Mental Math

Question 1

What is mental math?

Answer 1

Mental math is the act of doing arithmetic calculations using only your brain, without the help of calculators, computers, or pen and paper.

Question 2

Are you wondering why you need to learn mental math when calculators and pen and paper are available and allowed on the SAT test?

What is the point of doing mental calculations?

Answer 2

Always relying on computing tools makes your brain sluggish!

You stop noticing traps, short cuts, obvious answers and all you want to do is to start hitting buttons on your calculator. Constant use of calculators, we believe, takes away from your problem solving abilities.

Question 3

What are the benefits of using your mental math skills on the SAT?

Answer 3

• Learning mental math will help you to easily discover the relationships between numbers
• Your problem solving ability will benefit from thinking more* intuitively* about numbers as a result of practicing mental math
• Doing mental math helps to improve your concentration and your memory

Question 4

How many seconds do you have on the test to answer each math question?

Answer 4

On the SAT you only get 60 to 90 seconds to answer a question!

Anything you do to improve your timing will help you score higher. Here is another reason why learning different ways to do quick calculations can help.

Question 5

Which real life activity requires you to use your math skills?

(a) Making your favorite recipe
(b) Re-decorating your room
(c) Shopping
(d) Having dinner in a restaurant
(e) All of the above
(f) None of the above

Answer 5

(e) all of the above

Whether you are following your favorite recipe or re-decorating your room, you need math to figure out how much of each ingredient to use or what quantity of supplies you need to purchase.

Whether you are shopping or having dinner at a restaurant, you need to take a percent of a number to calculate the discounted price or the proper tip amount. Moreover, you need to learn to add up numbers fast to know what you can and can not afford to buy or to order.

Question 6

What percent numbers are the easiest to calculate mentally?

Answer 6

1%, 10%, 25% and 50%

If you have a good understanding of what a percent is, then calculating the percent numbers above of any number should be very easy.

Question 7

How do you take 1% or 10% of a number in your head?

Answer 7

1%: Mentally move the decimal point 2 places to the left.

10%: Mentally move the decimal point 1 place to the left.

Examples:
1% of 103 is 1.03
10% of 566 is 56.6

Question 8

How do you take 5% of a number in your head?

Answer 8

5%: Take 10% and halve the result.

Example:
5% of 566: take 10% (which is 56.6). One-half of 56.6 gives you 28.3 or 5%.

Question 9

How do you take 25% or 50% of a number in your head?

Answer 9

25%: Divide the number by 4.
50%: Divide the number by 2.

Examples:
50% of 566: One-half of 566 is 283.
25% of 228 is 57 = 228 ÷ 4

Question 10

Think about US currency. We have a penny, a nickle, a dime and a quarter. Take the same approach to think of percentages.

Any number can be expressed as a combination of 1%, 5%, 10%, 25%, 50% or 75% for easy calculations.

How would you express 74% in terms of the easy percent numbers above?

Answer 10

The easiest way to calculate 74% of a number is to express 74% as the difference of 75% and 1%.

To take 75% of a number, multiply that number by ³/₄. To figure out 1%, just move the decimal point 2 places to the left.

Question 11

To take 37% percent of a number in your head, you should express 37% as a sum of what percent numbers…?

Answer 11

37% = 3 x 10% + 5% + 2 x 1%

• To take 10%, move the decimal point to the left 1 place
• To take 5%, halve the result of taking 10%
• To take 1%, simply move the decimal point 2 places to the left

Question 12

Any number can be expressed as a combination of 1%, 5%, 10%, 25%, 50% or 75% for easy calculations.

What are some possible ways to express 65% as a sum of the “easy” percent numbers above?

Answer 12

• 65% = 10% x 6 + 5%
• 65% = 50% + 3 x 5%
• 65% = 50% + 25% - 10%
• 65% = 75% - 10%

Example: What is 65% of 236?

Third choice seems to be the easiest in this case. Halve the number to find 50% (118), halve that result to find 25% (59). Move the decimal one place to the left to find 10% (23.6). Add 50% and 25% and subtract 10%.

118 + 59 - 23.6 = 120 + 60 - 3 - 23.6 = 153.4

Question 13

Any percent number can be expressed as a sum of 1%, 5%, 10% and/or 25% for easy calculations.

Take 83% of 120 using this approach.

Answer 13

Express 83% as:

8 x 10% + 3 x 1%.

10% of 120 is 12.

1% of 120 is 1.20.

12 x 8 + 1.2 x 3 = 96 + 3.6 = 99.6

Question 14

It’s customary to leave 15% or 20% tip if you like the service at a restaurant.

How do you rapidly figure out the tip amount if your bill is $34?

Answer 14

Start with the easiest! Calculate 10% of the amount of your bill. Move the decimal point 1 place to the left.

10% of 34 is 3.4.

Express 15% as a sum of 10% and 5% (half of the 10%).

15% of 34 = 3.4 + 1.7 = 5.1

If you are leaving 20%, double the amount of 10%.

20% of 34 = 3.4 + 3.4 = 6.8

Question 15

When you are at a store, and your favorite pair of jeans is on sale for 35% off the original price, don’t wait for a cashier to tell you the discounted price.

How do you quickly estimate if the jeans become affordable on sale? Estimate the sale price if the original price is $94.

Answer 15

Express 35% as a sum of 3 x 10% + 5%. Figure out 10% by moving the decimal point one place to the left.

10% of 94 is 9.4….about $10.

30% would be about $30.

5% is about $5.

Total discount of 35% off the original price is about $35. Now, subtract the discount amount from the orginal price.

