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Flashcards in Mental Math Deck (108)
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1

What is mental math?

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

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?

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.

3

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

  • 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

4

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

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.

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

(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.

6

What percent numbers are the easiest to calculate mentally?

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.

7

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

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

8

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

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%.

9

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

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

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?

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.

11

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

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

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?

  • 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

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.

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

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?

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

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.

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

16

What numbers do we call complementary numbers?

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.

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?

7 is 33 away from 40.

14 is 76 away from 90.

43 is 117 away from 160.

18

How far apart are:       

67 and 100?

456 and 1,000?

3,782 and 10,000?

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

19

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

How would you use rounding in this example?

278 + 589 = ?

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.

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 = ?

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

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 = ?

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

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

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 = ?

It's easy to "subtract" 145 from 345, right?

345 - 145 = 200

Then, subtract "whatever remains":

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

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.

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

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

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

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

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.

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

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

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

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

28

This method is called rounding and adjusting

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

72 x 45

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

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?

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

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.

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