6th Grade · 6.NS.B.3

Dividing decimals in 6th grade

Dividing decimals means sharing or measuring with numbers like 7.56 and 0.6. In 6th grade your child learns to add, subtract, multiply and divide decimals with pencil-and-paper methods and to put the decimal point in the right place every time.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

6.NS.B.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

What it means, in plain words

A decimal is a number with a decimal point, such as 2.5 or 0.75. The digits after the point show tenths, hundredths and thousandths. In 5th grade, children worked with decimals to hundredths, often with drawings. In 6th grade they do all four decimal operations with the standard algorithms, the same pencil-and-paper methods they use for whole numbers.

To add or subtract, line up the decimal points so tenths sit under tenths. An empty place can be filled with a zero, because 6.4 is the same as 6.40. To multiply, your child multiplies as if there were no decimal points. Then the child counts the digits after the point in both numbers and gives the answer that many digits after its point. For example, $0.3 \times 0.2 = 0.06$ .

Dividing decimals has one main idea. When the divisor (the number you divide by) is a decimal, move its decimal point to the right until it is a whole number. Move the point in the dividend (the number being divided) the same number of places. This is the same as multiplying both numbers by 10 or 100, and it does not change the answer. Then divide as with whole numbers and put the decimal point in the answer straight above the one in the dividend.

Your child should be able to:

Add and subtract decimals by lining up the decimal points
Multiply decimals and place the decimal point in the answer
Divide a decimal by a whole number and by another decimal
Estimate first to check that the decimal point is in a sensible place

Example

What is $7.56 \div 0.6$ ?

Move the decimal point in 0.6 one place right to make 6. Move the point in 7.56 one place right too, to make 75.6.
Divide as with whole numbers: $75.6 \div 6$ gives the digits 1, 2 and 6.
Put the decimal point straight above the one in 75.6, so the answer is 12.6.
Check: $12.6 \times 0.6 = 7.56$ . So $7.56 \div 0.6 = 12.6$ .

Example

A 14.4-ounce bag of trail mix is shared equally into 8 snack bags. How much goes in each snack bag?

The divisor, 8, is a whole number, so divide as usual: $14.4 \div 8$ .
8 fits into 14 once, with 6 left over. Bring down the 4 to make 64. 8 fits into 64 exactly 8 times.
Put the decimal point straight above the one in 14.4, so the answer is 1.8.
Check: $8 \times 1.8 = 14.4$ . Each snack bag holds 1.8 ounces of trail mix.

Where children usually get stuck

For $7.56 \div 0.6$ , your child moves the point in 0.6 but not in 7.56 and then works out $7.56 \div 6 = 1.26$ .
The child learned "get rid of the decimal in the divisor" as a step on its own. The step works only because both numbers are multiplied by 10.
What helps. Ask how many 60-cent pencils you can buy with 7 dollars and 56 cents. The answer is about 12, so 1.26 cannot be right.
Your child lines up 3.5 and 2.75 by their right edges, adds them like 35 and 275, then writes 3.10.
Whole numbers line up on the right, so that habit carries over. With decimals, the decimal points have to line up instead. The right answer is 6.25.
What helps. Write 3.5 as 3.50, so both numbers have two digits after the point. Then the points and the right edges line up together, like 3 dollars 50 cents and 2 dollars 75 cents.
Your child writes $0.3 \times 0.2 = 0.6$ .
Three times two is six, and the child keeps the decimal point where it was in the numbers. A tenth of a tenth is a hundredth, so the answer is 0.06.
What helps. On graph paper, draw a 10 by 10 square. Shade 3 columns and 2 rows. The overlap is 6 small squares out of 100, which is 0.06.