$94 - $35 = $59

Question 16

What numbers do we call complementary numbers?

Answer 16

Complementary numbers are the numbers that add up to 10 or a power of 10 .

1 → 9, 2 → 8, 3 → 7, 4 → 6, 5 → 5

Example:
If you have to add 5 + 9 + 3 + 2 + 7 + 5 + 4 + 1 + 8, that can be rearranged as (5 + 5) + (9 + 1) + (3 + 7) + (2 + 8) + 4 for quick calculations.

Question 17

Your goal is to get really fast at figuring out complementary numbers using multiples of 10 (i.e. 30, 60, etc.) or powers of 10 (100; 1,000; 10,000).

Thinking of a number and asking yourself how far this number is from a hundred, fifty, etc. etc. is a good exercise for that.

How far apart is:

7 from 40?

14 from 90?

43 from 160?

Answer 17

7 is 33 away from 40.

14 is 76 away from 90.

43 is 117 away from 160.

Question 18

How far apart are:

67 and 100?

456 and 1,000?

3,782 and 10,000?

Answer 18

67 and 100 are 33 apart.

456 and 1,000 are 544 apart.

3,782 and 10,000 are 6,218 apart.

We simply subtract all digits of the number from 9 except the last digit on the right - from 10.

Another way to do it would be to turn 100 into 99 and subtract the number. Then, add 1. Do the same with 1,000 or 10,000.

99 - 67 = 32 + 1 = 33

9,999 - 3,782 = 6,217 + 1 = 6,218

Question 19

A simple way to quicky add or subtract numbers is to round them.

How would you use rounding in this example?

278 + 589 = ?

Answer 19

You can round 278 and 589 to:

(280 + 590) - 3 = 867
or
(300 + 600) - 22 = 867

Notice that you need to figure out how far 278 is from 300 and how far 589 is from 600. The “how far” practice really helps here.

Question 20

Partial sums is a method of mental calculations where you pick the numbers apart and add them separately.

Add using this method:

58 + 77 = ?

Answer 20

Picking apart numbers and adding partial sums is a simple technique and helps tremendously with mental calculations.

58 + 77 = (50 + 8) + (70 + 7)

= (50 + 70) + (7 + 8) = 120 + 15 = 135

You could have used the rounding method: 60 + 80 - 5 = 135

Question 21

Finding partial difference is a method of mental calculations where you pick the numbers apart and subtract them separately.

How would you subtract these numbers in your head?

873 - 135 = ?

Answer 21

Picking apart numbers and finding partial difference is a simple technique and helps tremendously with mental calculations.

873 - 135 = (800 - 100) + (73 - 35) = 738

Question 22

Subtraction is often faster in two steps than in one. This method will teach you to “subtract what’s easy, then subtract whatever remains”.

How would you apply this method to the example below?

345 - 187 = ?

Answer 22

It’s easy to “subtract” 145 from 345, right?

345 - 145 = 200

Then, subtract “whatever remains”:

200 - (187 - 145) = 200 - 42 = 158

Question 23

Determine the easiest way to add the numbers below:

1. 298 + 657
2. 86 + 27 + 33
3. 11 + 5 + 7 + 8 + 9 + 2 + 11 + 4

Don’t calculate the sums yet.

Answer 23

We would suggest to:

1. Round numbers in # 1
2. Pick apart numbers and add partial sums in # 2
3. Add up complementary numbers in # 3

Question 24

What methods of fast addition will help you to calculate the expressions below fast?

1. 1,503 + 398
2. 74 + 28 + 42
3. 22 + 16 + 18 + 19 + 2 + 1 + 4

Answer 24

1,503 + 398 → Rounding:

1,500 + 400 = 1,900 + 3 - 2 = 1,901

74 + 28 + 42 → Picking apart numbers:

(70 + 20 + 40) + (4 + 8 + 2) = 144.

22 + 16 + 18 + 19 + 2 + 1 + 4 → Adding up compliments:

(1 + 19) + (16 + 4) + (18 + 22) + 2 = 20 + 20 + 40 + 2 = 82

Question 25

When multiplying one 2-digit number by another 2-digit number, finding partial products to get the result would be difficult.

What easy method should you use for multiplying two 2-digit numbers?

42 x 23

Answer 25

42 x 23 = 946

• Multiply ones digits: 2 x 3 = 6
• Multiply tens digits: 4 x 2 = 8
• Write 8…..6 with space in between.
• Crossmultiply and add: (3 x 4) + (2 x 2) = 14
• Insert “crossmultiplication” result: 8…(14)….6
• Final answer is 946

Make sure to carry over whenever the number exceeds 9.

Question 26

When you multiply a 2 (or 3)-digit number by a 1-digit number, it is often easier to pick apart that number and find partial products.

Pick apart the numbers below and find the partial products:

• 48 x 7
• 682 x 7

Answer 26

Express 48 as a sum of 40 and 8.

(40 + 8) x 7

Use the distributive property. Then, add the partial products together.

280 + 56 = 336

Express 682 as a sum of 600, 80 and 2.

(600 + 80 + 2) x 7

4200 + 560 + 14 = 4,774

Question 27

Multiplying two 2-digit numbers can be daunting, but if you can factor and re-group those numbers, it becomes much easier.

How would you apply factoring and re-grouping method to the following example?

16 x 25

Answer 27

Using the factoring and re-grouping methods makes multiplication easily doable in your head.

16 x 25 = 4 x 4 x 5 x 5

Now, group the numbers whose product is easiest to multiply further. In this case, it’s definitely 4 and 5.

