Mental Calculations - Getting the result fast Flashcards

1
Q

Addition of 5

A

When adding 5 to a digit greater than 5, it is easier to first subtract 5 and then add 10.

For example,

7 + 5 = 12.
Also 7 - 5 = 2; 2 + 10 = 12.

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2
Q

Subtraction of 5

A

When subtracting 5 from a number ending with a a digit smaller than 5, it is easier to first add 5 and then subtract 10.
For example,

23 - 5 = 18.
Also 23 + 5 = 28; 28 - 10 = 18.

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3
Q

Division by 5

A

Similarly, it’s often more convenient instead to multiply first by 2 and then divide by 10.
For example,

1375/5 = 2750/10 = 275.

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4
Q

Multiplication by 5

A

It’s often more convenient instead of multiplying by 5 to multiply first by 10 and then divide by 2.
For example,

137×5 = 1370/2 = 685.

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5
Q

Division/multiplication by 4

A

Replace either with a repeated operation by 2.
For example,

124/4 = 62/2 = 31. Also,
124×4 = 248×2 = 496.
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6
Q

Division/multiplication by 25

A

Use operations with 4 instead.
For example,

37×25 = 3700/4 = 1850/2 = 925.

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7
Q

Division/multiplication by 8

A

Replace either with a repeated operation by 2.
For example,

124×8 = 248×4 = 496×2 = 992.

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8
Q

Division/multiplication by 125

A

Use operations with 8 instead.
For example,

37×125 = 37000/8 = 18500/4 = 9250/2 = 4625.

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9
Q

Product of two one-digit numbers greater than 5.

A

This is a rule that helps remember a big part of the multiplication table. Assume you forgot the product 7×9. Do this. First find the excess of each of the multiples over 5: it’s 2 for 7 (7 - 5 = 2) and 4 for 9 (9 - 5 = 4). Add them up to get 6 = 2 + 4. Now find the complements of these two numbers to 5: it’s 3 for 2 (5 - 2 = 3) and 1 for 4 (5 - 4 = 1). Remember their product 3 = 3×1. Lastly, combine thus obtained two numbers (6 and 3) as 63 = 6×10 + 3.

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10
Q

Product of numbers close to 100.

A

Say, you have to multiply 94 and 98. Take their differences to 100: 100 - 94 = 6 and 100 - 98 = 2. Note that 94 - 2 = 98 - 6 so that for the next step it is not important which one you use, but you’ll need the result: 92. These will be the first two digits of the product. The last two are just 2×6 = 12. Therefore, 94×98 = 9212.

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11
Q

Multiplying by 11.

A

To multiply a 2-digit number by 11, take the sum of its digits. If it’s a single digit number, just write it between the two digits. If the sum is 10 or more, do not forget to carry 1 over.

For example, 34×11 = 374 since 3 + 4 = 7. 47×11 = 517 since 4 + 7 = 11.

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12
Q

Faster subtraction.

A

Subtraction is often faster in two steps instead of one.

For example,

427 - 38 = (427 - 27) - (38 - 27) = 400 - 11 = 389.

A generic advice might be given as “First remove what’s easy, next whatever remains”. Another example:

1049 - 187 = 1000 - (187 - 49) = 900 - 38 = 862.

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13
Q

Faster addition.

A

Addition is often faster in two steps instead of one.

For example,

487 + 38 = (487 + 13) + (38 - 13) = 500 + 25 = 525.

A generic advice might be given as “First add what’s easy, next whatever remains”. Another example:

1049 + 187 = 1100 + (187 - 51) = 1200 + 36 = 1236.

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14
Q

Faster addition, #2.

A

It’s often faster to add a digit at a time starting with higher digits. For example,

583 + 645 = 583 + 600 + 40 + 5
= 1183 + 40 + 5
= 1223 + 5
= 1228.

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15
Q

Multipliply, then subtract.

A

When multiplying by 9, multiply by 10 instead, and then subtract the other number. For example,

23×9 = 230 - 23 = 207.

The same applies to other numbers near those for which multiplication is simplified:
 	23×51	= 23×50 + 23
		= 2300/2 + 23
		= 1150 + 23
		= 1173.
 	87×48	= 87×50 - 87×2
		= 8700/2 - 160 - 14
		= 4350 - 160 - 14
		= 4190 - 14
		= 4176.
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16
Q

Multiplication by 9, 99, 999, etc.

A

There is another way to multiply fast by 9 that has an analogue for multiplication by 99, 999 and all such numbers. Let’s start with the multiplication by 9.

To multiply a one digit number a by 9, first subtract 1 and form b = a - 1. Next, subtract b from 9: c = 9 - b. Then just write b and c next to each other:

9a = bc.

For example, find 6×9 (so that a = 6.) First subtract: 5 = 6 - 1. Subract the second time: 4 = 9 - 5. Lastly, form the product 6×9 = 54.

Similarly, for a 2-digit a:
bc	= 100b + c
 	= 100(a - 1) + (99 - (a - 1))
 	= 100a - 100 + 100 - a
 	= 99a.

Do try the same derivation for a three digit number. As an example,
543×999 = 1000×542 + (999 - 542)
= 542457.