# Calculation Shortcuts

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Knowing calculation shortcuts that make it easier to do mathematical calculations in your head is often a lot faster than having to get out your calculator (or other device). Here is a collection of useful calculation shortcuts:

## Multiplication Shortcuts

Break the problem up: One method of being able to do multiplication in you head is to break the problem up into more manageable parts. Suppose I asked you what 59 × 43 was. One way of solving this problem is to remember that 59 is (60 − 1), so 59 × 43 is just 60 × 43 − 1 × 43, or 10 × (6 × 43) − 43. Now the only complicated multiplication is 6 × 43, and it's not too hard—just think (6 × 40 + 6 × 3) which is 258. So the answer to the problem is 2,580 − 43 = 2,537. Of course, you'd do all of the above in your head. If I punch in 59 × 43 on my calculator, I get 2,537. Neat, eh?

Differences of squares: In some cases, when multiplying numbers whose difference is even, treating them as a difference of squares can help. This technique is based on the algebraic identity

(x + y)(xy) = x² − y²

So, say that you are calculating 61 × 59. This can be rewritten as (60 + 1)(60 − 1). As above, this equals 60² − 1² = 3,600 − 1 = 3,599.

Squares: When squaring a number, it can help to remember the algebraic identity

(x + y)² = x² + 2xy + y²

For example, say you're asked to find 31². This is (30 + 1)², which is 30² + 2(1)(30) + 1² = 900 + 60 + 1 = 961.

Multiplying by specific numbers:

• 5: To multiply by 5, you can divide by 2 and then multiply by 10 (see below for multiplying by 10).
• 10, 100, 1,000 etc.: To multiply a number by 10, take that number and add a zero to the end. So, 376 × 10 = 3,760. To multiply a number by 100, add two zeroes to the end. To multiply by 1,000, add three zeroes and so on.
• 11: Multiplying a number by 11 is fairly straightforward. The ones digit is the ones digit of the original number. The tens digit is the sum of the ones digit of the original number plus the tens digit of the original number (make a note of the carry). The hundreds digit is the sum of the tens digit of the original number and the hundreds digit, and so on. For example, take 2738 × 11. The ones digit is 8. The tens digit is 1 (3 + 8 = 11, so 1 carry the one). The hundreds digit is 1 (7 + 3 + 1 = 11, so 1 carry the 1). The thousands digit is 0 (2 + 7 + 1 carry = 10; carry the 1). The ten thousands digit is 3 (2 + 1 carry). So, the result is 30,118.
• 12: This technique is similar to the technique for 11, except that you double the left-hand digit in the sum. For example, take 7214 × 12. The ones digit is 2 × 4 = 8. The tens digit is the ones digit plus twice the tens digit (4 + 2 × 1 = 6). Similarly, the hundreds digit is 5, the thousands digit 6 (16, carry the 1), and the ten-thousands 7 + 1 carry = 8. So, the result is 86,568.

## Division Shortcuts

Cancellation: A problem in division is, of course, equivalent to a fraction. In other words, a ÷ b = a/b. Just as you can cancel out common factors from a fraction; similarly, you can cancel common factors from the dividend and divisor.

Break the problem up: Breaking the division problem up can be a useful shortcut. For example, if you don't know what 144 ÷ 9 is, you could break the problem up as (99 + 45) ÷ 9, which is 99 ÷ 9 + 45 ÷ 9 = 11 + 5 = 16.

Dividing by specific numbers:

• 10, 100, 1000, etc.: To divide by 10, just move the decimal place one place to the left (or remove the trailing zero from the dividend if it ends with zero). So, 460 ÷ 10 = 46, and 827 ÷ 10 = 82.7. For 100, move the decimal point two places; for 1000, three places, and so on.

Rearrange terms: When adding a long list of terms, one useful shortcut is to mentally rearrange the terms. As an example, say you're adding 57 + 67 + 43. This calculation can be rearranged as 57 + 43 + 67. 57 + 43 is 100, which makes the next calculation (100 + 67) simple.

Break the problem down: It can be helpful to break the problem down into a larger number of smaller additions. For example, we can treat 79 + 87 as (70 + 9 + 80 + 7). Using the "Rearrange terms" shortcut above, we can rearrange this as (70 + 80 + 9 + 7), which is 150 + 16, which is 166.

Casting out nines: See digital roots for more information on this technique.

Divisibility tests: See divisibility test and generalized divisibility rules for more information.

## General Tips

Laws of Arithmetic: Many of the above tips relating to breaking the problem up and rearranging terms are applications of the three laws of arithmetic. Being comfortable with mentally decomposing, re-composing, and rearranging can be useful. For example, if you're calculating 25 × 92, it may be easier to think of the problem as 25 × (4 × 23), which is equal to (25 × 4) × 23, which is 100 × 23, and at this point it's easy to see the answer is 2,300.

Memorizing: If you have finished elementary school, you likely have memorized the multiplication tables up to 12 or so. You might find it useful to memorize further. Here is a multiplication table that goes up to 20. As well, knowing other miscellaneous facts that tend to appear in your everyday calculations is useful, such as, perhaps, the powers of 2, or maybe the fact that 7 × 11 × 13 = 1,001.

Practice: In order to ensure that all of these shortcuts are mentally available when you need them, you need to practice. Take advantage of any opportunities you may have to try to work out calculations mentally instead of getting your calculator out.

Estimation: Often, in real-life situations, it is not necessary to perform an exact calculation; a good estimate is often good enough. Having good estimation skills is an important part of number sense.

## Practice

There is a worksheet on arithmetic available on Math Lair.