5th Grade · 5.NBT.B.7

Multiplying decimals in 5th grade

Multiplying decimals is one part of a bigger 5th grade goal: working with decimals in all four operations. A decimal is a number with a decimal point, such as 2.5 or 0.36. In 5th grade, your child adds, subtracts, multiplies and divides decimals to hundredths and explains each answer with place value.

What the standard says

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

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

What it means, in plain words

The digits after the decimal point are tenths and hundredths. In 0.36, the 3 means 3 tenths and the 6 means 6 hundredths. Adding and subtracting decimals works like adding and subtracting whole numbers, with one rule: line up the decimal points. Then tenths sit under tenths and hundredths sit under hundredths. An empty place can be filled with a 0, so 7.2 can be written as 7.20.

Multiplying decimals starts with whole numbers. To find $0.3 \times 0.2$ , your child first finds $3 \times 2 = 6$ . Then place value decides where the decimal point goes. Tenths times tenths gives hundredths, so the answer is 6 hundredths, or 0.06. A quick estimate also catches a misplaced point: $3.1 \times 4.9$ is close to $3 \times 5 = 15$ , and the exact answer is 15.19.

Dividing decimals uses the same ideas. $4.8 \div 4 = 1.2$ , because 4 groups of 1.2 make 4.8. To divide by a decimal, children think about how many groups fit: $3 \div 0.5 = 6$ , because six halves make 3. Drawings, such as a grid of 100 small squares, come before any shortcut.

Your child should be able to:

Add and subtract decimals to hundredths by lining up the decimal points
Multiply decimals and place the decimal point using place value or an estimate
Divide a decimal by a whole number, and a whole number by a simple decimal such as 0.5
Show a decimal calculation with a drawing, such as a grid of 100 squares

Example

What is $1.6 \times 0.3$ ?

Multiply as if there were no decimal points: $16 \times 3 = 48$ .
1.6 is 16 tenths and 0.3 is 3 tenths. Tenths times tenths gives hundredths, so the answer is 48 hundredths.
48 hundredths is written 0.48.
Check with an estimate: 1.6 is less than 2, and $2 \times 0.3 = 0.6$ , so the answer must be less than 0.6.
So 1.6 times 0.3 is 0.48.

Example

What is $7.2 - 3.45$ ?

Line up the decimal points. Write 7.2 as 7.20 so both numbers have hundredths.
Subtract as with whole numbers, regrouping where needed: $7.20 - 3.45 = 3.75$ .
Check by adding: $3.75 + 3.45 = 7.20$ .
So 7.2 minus 3.45 is 3.75.

Where children usually get stuck

To add $4.5 + 2.36$ , your child lines up the last digits and gets 2.81.
With whole numbers, the right ends always line up. With decimals, that puts 5 tenths on top of 6 hundredths.
What helps. Line up the decimal points instead and write 4.5 as 4.50. The sum is 6.86. A quick estimate (about 4 plus 2) also shows that 2.81 is far too small.
Your child writes $0.4 \times 0.2 = 0.8$ .
The child treats multiplying like adding, where tenths plus tenths stay tenths. Tenths times tenths gives hundredths, so the answer is 0.08.
What helps. Draw a 10 by 10 grid. Shade 4 columns for 0.4. Then shade 2 rows in another color for 0.2. The squares shaded twice, 8 out of 100, show 0.08.
For $3.2 \times 4.1$ , your child writes 131.2 or 1.312.
The child found $32 \times 41 = 1{,}312$ correctly but guessed where the decimal point goes.
What helps. Estimate first: about $3 \times 4 = 12$ . The only placement close to 12 is 13.12.

Try four practice questions

Real 5th 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

Maya is finding

6.48 \div 2

. Her work is shown below.

	$\displaystyle 6.48 \div 2$
Break apart and divide	$\displaystyle = 6 \div 2 + 0.4 \div 2 + 0.08 \div 2$
Simplify	$\displaystyle = 3 + 0.2 + 0.04$
Final answer	$\displaystyle = 32.4$

Select the statement that best describes Maya's work.

Maya placed the decimal point too far to the right.

Maya left out the hundredths part of the dividend.

Maya divided one place-value part incorrectly.

Every step shown is correct.

Show the answer and how to solve it

The correct statement says Maya placed the decimal point too far to the right; the correct quotient is 3.24, not 32.4.