4 x 5 x 4 x 5 =20 x 20 = 400

Question 28

This method is called rounding and adjusting.

Looking at the following example, which number lends itself to rounding off?

72 x 45

Answer 28

It makes sense to round down 72 since it’s close to 70.

Turn 70 into 7 and break up 45 into 40 and 5.

7 x (40 + 5) = 280 + 35 = 315

Add a zero at the end of 315 because we removed a 0 from 70. The partial product is 3,150.

We rounded down by 2, so we have to adjust the partial product of 70 and 45.

2 x 45 = 90

Add up partial products:

3,150 + 90 = 3,240

Question 29

Sometimes you need to use a combination of different methods to find the result fast.

How would you use factoring and rounding when multiplying 75 x 15?

Answer 29

First, factor both numbers.

75 x 15 = 25 x 3 x 5 x 3 = 5 x 5 x 3 x 5 x 3

Then, group the factors.

5³ x 3² = 125 x 9

Lastly, round up. Multiply 125 by 10 instead of 9. Then, subtract 125 from the product.

125 x 10 = 1250 - 125 = 1125

Question 30

Let’s review different methods of multiplying two numbers in your head.

• Round and Adjust
• Find Partial Products
• Factor out
• Multiply and Crossmultiply

Think of one example for each method and practice on your own.

Answer 30

With practice, you will know what method to apply when multiplying any two 2-digit numbers.

Example:
55 x 88 is easier calculated by factoring out both numbers.
5 x 11 x 8 x 11 = 40 x 11² = 40 x 121 = 4,840

Example:
22 x 56 is easier calculated by rounding and adjusting.
(20 x 56) + (2 x 56) = 1,120 + 112 = 1,232

Question 31

Which expression matches

(50 x 30) + (50 x 9) + 30 + 9?

(a) 59 x 31
(b) 51 x 39
(c) 50 x 9 + 39
(d) 30 x 59 + 39

Answer 31

(b) 51 x 39

Examine all choices by breaking the numbers apart and writing partial products.

(50 + 1) x (30 + 9)

Use FOIL rule to make it look like the answer.

(50 x 30) + (50 x 9) + (1 x 30) + (1 x 9) =

= (50 x 30) + (50 x 9) + 30 + 9

Question 32

Multiplying by 10 is very simple. Just add a zero to the number.

What is one fast and simple way to multiply by 5?

Answer 32

We mentioned multiplying by 10 to lead you to the right answer.

To multiply a number by 5, take that number and multiply by 10, then divide by 2.

Example:
380 x 5 = (380 x 10) ÷ 2 = 1,900

Note: It works for power of 10.
76 x 50 = (76 x 100) ÷ 2 = 3,800

Question 33

This easy method of multiplying by 5 is different for even and odd numbers.

How do you multiply an even number by 5?

Answer 33

Halve the number you are multiplying and add a zero to the result.

Example: 144 x 5 = 720

Half of 144 is 72. Add a zero for an answer of 720.

Question 34

This easy way of multiplying by 5 is different for even and odd numbers.

How do you multiply an odd number by 5?

Answer 34

Subtract 1 from the number, then halve the result and add a 5 at the end of the resulting number.

Example:

37 x 5 = ?
37 - 1 = 36
36 ÷ 2 = 18

Add a 5 at the end of this number to get the final result of 185.

Question 35

How do you quickly multiply a 2-digit number by 11?

36 x 11 = ?

Answer 35

To multiply any 2-digit number by 11, we just re-write the two digits with the space in between them.

36 x 11 = 3…..6

In that space you insert the sum of the two digits.

3 + 6 = 9 so, the answer is 396.

Note: If the sum of two digits is greater than 9, it will involve carry over figure. You simply add the carry over to the first digit.

57 x 11 = 5….7

5 + 7 = 12; add 1 to 5 ⇒ 57 x 11 = 627

Question 36

Can you apply your knowledge of how to multiply a 2-digit number by 11 to a 3-digit number?

413 x 11 = ?

Answer 36

Use the same technique as with the 2-digit numbers.

Write the first and the last digit at the ends with space in between. Then, insert the sum of the 1st and the middle digits and the middle and the 3rd digit.

413 x 11 = 4….3

4 + 1 = 5; 1 + 3 = 4

The answer is 4,543.

Question 37

Remember how you multiply a 2-digit number by 11?

Similarly, you can multiply a 2-digit number by 13 with just one extra step.

Can you figure out that extra step?

Answer 37

To multiply a 2-digit number by 13, start the same way as if you were multiplying the number by 11. Put the tens digit on the left, units digit on the right and their sum in the middle.

Extra step: double the number and add to the result.

Example:

23 x 13
2(2+3)5 = 253
253 + (23 x 2) = 299

Question 38

Can you multiply a 2-digit number by 16 without a calculator? You might be surprised how easy it is.

42 x 16 = ?

Answer 38

To multiply a 2-digit number by 16:

Multiply the number by 10.

42 x 10 = 420

Take half of the resulting number.

1/2 x 420 = 210

Add those two results to the number ifself to get the final answer.

420 + 210 + 42 = 672

Question 39

Practice what you’ve learned about quick ways of multiplying by 5.

• 461 x 5 = ?
• 1222 x 5 = ?

You may use whatever method you like.

Answer 39

461 x 5 = 2,305

461 is an odd number. Subtract 1, halve that number and add 5 at the end of the resulting number.

(461 - 1) ÷ 2 = 230…

Add 5 at the end to get the result of 2,305.

1,222 x 5 = 6,110

1222 is an even number. Divide by two and add a zero to the resulting number.

