DIVISION

What is Division?

Multiplication is repeated addition. Division is the inverse operation (opposite action) of multiplication, so -- division is repeated subtraction. If 4 of us share the cost of a $12 pizza, we each pay $3 because 4 × 3 = 12, therefore 12 ÷ 4 = 3. This means we could separate $12 into 4 "equal 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.

Fractions mean division. When we take ½ of anything, we divide it by 2. When we find ¼, we divide by 4. And, since ¾ is the same as 3 × ¼ , when we take ¾ of something, we divide it by 4 and multiply it by 3.

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 name 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 subtract all the whole-number "divisor-sized packets" we can, from the dividend. When we divide an odd number by an even one, there will always be a remainder of 1, since any odd number is 1 bigger than the preceding even number.

Say 3 of us want to share 7 candy bars equally. After each of us takes 2 whole candy bars, there will be one left over. That's why it's called the remainder. We each get one third of that last remaining bar, so a fair share of 7 candy bars for 3 people is 2 and a third bars each.

If our pizza had cost $13 instead of $12, an equal share of the cost would be 13 ÷ 4 = 3¼ or 3.25. Once we each pay $3, there will be $1-more left to pay. So, each of us contributes another $¼ or 25¢, and the 13th 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 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 ways to deal with the remainder. Either we express the quotient as a mixed number with fraction part made up of the remainder over the divisor, or we write " R " and the remainder value next to the quotient -- shown above in the example on the right.

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Zeros in Quotient,

When we divide a large number by a 1- or 2-digit divisor, sometimes we have to put a zero (0) in the quotient because the divisor is bigger than the partial dividend. In such a case, we have to bring down 2 numbers at a time to continue the division. Let's look at an example.

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Estimating Quotients

Remember that division is the inverse operation of multiplication, so when we divide one number into another, we want to find 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-fifths 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 or 6 R9.

Sometimes our first trial quotient or estimate is too big due to the rounding. When we multiply it by the divisor, the product is bigger than the dividend, so we try a smaller number.

Example:

Estimate by rounding, then find the quotient 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.

Checking our Division

When we do math questions, we must always check our work, because mathland is a perfect place where coming close is no good at all. When we do math, we aim for perfection.

To check a division question:

multiply quotient × divisor then add the remainder.

Example:

In the example above 92 ÷ 32, quotient = 2, remainder = 28.
To check : 2 (quotient) × 32 (divisor) + 28 (remainder) = 64 + 28 = 92 (dividend).

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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 order 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 that is smaller than the divisor or equal to zero (0), and no numbers left undivided in the original dividend. If our dividend in this last example was 1890 instead of 189, we would continue the division by bringing down the 0, to meet the 2 in the remainder, and then we would divide 20 by 17.

To check our work in 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.

4. To check, multiply quotient by divisor, add the remainder, get the dividend.

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.

There's a link to the multiplication table to help you work.

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Practice

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

Example:

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 =

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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 =

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 =

( Primary MathRoom Index )

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MULTIPLICATION TABLE

× 1 2 3 4 5 6 7 8 9
1 1 2 3 4 5 6 7 8 9
2 2 4 6 8 10 12 14 16 18
3 3 6 9 12 15 18 21 24 27
4 4 8 12 16 20 24 28 32 36
5 5 10 15 20 25 30 35 40 45
6 6 12 18 24 30 36 42 48 54
7 7 14 21 28 35 42 49 56 63
8 8 16 24 32 40 48 56 64 72
9 9 18 27 36 45 54 63 72 81