cost of a telephon call using a carrier A is $1.00 for any time up to and including 20 mins and $0.07 per minute thereafter. The cost using long-distance carrier B is flat rate of $0.06 per minute. For a call that lasts t minutes, the cost of using carrier A is the same as the cost of using carrier B. If t is a positive integer greater than 20, what is the value of t?