1222 ÷ 2 = 611

Aadd a zero at the end to get the result of 6,110.

Or simply multiply 461 and 1222 by 10 and divide by 2.

Question 40

Multiply the numbers below in your head:

• 64 x 11 = ?
• 57 x 13 = ?

Answer 40

64 x 11 = 704

Write the tens digit on the left, the units digit on the right and their sum in the middle. Carry over.

64 x 11 = 6…(6 + 4)…4 = 704

57 x 13 = 741

The first three steps are just like multiplying by 11. Extra step: double 57 and add to the resulting number.

57 x 13 = 5..(5 + 7)..7 = 627

627 + 57 x 2 = 627 + 114 = 741

Question 41

Practice the easy way of multiplying by 16 reviewed in this deck.

38 x 16 = ?

Answer 41

38 x 16 = 608

Multiply the number by 10, halve the result, then add both numbers to the original number (multiplicand).

38 x 10 = 380
380 ÷ 2 = 190
380 + 190 + 38 = 608

*** This is a good practice to add numbers fast as well. Round 190 to 200 and 38 to 40. Then, subtract the “overage.”

380 + 200 + 40 = 620 - 12 = 608

Question 42

Practice the easy way of multiplying by 16 reviewed in this deck.

87 x 16 = ?

Answer 42

87 x 16 = 1,392

Multiply the number by 10, halve the result, then add both numbers to the original number (multiplicand).

87 x 10 = 870
870 ÷ 2 = 435
870 + 435 + 87 = 1392

*** You can practice adding these by breaking the numbers apart and finding partial sums.

(800 + 400) + (70 + 30 + 80) + (5 + 7)

Question 43

How do you quickly multiply any number by:

• 9?
• 99?
• 999?

Answer 43

Multiply by 10 and subtract multiplicand (the original number) from the result.

Example:
256 x 9
256 x 10 = 2,560 - 256 = 2,304

NOTE: Similarly, to multiply by 99 or 999, instead multiply by 100 or 1,000. Then, subtract the original number.

Question 44

Multiplying by 5 is the same as multiplying by 10 and dividing by 2.

What numbers can you use to make multiplication by 25 or 125 easier?

Answer 44

When you multiply a number by 25, multiply by 100 instead, then divide by 4.

Similarly, to multiply by 125, first multiply by 1,000, then divide by 8.

Example:
64 x 25 = (64 x 100) ÷ 4 = 1,600
16 x 125= (16 x 1,000) ÷ 8 = 2,000

Question 45

Remember how you multiply numbers by 5, 25 or 125 fast and without a calculator?

How do you mentally divide numbers by

• 5?
• 25?
• 125?

Think of opposite mathematical operations….

Answer 45

To multiply a number by:

5, multiply by 10 and divide by 2.
25, multiply by 100 and divide by 4.
125, multiply by 1000 and divide by 8.

Division is the opposite of multiplication so, use opposite operations to divide the number by 5, 25, or 125.

To divide a number by:

• 5, multiply by 2 and divide by 10.
• 25, multiply by 4 and divide by 100.
• 125, multiply by 8 and divide by 1,000.

Question 46

How do you multiply a 2-digit number by 4 or 8?

• 35 x 4 = ?
• 35 x 8 = ?

Answer 46

Multiplying by 4 or 8 is the same as doing multiplication by 2 two or three times, respectively.

35 x 4 = (35 x 2) x 2 = 70 x 2 = 140

To multiply 35 by 8, take the result above and double it.

35 x 8 = (35 x 4) x 2 = 140 x 2 = 280

Question 47

How do you divide a 2-digit number by 4 or 8?

• 176 ÷ 4 = ?
• 176 ÷ 8 = ?

Answer 47

Dividing by 4 or 8 is the same as dividing by 2 two or three times, respectively.

176 ÷ 4 = (176 ÷ 2) ÷ 2 = 88 ÷ 2 = 44

176 ÷ 8 = (176 ÷ 4) ÷ 2 = 44 ÷ 2 = 22

Question 48

Breaking numbers apart works for division as well as multiplication.

How do you apply the method of finding partial quotients to:

1236 ÷ 6 = ?

Answer 48

• Break up 1,236 into 1,200 and 36
• Divide both numbers by 6 to find partial quotients
• Add the partial quotients together

1236 ÷ 6 =

1,200 ÷ 6 + 36 ÷ 6 = 200 + 6 = 206

Question 49

The factoring method that we reviewed for multiplication works for division as well.

How would you factor numbers in this expression?

154 ÷ 14 = ?

Answer 49

• Factor 154 into 77 and 2, and further into 11, 7 and 2
• Factor 14 into 7 and 2
• Eliminate common factors

154 ÷ 14 = 11

Question 50

There are several tricks to see if a number is divisible by 7 but applying them takes a long time.

What would be one easy way to check if a number is a multiple of 7?

Hint: Get help from the neighbors.

Answer 50

You can always rely on a good, old nearest neighbor and either count down or count up from it.

Example:

Is 198 divisible by 7?

Start with finding the nearest “easy” multiple. It’s 210 (30 x 7). Count down from 210 by sevens. 210 (30 sevens), 203 (29 sevens), 196 (28 sevens).

As you see, 198 is not divisible by 7.

Question 51

Refresh your memory of divisibility rules.

How do you know if an integer is a multiple of 3 or 9?

Answer 51

An integer is divisible by 3 if the sum of its digits is divisible by 3.
An integer is divisible by 9 if the sum of its digits is divisible by 9.

Example:

Is 1,256,433 divisible by 3 or by 9?

