What is Division?

Just as multiplication is repeated addition, division -- its inverse operation (opposite action) -- is repeated subtraction. If 4 of us share the cost of a \$12 pizza, we each pay \$3 because 12 ÷ 4 = 3. This means that we could separate \$12 into 4 "packets" of \$3 each.

12 - 3 - 3 - 3 - 3 = 0

But since division is the inverse operation of multiplication, and repeated subtraction could become extremely time consuming once the numbers are big -- we use the multiplication tables, combined with our knowledge of factors and estimating, to help us divide numbers efficiently.

A fraction indicates division. Three-quarters means 3 ÷ 4 which of course is the same as
¾ or 0.75. This tells us that division changes or converts a fraction into a decimal or a whole or mixed number. When we divide a big number by a small one, we get either a mixed number or a whole number. When we divide a small number by a big number, we get a fraction or a decimal.

Examples Division Vocabulary

Since math is a language, we have to pay special attention to the words we use to name the different numbers and symbols in our work. We use 3 terms -- multiplier, multiplicand and product -- to name the parts of a multiplication expression. We use 4 terms to define the parts of a division expression.

The dividend is the number to be divided. If the division is presented as a fraction, it is the numerator. The number doing the dividing is called the divisor, or the denominator (in the fraction). The result or answer is called the quotient. At times, there is a remainder -- which, like its name says -- is left over once we've found the whole number of divisor-sized packets we can separate from the dividend. Whenever we divide an odd number by an even one, there will be a remainder of 1 because any odd number is 1 bigger than the even number that precedes it.

In the first example above:
the dividend = 7, the divisor = 3, the quotient = 2 and the remainder = 1.

Say 3 of us want to share 7 candy bars equally. After each of us has taken 2 candy bars, there will be one left over. That's why it's called the remainder. Each of us will get one third of that last remaining bar making a fair share of 7 candy bars = 2 and a third bars each. If our pizza cost \$13 instead of \$12, an equal share of the cost would be 13 ÷ 4 = 3¼ or 3.25. Once we pay \$3 each, there would still be \$1-more left to pay. When we each pay another \$¼ or 25¢, the last dollar is paid. In the 2nd fraction above -- the one with 15 + 2 in the numerator.
We rewrote or regrouped 17 as 15 + 2 because 15 = 5 × 3 -- a multiple of 5 and therefore divisible by 5 -- and since we're trying to divide by 5 -- that's a good thing!! 15 is also the greatest whole number multiple of 5 that we can subtract from 17 and that's what we want. When we divide the 15 by 5, we get 3 -- the whole number part of the quotient. We have a remainder of 2 that needs to be divided by 5 -- so our answer is . Notice the remainder of 2 on the right side.

If 5 of us shared a cab that cost \$17, we'd each have to contribute \$3.40
because two-fifths of a dollar = 40¢.

what to do with the remainder?

There are 2 different ways to deal with the remainder. We can express the quotient as a mixed number with fraction part of the remainder over the divisor, or we can write R2 next to the quotient as in the example on the right above.

Estimating Quotients

Remember that division is the inverse operation of multiplication, so when we divide one number into another, we're looking for the biggest whole number multiple of the divisor that we can subtract from the dividend. This means we'd better know our multiples. In the example above, we subract 15 from 17 because we know there are 3 whole packets of 5 in 17. The 2 remaining parts we break into 10-fifths, which we then divide into 5 packets of 2/5 each.

In the examples above, the divisors and dividends are small numbers. When we work with bigger numbers, it often helps to round both of them so we can estimate or guess how many times one divides into the other.

Example:

Estimate by rounding, then find the quotient of 123 ÷ 19

We round 123 down to 120 -- we round 19 up to 20.
Now we think 120 ÷ 20 = 6. This is our trial quotient.
We multiply 6 × 19 to get 114.
This we subtract from 123 for a remainder of 9
and since 9 is less than 19 -- the divisor, it remains the remainder.
Our trial quotient (estimate) is 6, the actual quotient is .

Sometimes our first trial quotient or estimate is too big because of the rounding. Once we multiply it by the divisor, we see that the product will be bigger than the dividend, so we reduce it to a lower number.

Example:

Estimate by rounding, then find the quotient of 92 ÷ 32

To estimate, we round 92 down to 90 and 32 down to 30.
Now we think 90 ÷ 30 = 3. So, our estimate or trial quotient is 3.
But, when we multiply 3 × 32, we get 96 which is bigger than 92.
This means
3 is too big. The quotient must be less than 3. It must be 2. Divisibility Rules

These rules help us find the factors of given numbers:

A number is divisibe by:

 2 if it's an even number (ends in 0, 2, 4, 6 or 8) 3 if the sum of its digits is divisible by 3 4 if the last two digits are divisible by 4 5 if it ends in 5 or 0 6 if it is divisible by 2 and by 3 (the number must be even to divide by 2) 8 if the last three digits are divisible by 8 9 if the sum of its digits is divisible by 9 10 if it ends in 0

examples

 a) 531 is divisible by 3 since 5 + 3 + 1 = 9 b) 9516 divides by 4 since 16 divides by 4 c) 1128 is divisible by 6 since it is even and the sum of its digits = 12 (divides by 3) d) 5024 is divisible by 8 since 024 divides by 8. e) 576 is divisible by 9 since the digit sum = 18 which is divisible by 9.

.

Dividing Large Dividends by 2-Digit Divisors

Say we have to divide 189 by 17. We can see that the quotient will be bigger than 10
because 10 × 17 = 170. In such a case, we must pay close attention to the place value of
the numbers in the quotient. Here's what we do: The sequence of operations is:

DIVIDE, MULTIPLY, SUBTRACT

This gets us to the first remainder. If there are still numbers at the right end in the dividend, we bring one of them down to join the first remainder. This becomes the new dividend. Now we repeat the sequence of operations until we're left with a remainder smaller than the divisor, and no numbers left undivided in the initial dividend. If our dividend here been 1890, we would continue the division by bringing the 0 down to meet the 2 in the remainder and then we would divide 20 by 17.

To check our work when doing division, we multiply the quotient by the divisor, then add the remainder to get the dividend.

One more example and we're done: Summary of Steps in Division

1. For single-digit divisors, use a trial quotient, divide, multiply and subtract.

2. For double-digit divisors, keep number columns lined up, estimate, then divide as in #1.

3. Express remainders as fractions or with uppercase " R " and the remainder's value.

Now get a pencil, an eraser and a note book, copy the questions,
do the practice exercise(s), then check your work with the solutions.
If you get stuck, review the examples in the lesson, then try again. Practice

1) Factor the dividend as the highest multiple of the divisor + something and find the quotient.
Express the answers as mixed numbers.

 a) 25 ÷ 7 = b) 39 ÷ 8 = c) 125 ÷ 11 = d) 56 ÷ 5 =

2) Divide. Show all your work.

 a) 2 563 ÷ 14 = b) 3 494 ÷ 28 = c) 12 540 ÷ 11 = d) 756 ÷ 35 =

.

Solutions

1) Factor the dividend as a multiple of the divisor plus something, then find the quotient.
Express the answers as mixed numbers.

 a) 25 ÷ 7 = (21 + 4) ÷ 7 = b) 39 ÷ 8 =(32 + 7) ÷ 8 = c) 125 ÷ 11 =(121 + 4) ÷ 11 = d) 56 ÷ 5 =(55 + 1) ÷ 5 = 2) Divide. Show all your work.

 a) 2 563 ÷ 14 = b) 3 494 ÷ 28 = c) 12 540 ÷ 11 = d) 756 ÷ 35 =  