Try four practice questions

Real 6th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Look at the multiplication below.
$\displaystyle 4.2 \times 3$
Select the correct product.
A
$\displaystyle 1.26$
B
$\displaystyle 12.6$
C
$\displaystyle 11.6$
D
$\displaystyle 12.2$
Show the answer and how to solve it
The answer is $12.6$ because $42 \times 3 = 126$ , and then the decimal goes $1$ place from the right.
1. 1First, read the multiplication.
2. 2Ignore the decimal for a moment and multiply $42 \times 3$ .
3. 3 $4.2$ has $1$ digit to the right of the decimal, so put the decimal $1$ place from the right in $126$ .
4. 4That gives $12.6$ , so it is the correct choice.
Common mistake. A common mistake is writing $1.26$ because the digit product $126$ is correct but the decimal is placed one position too far left. A quick estimate helps: $4.2 \times 3$ should be a little more than $12$ , not about $1$ .
The idea underneath. Multiply the digits as a whole number first, then place the decimal so the product has the same number of decimal places as the decimal factor.
2On level
Find the quotient.
$\displaystyle 5.4 \div 1.2$
Enter the quotient.
Quotient:
Show the answer and how to solve it
Answer. 4.5
The quotient is 4.5 because moving both decimals one place gives 54 divided by 12, which equals 4.5.
1. 1The divisor is 1.2. Move the decimal 1 place to the right in both numbers so the divisor becomes a whole number.
2. 2The total changes from 5.4 to 54, and each group size changes from 1.2 to 12. Both are 10 times as large, so the same number of groups fits. The quotient stays the same.
3. 312 fits into 54 four whole times with a remainder of 6.
4. 4The remaining 6 is 0.5 of 12, so add 0.5 to the 4 whole groups.
Common mistake. A common mistake is moving the decimal in 1.2 to get 12 but leaving 5.4 unchanged. If you change the divisor, you must change the dividend by the same factor — the rewritten dividend is 54, not 5.4.
The idea underneath. Division asks how many groups of the divisor fit in the dividend. Multiplying the total and each group size by the same number leaves the number of groups unchanged. Use 10 or 100 to make the divisor a whole number.

3On level

Look at the work for

2.73 \times 14

shown below.

	Whole-number work:
Step 1	$\displaystyle 273 \times 14 = 1092 + 273$
Step 2	$\displaystyle = 1365$
Step 3	$\displaystyle \text{Decimal product: }13.65$

Select the statement that best describes the work.

The decimal point is misplaced.

The tens partial product was not shifted one place left.

A digit multiplication fact, a carry, or the addition is wrong.

The work is correct.

Show the answer and how to solve it

The tens partial product should be 2730, not 273. With the correct partial products, $2.73 \times 14 = 38.22$ .

1Look at Step 1 in the work. It shows two partial products: 1092 and 273.
2Check the ones product. The ones digit of 14 is 4, so ignoring the decimal point the ones partial product should be $273 \times 4 = 1092$ . That matches.
3Check the tens product. The tens digit of 14 is 1, which represents 1 ten. So the tens partial product should be $273 \times 10 = 2730$ , not 273.
4Using the correct tens partial product, the corrected integer sum and final decimal product are:
5The first mistake is in Step 1: the tens partial product was not shifted one place left, so 273 appears instead of 2730.

Common mistake. A common mistake is blaming the decimal because 13.65 looks too small. The real error comes earlier: the tens digit of 14 should create a partial product ten times as large, giving 2730 instead of 273.

The idea underneath. In standard multiplication, each partial product keeps the place value of the factor digit, so a tens digit creates a product that is 10 times as large as the same digit in the ones place.

4Stretch
Find the quotient for the expression below.
$\displaystyle 0.1 \div 0.4$
Select the correct quotient.
A
0.25
B
0.26
C
2.5
D
25
Show the answer and how to solve it
The answer is $0.25$ because $0.1 \div 0.4$ can be rewritten as $1 \div 4$ , and $1 \div 4 = 0.25$ .
1. 1The divisor is $0.4$ , which has $1$ decimal place, so move the decimal point $1$ place right in both numbers to make the divisor a whole number.
2. 2Rewrite the division. The quotient does not change.
3. 3Since $4$ does not divide into $1$ as a whole number, find the tenths digit first, then the hundredths.
4. 4Combine the parts: $0.2 + 0.05 = 0.25$ .
Common mistake. A common mistake is thinking that multiplying both numbers by $10$ also multiplies the quotient by $10$ . Scaling both the dividend and divisor by the same factor keeps the quotient unchanged, so the answer is $0.25$ , not $2.5$ or $25$ .
The idea underneath. You can make a decimal divisor into a whole number by multiplying both numbers by the same power of 10, and that does not change the quotient.