This integer is divisible by 3, but not divisible by 9 since the sum of its digits is divisible by 3, but not by 9.

1 + 2 + 5 + 6 + 4 + 3 + 3 = 24

Question 52

Refresh your memory of divisibility rules.

How do you know if an integer is a multiple of 2, 4 or 8?

Answer 52

An integer is divisible by 2 if the last digit is even.
An integer is divisible by 4 if the last two digits are divisible by 4.
An integer is divisible by 8 if the last three digits are divisible by 8.

Example:

Is 55,619,864 divisible by 2, 4, and/or 8?

Yes, it’s divisible by 2, 4 and 8 since 4 is an even number, 64 is divisible by 4, and 864 is divisible by 8.

Question 53

Refresh your memory of divisibility rules.

How do you know if an integer is a multiple of 5 or 10?

Answer 53

An integer is divisible by 5 if the last digit is 5 or 0.

An integer is divisible by 10 if the last digit is 0.

Example:

387,875

This integer is a multiple of 5, but not a multiple of 10.

Question 54

When is an integer divisible by 12?

Answer 54

An integer is divisible by 12 when it is divisible by both 3 and 4.

Example:

Is 12,456 divisible by 12?

The sum of all digits of this number equals 18 and is divisible by 3. The last two digits form a number divisible by 4. So, it meets both conditions. 12,456 is divisible by 12.

Question 55

If an integer N is divisible by 3, 5 and 12, what is the next integer that is divisible by all three numbers?

N + ? = next integer

Answer 55

The next integer that is divisible by 3, 5, and 12 is N + 60.

Since N is divisible by 3, 5 and 12, it must be a multiple of these three numbers. Their least common multiple (LCM) is 60.

Question 56

Which of the numbers below is divisible by 12?

(a) 136
(b) 244
(c) 322
(d) 348
(e) 376

Answer 56

(d) 348

An integer is divisible by 12 when it’s divisible by both 3 and 4. The only number here that meets both conditions is 348.

Question 57

What is the equivalent of dividing a number by another number?

Answer 57

The equivalent is multiplying that number by the reciprocal of the second number.

Question 58

Since division is equivalent to multiplication by the reciprocal of the number, sometimes it is faster and easier to do exactly that when dividing.

How do you quickly divide 99 by 0.6?

Answer 58

0.6 equals ⁶/₁₀ or ³/₅. Dividing by ³/₅ is the same as multiplying by ⁵/₃.

99 ÷ 0.6 = 99 x ⁵/₃ = (33 x 3) x ⁵/₃ = 33 x 5 = 165

Finding common factors and eliminating them helps you avoid long calculations and minimizes mistakes!

Question 59

Since division is equivalent to multiplication by the reciprocal of the number, sometimes it is faster and easier to do exactly that when dividing.

How do you divide 55 by 2.2 using the reciprocal method?

Answer 59

2.2 equals 2²/₁₀ or 2¹/₅. Convert the number into an improper fraction ¹¹/₅. Dividing by ¹¹/₅ is the same as multiplying by ⁵/₁₁.

55 ÷ 2.2 = 55 x ⁵/₁₁ = (11 x 5) x ⁵/₁₁ = 5 x 5 = 25

Question 60

Practice mental division by using “multiplying by the reciprocal” method on these problems:

• 7 ÷ 2.8 = ?
• 18 ÷ 0.75 = ?

Answer 60

7 ÷ 2.8 = 2.5

Turn 2.8 into an improper fraction… ¹⁴/₅. Flip it and multiply 7 by ⁵/₁₄. Dividing both the numerator and the denominator by 7, we get ⁵/₂ which is 2.5.

18 ÷ 0.75 = 24

0.75 is ³/₄. Multiply by the reciprocal and reduce the fraction to its lowest term.

18 x ⁴/₃ = 6 x 4 = 24

Question 61

If a number ends in 5, its square always ends in 25. So, you’ve got two digits of your answer figured out. How do you figure out the rest?

35 x 35 = ?

Answer 61

We know that the last two digits of the square of 35 are 25. To figure out the first two digits of the result, we multiply 3 by one more than 3.

3 x (3 + 1) = 3 x 4 = 12

Stick together 12 and 25 to get the correct answer.

35 x 35 = 1,225

Question 62

46 is “one away” from the perfect square you might already know.

How do you mentally square the number that is “one away” from a round number or a number ending in 5?

46 x 46 = ?

Answer 62

This is the “one away” rule.

Square the nearest round number or the number ending in 5.

45² = 2025

Add the number to itself minus one.

46 + (46 - 1) = 91

Now, add partial results.

46² = 2025 + 91 = 2,116

Question 63

How do you mentally square the number that is “one away” from a round number?

19 x 19 = ?

Answer 63

Square the nearest round number.

20² = 400

If your number is “one up” from the round number, add the number to itself minus one. If the number is “one down”, it’s itself plus one.

19 + (19 + 1) = 39

If the number is “one up” from the round number, add the partial sums. If the number is “one down”, subtract the partial sums.

19² = 400 - 39 = 361

Question 64

We recommend that you memorize the squares of numbers up to at least 15.

You should already know the squares of numbers from 1 to 10. Do you know the squares of numbers from 11 to 15?

Answer 64

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

Question 65

Squares of 2-digit numbers not ending in 0 or 5 you can calculate in your brain using this simple method….

48 x 48 = ?

Answer 65

Square each digit to get a partial answer.

4² = 16 8² = 64

Stick these numbers together.

1,664

Multiply the two digits of a number you are squaring together, double the result and add a zero at the end.

