Divide Decimals and Explain the Scaling | Grade 6 Math | LevelKite

You already know how to multiply a decimal by a whole number. For example, $4 \times 2.1 = 8.4$ . Division asks about the same fact in reverse. The division $8.4 \div 4$ means: what number times $4$ gives $8.4$ ? The answer is $2.1$ .

Three words name the parts of a division. In $8.4 \div 4 = 2.1$ :

The dividend is the number being divided, $8.4$ .
The divisor is the number you divide by, $4$ .
The quotient is the result, $2.1$ .

The divisor is never zero in this lesson. Division by zero is not defined. For example, no number times $0$ gives $8.4$ .

First you will divide a decimal by a whole number. Then you will divide by a decimal.

Start with a whole-number divisor, as in $8.4 \div 4$ . Picture sharing $8.4$ equally among $4$ parts. Think in place-value units: a tenth is $0.1$ and a hundredth is $0.01$ . Because $1$ equals $10$ tenths, $8.4$ is $80 + 4 = 84$ tenths. Sharing $84$ tenths among $4$ parts gives $84 \div 4 = 21$ tenths each, and $21$ tenths is $2.1$ .

The digits come from a whole-number division. The units tell you where the decimal point goes. So, when the divisor is a whole number other than zero:

Divide one place at a time, from left to right.
Each quotient digit keeps the place of the units you are sharing. Sharing tenths gives a tenths digit, so the decimal point of the quotient lines up with the decimal point of the dividend.
If part of a unit is left, trade it for $10$ of the next smaller unit and keep dividing. You may write zeros after the last decimal digit, because $7.4$ and $7.40$ are the same amount.

Decimal amounts can be cut into smaller parts, so you keep sharing the leftover instead of setting it aside.

Worked example

A decimal divided by a whole number

What is $7.4 \div 4$ ?

Share the ones. 7 ones among 4 equal parts gives 1 one in each part, because 4 times 1 is 4, and 4 times 2 is 8, which is too many. Write 1 in the ones place of the quotient. Subtract to find what is left: 3 ones.
$7 - 4 \times 1 = 7 - 4 = 3$
Trade the 3 ones for 30 tenths, because 1 one equals 10 tenths. With the 4 tenths from the dividend, you have 34 tenths. You are now sharing tenths, so write the decimal point in the quotient. 34 tenths among 4 parts gives 8 tenths each, because 4 times 8 is 32. That leaves 2 tenths.
$34 - 4 \times 8 = 34 - 32 = 2$
Trade the 2 tenths for 20 hundredths, because 1 tenth equals 10 hundredths. Writing 7.4 as 7.40 shows the hundredths place without changing the amount. 20 hundredths among 4 parts gives 5 hundredths each, because 4 times 5 is 20. Nothing is left.
$20 - 4 \times 5 = 20 - 20 = 0$
Put the shares together: 1 one, 8 tenths and 5 hundredths. Nothing is left over and nothing was rounded, so 1.85 is the exact quotient.
$7.4 \div 4 = 1.85$
Check with multiplication. The divisor 4 is not zero and the quotient is exact, so the quotient times the divisor must equal the dividend. Multiply each place of 1.85 by 4 and add.
$1.85 \times 4 = 4 + 3.2 + 0.2 = 7.4$

Answer

$7.4 \div 4 = 1.85$ . Each of the $4$ equal parts is exactly $1.85$ , with nothing left over.

Check a quotient by multiplying. Division asks what number times the divisor gives the dividend. So, for a divisor that is not zero:

An exact quotient (nothing left over, nothing rounded) times the divisor equals the dividend. Here, $1.85 \times 4 = 7.4$ .
A rounded quotient gives only an approximate check. Round $1.85$ to $1.9$ , and $1.9 \times 4 = 7.6$ is close to $7.4$ but not equal.
A remainder is an amount left over and not shared. It appears when whole things cannot be cut. Sharing $7$ marbles among $4$ friends gives $1$ each with $3$ left. The check is divisor times whole-number quotient plus remainder: $4 \times 1 + 3 = 7$ . Compare $7.4 \div 4 = 1.85$ : no remainder, because the leftover is shared too.

Now let the divisor be a decimal, as in $7.2 \div 0.4$ . It is hard to picture sharing among $0.4$ of a part. Use the other meaning of division: how many groups of $0.4$ fit into $7.2$ ?

Count both numbers in the same unit, tenths. The dividend $7.2$ is $72$ tenths, and the divisor $0.4$ is $4$ tenths. Now the question is: how many groups of $4$ tenths fit into $72$ tenths? That is the same count as how many fours fit into $72$ , which is $18$ .

So $7.2 \div 0.4 = 72 \div 4 = 18$ . Counting in tenths instead of ones multiplies both numbers by $10$ . The unit changed, but the number of groups did not.

A number you multiply by is a factor, and multiplying by a factor is called scaling. The numbers $10$ , $100$ , $1000$ and so on are powers of ten. The table shows what happens to the quotient when you scale the numbers in this division.

Scaling both numbers by the same power of ten keeps the quotient at 18

Division	Change from the first row	Quotient
$7.2 \div 0.4$	none (the starting problem)	$18$
$72 \div 4$	dividend $\times 10$ and divisor $\times 10$	$18$
$720 \div 40$	dividend $\times 100$ and divisor $\times 100$	$18$
$7.2 \div 4$	divisor $\times 10$ only	$1.8$

Key concept

Scale both numbers to make the divisor whole

To divide by a decimal, multiply the dividend and the divisor by the same power of ten. Choose the power that makes the divisor a whole number: $10$ for one decimal place, $100$ for two, and so on. The quotient stays the same, because both numbers are simply counted in a smaller unit. Then divide by the whole number, one place at a time.

