Math Explorer's Club
Tips for Mental Computations
Introduction
Lesson 1: Addition
Lesson 2: Subtraction
Lesson 3: Multiplication
Lesson 4: Division
Lesson 5: Calendar computations
Lesson 6: Guessing a number and other tricks

Division tips

Division is perhaps the hardest of the elementary operation, since one does not always gets integers. For instance, 100 ÷ 23 = 4 + 8/23 (4 + 8/23 × 23 = 100). Sometimes it may be enough for our purposes to estimate the answer by rounding off. For instance, 100 ÷ 23 lies between 100 ÷ 25 = 4 and 100 ÷ 20 = 5.

Simplifying the problem

Suppose we want to divide a by b, when a and b have a common factor c. Then we can divide both numbers a and b by c to simplify the problem. For instance, suppose that we are trying to divide two even numbers, we can divide both by 2 to get a simpler problem. For example,
46 ÷ 26 = 1 + 20/26.
since both are even numbers, 23 ÷ 13 = 1 + 10/13 gives the same answer, which is easy to see.

Using fractions

First recall that dividing any number by a power of 10 is very easy: simply move the decimal point as many positions to the left as zeroes in the power of 10. For instance, 847 ÷ by 10 is just 84.7 (we moved the decimal point one place to the left) and 847 ÷ 100 = 8.47.

Suppose we want to divide a number x by 25. Since 25 = 100 ÷ 4, we get that x ÷ 25 = (x × 4) ÷ 100. For instance,

374 ÷ 25 = (374 × 4) ÷ 100

Now 374 × 4 is easy to do: multiply by 2 twice. 374 × 2 = 748, and 748 × 2 = 1496. Now, we divide by 100 to get 1496 ÷ 100 = 14.96. So 374 ÷ 25 = 14.96.

We can use the same idea to divide by 5 = 10 ÷ 2. For instance to divide 284 by 5, we multiply by 2 and then divide by 10. We get 214 × 2 = 428, and 428 ÷ 10 = 42.8. Thus 214 ÷ 5 = 42.8.

What if we are asked to divide by 0.5. This is the same as multiplying by 2, since 0.5 = 1/2. What about dividing by 0.2? Since 0.2 = 2/10 = 1/5, we can just multiply by 5. So when dividing by some decimals it may be useful to know the decimal expressions of fractions with small denominators. For instance,
  1. 1/5 = 0.2
  2. 2/5 = 0.4
  3. 3/5 = 0.6
  4. 4/5 = 0.8
and
  1. 1/8 = 0.125
  2. 2/8 = 0.25
  3. 3/8 = 0.375
  4. 4/8 = 0.5
  5. 5/8 = 0.625
  6. 6/8 = 0.75
  7. 7/8 = 0.875
Using the table above we can do 42 ÷ 0.375 very easily: multiply by 8, and then divide by 3. We get 42 × 8 = 336, and 368 ÷ 3 = 112. So 42 ÷ 0.375 = 112.

Activity: Dividing with fractions
    Perform the following divisions in your head (if possible):
  1. 345 ÷ 25.
  2. 173 ÷ 25.
  3. 825 ÷ 5.
  4. 69 ÷ 5.
  5. 423 ÷ 5.
  6. 36 ÷ 0.125.
  7. 90 ÷ 0.6.

x ÷ 9, where x is a two-digit number

Here is a very simple rule to get x ÷ by 9, where x is any two digit number.
  1. If the sum of the digits of x is less than 9, then the quotient is x and the remainder is the sum of the digits of x.
  2. If the sum of the digits of x is at least 9, then the quotient is x + 1 and the remainder is the sum of the digits of x module 9.
For instance,
23 ÷ 9 = 2 with remainder 5

with 2 being the tens digit of 23, and 5 being the sum of the digits of 23.

Now consider 78 ÷ 9. Since the sum of the digits is 15 > 9, the quotient is 8. The remainder is 7 + 8 &equiv 6 (mod 9).

Activity: Dividing by 9
    Perform the following divisions in your head (quotient and remainder):
  1. 81 ÷ 9.
  2. 13 ÷ 9.
  3. 75 ÷ 9.
  4. 69 ÷ 9.
  5. 76 ÷ 9.
Why does the method work?

Checking your answer by casting out nines

We can use casting out nines to check the answer we have obtained. For instance, consider we try to divide 233464 by 32, and we obtain a quotient of 7295, and a remainder of 24 (which is the right answer). To verify the result we proceed as follows:
  1. Cast out nines (get the equivalence module 9) from the divisor. In our case 32 &equiv 3 + 2 &equiv 5 (mod 9).
  2. Cast out nines from the quotient. In our case we obtain 7 + 2 + 9 + 5 &equiv 5 (mod 9).
  3. Multiply the remainder in the previous 2 steps, and cast out nines. In our case we get 5 × 5 = 25 &equiv 7 (mod 9).
  4. Cast out nines from the remainder. In our case 24 &equiv 6 (mod 9).
  5. Cast out nines from the dividend. 233464 &equiv 2 + 3 + 3 + 4 + 6 + 4 &equiv 4 (mod 9).
  6. Here is the test: Add the result of steps 3 and 4. If the result is not the same as that obtained in step 5 module 9, we made a mistake. In our case, 7 + 8 = 15 &equiv 6 (mod 9).

As before, if the test fails, we can be sure that we have made a mistake somewhere in our computation (or while casting out nines). If our result passes the test, we have reason to believe our answer is right.
For instance, suppose we divide 1532 by 17, and we obtained a quotient of 90 and a remainder of 3. Let's see if this makes sense:
  1. Cast out nines from the divisor. In our case 17 &equiv 1 + 7 &equiv 8 (mod 9).
  2. Cast out nines from the quotient. In our case we obtain 9 + 0 &equiv 0 (mod 9).
  3. Multiply the remainder in the previous 2 steps, and cast out nines. In our case we get 8 × 0 &equiv 0 (mod 9).
  4. Cast out nines from the remainder. In our case 3 &equiv 3 (mod 9).
  5. Cast out nines from the dividend. 1532 &equiv 2 (mod 9).
  6. Since 0 + 3 is not 2 (mod 9), we conclude that we made a mistake.
Indeed, the right remainder is 2.

Activity: Checking your answer by casting out nines
    Perform the following subtractions in your head:
  1. Do the following divisions and verify your result by casting out nines.
    1. 2357 ÷ 19.
    2. 8463 ÷ 25.
    3. 36373 ÷ 43.
  2. Provide an example where an answer to a division problem (quotient and remainder) is not correct, but it passes the casting-out-nines test.
Cornell University, Department of Mathematics