4 x 8 = 32
32 x 2 = 64
add a zero at the end … 640

Add 640 to the partial product 1,664.

640 + 1,664= 2,304

Question 66

Apply your skills to square the numbers below:

• 14² = ?
• 31² = ?
• 85² = ?

Answer 66

14² = 196

You should have memorized the perfect square table upto 15.

31² = 961

Use “one away” rule.
30² = 900 + (31 + (31 - 1))

85² = 7,225

Use the trick for the numbers ending in 5. Find the partial products and stick them together.

5 x 5 = 25

8 x (8 + 1) = 72

Question 67

How do we multiply decimal numbers without a calculator?

0.24 x 0.03 = ?

Answer 67

Pay no attention to the decimal points at first. Multiply as if the numbers were whole numbers. Add up the number of decimal places in each number multiplied. The answer will have as many decimal places as that sum. Remember to start counting at the right.

Example:

0.24 x 0.03

24 x 3 = 72

The answer should have 4 decimal places. Start at the right and move decimal point 4 places to the left.

0.24 x 0.03 = 0.0072

Question 68

It might be easier to use role reversal to multiply decimal numbers.

How would you reverse these numbers:

0.26 x 50?

Answer 68

Reverse them this way:

0.26 x 50 = 0.50 x 26

Much easier, right? For some of you, thinking of 0.50 as ¹/₂ will make the problem even easier.

¹/₂ x 26 = 13

Question 69

How do we mentally divide decimal numbers when they have the same number of decimal places?

0.24 ÷ 0.06 = ?

Answer 69

If the dividend and the divisor have the same number of decimal places, completely disregard decimal points. Divide as if the numbers were whole numbers.

Example:
0.24 ÷ 0.06
24 ÷ 6 = 4

Question 70

How do we mentally divide decimal numbers when they have a different number of decimal places?

2.4 ÷ 0.06 = ?

Answer 70

If the dividend and the divisor have different number of decimal places, get rid of the decimal point in the divisor.

Move that decimal point to the right till the number becomes a whole number. Then, move the decimal point in the dividend the same number of places. Essentially, you are multiplying both numbers by 10 or a power of 10.

Example:
2.4 ÷ 0.06
0.06 becomes 6 by moving the decimal point 2 places to the right (i.e. multiplying by 100). 2.4 becomes 240.
240 ÷ 6 = 40

Question 71

Remembering fractional equivalents of percentages will help you solve percent problems faster.

What are fractional equivalents of 12.5%, 25%, 37.5%, 50%, 62.5%, 75%, 87.5%?

Notice anything about these numbers?

Answer 71

Actually, you don’t have to memorize them! If you remember one, you can figure out the rest.

The percentages on the face of the card are 12.5% apart from each other. Which also means, when converted to fractions, they are ¹/₈ apart from each other.

So, if you know that 12.5% is ¹/₈, you can calculate quickly that

• 25% = ¹/₄
• 37.5% = ³/₈
• 50% = ¹/₂
• 62.5% = ⁵/₈
• 75% = ³/₄
• 87.5% = ⁷/₈

Question 72

What are fractional equivalents of 16.66…%, 33.33…%, 50%, 66.66…%, 83.33…%?

Notice anything about these numbers?

Answer 72

You don’t have to memorize fractional equivalents of the percentages on the face of the card. They are 16.66% apart from each other. Which also means, when converted to fractions, they are ¹/₆ apart from each other.

So, if you know that 16.66…% is ¹/₆, you can calculate quickly that:

• 33.33…% = ¹/₃
• 50% = ¹/₂
• 66.66…% = ²/₃
• 83.33…% = ⁵/₆

Question 73

How do you use your knowledge of algebra, i.e. the distributive property to calculate the expression below?

6 x 7 + 7 x 14 + 8 x 7

Answer 73

Use the distributive property to turn

6 x 7 + 7 x 14 + 8 x 7

into

7 x (6 + 14 + 8) = 7 x 28 = 196

Round up 28 to 30 and multiply 7. Subtract 14 to get the final result.

Question 74

What is the fastest way to calculate the expression below mentally?

24² - 16²

Hint: Use your algebra skills.

Answer 74

Recognize the difference of squares formula from algebra! Use the formula to convert
24² - 16² to (24 + 16) x (24 - 16) = 40 x 8 = 320

Question 75

Calculating the sum of all integers between 1 and 100 (inclusive) may be a daunting task…. unless you just love crunching numbers.

Rather than adding

1 + 2 + 3 + 4 + 5 + 6 …. + 100

is there a short cut to the right answer?

Answer 75

Yes, there is!

There are 100 integers between 1 and 100 inclusive. Use the formula to find the sum:

Sum = ¹/₂ N (N + 1)

where N is number of elements in the series.

Sum = ¹/₂ x 100 x 101 = 50 x 101 = 5,050

Question 76

You already know how to quickly find the sum of the first 100 consecutive integers.

¹/₂ N (N + 1) = ¹/₂ x 100 x 101 = 5,050.

where N is the number of terms in a set.

What if the series of your numbers doesn’t start with 1? What if it starts with 5?

What is the sum of:

5 + 6 + 7 + … + 100?

Answer 76

Sum = ¹/₂ N (N + 1)

The sum of all integers from 1 to 100 is 5,050. But, in our problem, the first term is 5, not 1. Now, you need to subtract off the numbers you don’t want.

Using the same formula, the sum of integers from 1 to 4 is ¹/₂ x 4 x 5 = 10

Sum (1-100) - Sum (1-4) = 5,050 - 10 = 5,040

Question 77

How do you use the average of a set of evenly spaced numbers to figure out the sum of all numbers?