Scale both numbers. In the table, scaling only the divisor changed the quotient from $18$ to $1.8$ .

In the formula, $a$ is the dividend, $b$ is the divisor (never zero), and $k$ is the power of ten you choose.

a \div b = (a \times k) \div (b \times k)

Worked example

A divisor in hundredths

A wire is $4.5$ meters long. You cut it into pieces that are each $0.25$ meter long. How many pieces do you get?

Decide what to compute. You want the number of groups of 0.25 meter that fit into 4.5 meters. That is a division with a decimal divisor.
$4.5 \div 0.25$
Make the divisor a whole number. It has two decimal places, so multiply both numbers by 100. This is like measuring in centimeters, because 1 meter equals 100 centimeters. Each piece is 25 centimeters and the wire is 450 centimeters. Notice the zero that fills the ones place of 450.
$0.25 \times 100 = 25 \quad \text{and} \quad 4.5 \times 100 = 450$
Divide the scaled numbers. The number of pieces is the same in either unit. Start with the tens: 45 tens divided by 25 is 1 ten, because 25 times 1 is 25. That leaves 20 tens. Trade them for 200 ones. 200 ones divided by 25 is 8 ones, because 25 times 8 is 200. Nothing is left and nothing was rounded, so 18 is the exact quotient.
$4.5 \div 0.25 = 450 \div 25 = 18$
Check in the original problem. The divisor 0.25 is not zero and the quotient 18 is exact, so the quotient times the divisor must equal the dividend. Multiply each place of 0.25 by 18 and add.
$18 \times 0.25 = 18 \times 0.2 + 18 \times 0.05 = 3.6 + 0.9 = 4.5$

Answer

$4.5 \div 0.25 = 18$ . You get exactly $18$ pieces, and no wire is left over.

Look again at $4.5 \div 0.25 = 18$ . The quotient is greater than the dividend, and that is correct. Each piece is shorter than $1$ meter, so a wire of $4.5$ meters gives more than $4.5$ pieces.

How the quotient compares with the dividend depends on the divisor. The table below is for a positive dividend (a number greater than $0$ ) and a positive divisor. Its multiplication column gives the reason for each row:

A factor greater than $1$ increases a positive number.
A factor of exactly $1$ leaves a number unchanged.
A factor between $0$ and $1$ decreases a positive number.

Division undoes the multiplication, so the size changes in the opposite direction.

Zero is a special case. $0$ times any factor is $0$ , and $0$ divided by any number except $0$ is $0$ .

A positive dividend divided by a positive divisor

Size of the divisor	Example	Matching multiplication	Quotient compared with the dividend
greater than $1$	$6 \div 1.5 = 4$	$4 \times 1.5 = 6$ : the factor $1.5$ increases $4$ to $6$	smaller: $4$ is less than $6$
exactly $1$	$6 \div 1 = 6$	$6 \times 1 = 6$ : the factor $1$ leaves $6$ unchanged	equal: $6$ equals $6$
between $0$ and $1$	$6 \div 0.5 = 12$	$12 \times 0.5 = 6$ : the factor $0.5$ decreases $12$ to $6$	greater: $12$ is more than $6$

Worked example

A decimal divisor greater than 1

What is $4.32 \div 1.2$ ?

Predict the size. The dividend 4.32 is positive and the divisor 1.2 is greater than 1, so the quotient will be smaller than 4.32.
The divisor has one decimal place, so multiply both numbers by 10. Scaling both by the same power of ten keeps the quotient the same, so the problem becomes 43.2 divided by 12.
$1.2 \times 10 = 12 \quad \text{and} \quad 4.32 \times 10 = 43.2$
Divide 43.2 by the whole number 12, one place at a time. 43 ones divided by 12 is 3 ones, because 12 times 3 is 36. That leaves 7 ones. Trade them for 70 tenths and add the 2 tenths to get 72 tenths. Write the decimal point after the 3, because the next digit counts tenths. 72 tenths divided by 12 is 6 tenths, with nothing left.
$4.32 \div 1.2 = 43.2 \div 12 = 3.6$
Check the scaled division, which has the same quotient. The divisor 12 is not zero and the quotient is exact, so their product must equal 43.2. The quotient 3.6 is also smaller than 4.32, as predicted.
$3.6 \times 12 = 36 + 7.2 = 43.2$

Answer

$4.32 \div 1.2 = 3.6$ . The quotient is exact and, as predicted, smaller than the dividend.

Here is the idea to keep. With a whole-number divisor, divide one place at a time and keep each quotient digit in its place. With a decimal divisor, first multiply the dividend and the divisor by the same power of ten, so the divisor becomes a whole number. The quotient does not change, and you divide as before.

Check your answer in two ways. An exact quotient times the divisor, which is never zero, equals the dividend. A rounded quotient gives only an approximate check. Then compare sizes for a positive dividend. A divisor greater than $1$ gives a quotient smaller than the dividend. A divisor of $1$ gives the dividend itself. A divisor between $0$ and $1$ gives a quotient greater than the dividend.

Next, you will use these skills to solve word problems and to find mistakes in decimal multiplication and division.