Answer 77

This is a set of evenly spaced numbers:

{1, 6, 11, 16, 21}

In such a set, the mean (average) is equal to the median (the middle term). 11 is both the mean and the median.
Can we multiply the mean of the set by number of elements to get the sum of all the elements? Yes, absolutely.

Sum = Mean x Number of elements

Sum = 11 x 5 = 55

Question 78

How do you find the sum of a set of consecutive integers below using the average?

{ 31, 32, 33,….., 75}

Answer 78

Find the average of the first and the last term of the set and multiply by the number of terms in this set.

Sum = ^{(First + Last)} / ₂ * # of Terms

Example:

Find the sum of all integers between 31 and 75 inclusively.

Sum = ^{(31 + 75)}/₂ * 45 = 53 * 45

Remember how to do it fast without a calculator? 53 * 45 = ?

• Multiply tens digits: 5 * 4 = 20
• Multiply ones: 3 * 5 = 15
• Write 2…… 5
• Crossmultiply and add: 5 * 5 + 3 * 4 = 37
• Carry over 1 from 15 to 37.
• The result is 2,385

Question 79

How do you compare fractions?

Answer 79

Depending on what fractions you are comparing, you can use these methods:

• Crossmultiplying
• Finding LCD
• Turning fractions into decimals
• Approximating

Question 80

How do you use crossmultiplication to compare fractions?

Answer 80

Find two cross-products.

• 1st fraction’s numerator and 2nd fraction’s denominator. Write the result on the left.
• 2nd fraction’s numerator and 1st fraction’s denominator. Write the result on the right.
• Compare two numbers.

If cross-products are equal, then fractions are equal. If the number on the left is greater than the number on the right, the 1st fraction is greater than the 2nd. And vice versa.

Example: Compare ³/₈ and ⁵/₁₃

The cross-product of 3 and 13 is smaller than the cross-product of 8 and 5. Therefore, ³/₈ is smaller than ⁵/_13.

Question 81

Sometimes, it’s easier to compare fractions by turning them into decimal numbers. The denominators of the fractions that land themselves to this method should be factors of 10, 100 or 1,000.

How do you compare:

³/₈ and ³⁶/₁₂₅?

Answer 81

Since both denominators are factors of 1,000, it’s easy to turn these fractions into decimal numbers and then compare.

³/₈ = 0.375

³⁶/₁₂₅ = 0.288

0.375 > 0.288

Question 82

What number do you need to add to 485 to get to the product of 72 and 8?

? + 485 = 72 x 8

Answer 82

72 x 8 = 576

576 - 485 = 91

91 + 485 = 72 x 8

Question 83

The expression 9 x 98 has the same value as:

(a) 9 x 89
(b) 898 x 1
(c) 980 - 98
(d) 989 - 89

Answer 83

(c) 980 - 98

Learn to see short cuts. Do not multiply 9 x 98 or calculate any of the answer choices. You need to realize that ….

9 x 98 = (10 x 98) - (1 x 98) = 980 - 98

Question 84

Express 720 as a product of two numbers one of which is 20 times larger than the other one.

Answer 84

x + 20x = 720

Solve for x.

6 x 120 = 720

Question 85

There are 48,048 houses in the Town A. Every twelfth house has a circular driveway. How many circular driveways are there in Town A?

Answer 85

¹/₁₂ x 48,048 = 4,004

Question 86

Find the sum and the difference of these two numbers using just your brain:

1. 532 + 364 = ?
2. 532 - 364 = ?

Answer 86

1. The sum is 896.
2. The difference is 168.

Question 87

Increase 12 fifteen times, divide by 6 and add 48 to the quotient.

Answer 87

12 x 15 = 180

180 ÷ 6 = 30

30 + 48 = 78

Question 88

The product of 11 and 7 increase by 43. The result divide by 6. Then, multiply the quotient by 5.

Answer 88

(11 x 7) + 43 = 77 + 43 = 120

120 ÷ 6 = 20

20 x 5 = 100

Question 89

Which is greater:

one-sixth of 102

or

two-eighths of 84?

Answer 89

¹/₆ ^{of 102 is smaller than 2/₈ of 84.}

¹/₆ * 102 = 17

²/₈ * 84 = ¹/₄ * 84 = 21</sup></sup>

Question 90

What number is 140 times smaller than 840?

Answer 90

6 is 140 times smaller than 840.

Question 91

What do you need to add to the sum of 39, 40 and 41 to get the same result as in the sum of 40, 41 and 42?

40 + 41 + 42 = 39 + 40 + 41 + x

Do not calculate!

Answer 91

You need to add 3 to the sum of 39, 40 and 41 to get the same result.

X = 3

It should have taken a second for you to answer it. Notice that 40 and 41 are on both sides of the equation. You can disregard them and just compare 39 and 42.

Question 92

( 2 x 3 x 4 x 5 x 6) ÷ 45 = ?

Do not calculate the product of 2, 3, 4, 5 and 6 and divide by 45! Use techniques you learned in this deck.

Answer 92

16

The technique you want to use in this example is factoring. Factor out 45 ( 5 x 3 x 3) and eliminate common factors from both the numerator and the denominator. What’s left is 2 x 4 x 2.

Question 93

What is the average and the sum of elements of this set of numbers:

{1, 11, 21, 31, 41, 51, 61, 71, 81, 91 and 101}?

Answer 93

The average is 51.

In the evenly spaced set of numbers, the average is the median. Another way to find it would be to take an average of 1st and last elements of the set.

The sum is 561.

The sum of evenly spaced terms of a sequence is its median/average multiplied by number of terms.

51 x 11 = 561

Remember the trick of multiplying by 11? If not, go back to that card.

Question 94

The product of a number and twice that number is 72. What is the third of that number?

Answer 94

The third of that number is 2.

N x 2N = 72

2N² = 72 ⇒ N = 6 ⇒ ¹/₃ N = 2

Question 95

Which of the following has the greatest value?

(a) 20% of 79
(b) 25% of 65
(c) 30% of 49
(d) 35% of 35

It’s only fair of us to ask you to not use a calculator.

Answer 95

(b) 25% of 65

Learn to estimate and eliminate wrong answers. It would be easier to round up 79 to 80 and 49 to 50.

(c) and (d) options are clearly less than (a) and (b) so they are out.

(a) is just under 16 (one-fifth of 80). (b) is just over 16 (one-fourth of 64).

Question 96

8777 - 7778 = 333 + ?

(a) 456
(b) 566
(c) 666
(d) 766

Answer 96

(c) 666

Notice that 7,778 is 1 greater than 7,777.

8,777 - (7,777 + 1) = 999

Use “how far” method to find a compliment number to 333. The sum of two numbers has to equal 999.

Question 97

75% of 40 is 25% of what number?

Answer 97

120

It’s easier to substitute percentages with equivalent fractions here. 75% (or ³/₄) of 40 is 30.

Now, 30 represent 25% (or ¹/₄) of some other number.

? x ¹/₄ = 30 ⇒ ? = 120

Question 98

Dividing ¹/₃ by ¹/₃ yields the same result as multiplying ²/₃ by ?

¹/₃ ÷ ¹/₃ = ²/₃ x ?

Answer 98

³/₂

Dividing by a number is the same as multiplying by a reciprocal of that number.

The product of a fraction and its reciprocal is always 1.

Question 99

If the product of all prime numbers between 1 and 330 is divided by 330, the remainder is?

(a) 2
(b) 3
(c) 5
(d) 11
(e) 0

Answer 99

(e) 0

Recognize that 330 is a product of primes 2, 3, 5 and 11. Therefore, dividing the product of all primes by the product of 2, 3, 5 and 11 will yield zero remainder.

Question 100

The product of 0.08 and 0.8 equals 6.4 divided by what number?

0.08 x 0.8 = 6.4 ÷ ?

(a) 1
(b) 10
(c) 100
(d) 1,000
(e) 10,000

Answer 100

(c) 100

0.08 x 0.8 = 0.064

Multiply 8 by 8 and move the decimal point three places to the left.

6.4 ÷ ? = 0.064

You need to move the decimal point two places to the left so the answer is 100.

Question 101

What is the product of 20% and 60%?

20% x 60% = ?

(a) 12%
(b) 120%
(c) 1,200%
(d) 12,000%

Answer 101

(a) 12%

The biggest mistake would be to just multiply 20 by 60 and stick a percent sign at the end.

The correct way is to turn percents into decimals, multiply, then convert the product back to percents.

0.2 x 0.6 = 0.12 = 12%

Question 102

There are as many seconds in 2 hours as there are minutes in…?

(a) 24 hours
(b) 48 hours
(c) 60 hours
(d) 72 hours
(e) 120 hours

Answer 102

(e) 120

Notice that there are 60 x 60 x 2 seconds in 2 hours. It’s as many minutes as in 120 hours…. 60 x 120

*** Sometimes, analyzing factors is easier than multiplying and analyzing the product.

Question 103

If ♦mnop♦ = (m x n) + (o x p),

figure out the value of ♦9786♦.

Answer 103

9 x 7 + 8 x 6 = 63 + 48 = 111

Question 104

What is the least possible sum of 6 different positive multiples of 5?

Answer 104

105

Start with the smallest positive multiple of 5.

5 + 10 + 15 + 20 + 25 + 30 = 105

Question 105

Solve:

• 23 x 46 = ?
• 10,000 - 476 = ?

Use techniques learned in this deck.

Answer 105

23 x 46 = 1,058

If you notice that 46 is 2 x 23, then you can square 23 and multiply by 2. If not, use the “multiply and crossmultiply” method for finding a product of two 2-digit numbers.

10,000 - 476 = 9,524

Use the “how far away” method. Turn 10,000 into 9,999 + 1.

Question 106

Which of the following is less than

0.11 x 0.48?

(a) 0.48 - 0.11
(b) 0.48 + 0.11
(c) 0.01148
(d) 11 x 4.8

Answer 106

(c) 0.01148

Multiply 11 by 48. Move the decimal 4 places to the left in the product. Round up. You are looking for the number that is less than 0.05.

Question 107

What operation does the symbol below represent?

756 ♦ 88 = 747 ♦ 79

(a) addition
(b) subtraction
(c) multiplication
(d) division

Answer 107

(b) subtraction

Have you noticed that the numbers on the left are 9 less than the numbers on the right?

Question 108

Solve:

(76 - 19) x 9 + (36² - 34²) x ¹/₇

Of course, without the calculator!

Answer 108

(76 - 19) x 9 + (362 - 342) x 1/7 = 533

Use PEMDAS. Start with what’s in parenthesis.

1. 76 - 19 = 76 - (20 - 1) = 57
2. 36² - 34² = (36 + 34) x (36 - 34) = 70 x 2 = 140
3. 57 x 9 = 57 x 10 - 57 = 513
4. 140 x ¹/₇ = 20
5. 513 + 20 = 533