1Look at the division line. Maya breaks 6.48 into 6, 0.4, and 0.08, then divides each part by 2. Check each partial quotient.
2Every partial quotient is correct. Adding them gives the correct sum.
3The partial quotients add up to 3.24. But the final line shows 32.4 — the decimal point moved one place to the right.
4Because no part was omitted and each division was correct, the only error is on the last line: the decimal point is too far to the right. The statement saying Maya placed the decimal point too far to the right correctly describes her work.

Common mistake. A common mistake is choosing 'divided one place-value part incorrectly' because 32.4 looks wrong. But $3 + 0.2 + 0.04 = 3.24$ is correct — only the final decimal placement is wrong.

The idea underneath. Sharing a total among equal groups is the same as sharing each part among those groups. Add the shares in one group to find that group's total.

2On level

Marco is placing the decimal point in a product. His work is shown below.

Product	$\displaystyle 0.53 \times 0.4$
Raw digits	$\displaystyle 53 \times 4 = 212$
Decimal places	$\displaystyle 2\text{ decimal places} + 1\text{ decimal place} = 3\text{ decimal places}$
Marco's answer	$\displaystyle 212\text{ with 3 decimal places} = 2.12$

Select the statement that best describes Marco's reasoning.

Marco is correct; the product is 2.12.

Marco counted 3 decimal places correctly, but he placed the decimal so the product has only 2 decimal places; it should be 0.212.

Marco forgot to use the decimal places in both factors and left the product as 212.

Marco should multiply the whole-number product 212 by 10 because the decimal points were removed before multiplying.

Show the answer and how to solve it

The correct statement is that Marco counted 3 decimal places correctly, but his final product should be 0.212, not 2.12.

1Check Marco's work line by line.
2The raw multiplication is correct: ignoring the decimal points gives $53 \times 4$ .
30.53 is 53 hundredths, and 0.4 is 4 tenths. The factors are 53 divided by 100 and 4 divided by 10. So divide 212 by 100, then by 10. The product is 212 thousandths, or 0.212.
4Place the decimal point so that 3 digits fall after it. The raw digits 212 become 0.212.
5Marco wrote 2.12, which has only 2 decimal places after the point. His decimal point is one position too far to the right. The correct product is 0.212.

Common mistake. A common mistake is to count 3 decimal places correctly but then place the decimal one position too far right, writing 2.12 (only 2 decimal places after the point) instead of 0.212 (3 decimal places). After placing the point, always recount the digits after it to confirm they match the total.

The idea underneath. The decimal places show the size of each unit. Dividing by 100, then by 10, means dividing by 1,000. That is why 2 + 1 decimal places give 3 decimal places in this product.

3On level
The two decimals below are lined up for addition. Use each dropdown to label the column below it.
3 . 7 2
+ 5 . 4 6
Show the answer and how to solve it
From left to right, the column labels are ones, decimal points, tenths, and hundredths.
1. 1Start by finding the decimal point. It is the column with the dot in both addends. For that column, select 'decimal points' — this is col-2.
2. 2The column immediately to the left of the decimal point holds whole-number digits (3 and 5). That column is the ones place — select 'ones' for col-1.
3. 3The first column to the right of the decimal point (7 and 4) is the tenths place — select 'tenths' for col-3. The next column to the right (2 and 6) is the hundredths place — select 'hundredths' for col-4.
Common mistake. A common mistake is choosing 'tens' for the leftmost digit column (col-1). Because 3.72 has only one digit to the left of the decimal point, that digit is in the ones place, not the tens place. Always use the decimal point's position — not the digit's distance from the edge — to determine the place value.
The idea underneath. The decimal point separates whole-number places from fractional places. Use it to find the place value of each digit column.
4Stretch
Use compensation to find the exact sum.
$\displaystyle 3.97 + 4.56$
Choose an adjustment that keeps the sum the same and gives at least one number a 0 in the hundredths place. Then type the exact sum.
Adjustment Exact sum
Show the answer and how to solve it
Choose 1st +0.03; 2nd -0.03 and enter 8.53 because $3.97 + 4.56$ has the same total as $4.00 + 4.53$ .
1. 1Start with the original addition: $3.97 + 4.56$ .
2. 2Add 0.03 to the first addend and subtract 0.03 from the second addend. The total stays the same. $4.00$ has a 0 in the hundredths place.
3. 3Add the adjusted decimals to get the exact sum.
4. 41st -0.03; 2nd +0.03 gives $3.94 + 4.59$ . The sum stays 8.53, but neither number has a 0 in the hundredths place. Choose 1st +0.03; 2nd -0.03, and enter 8.53.
Common mistake. 1st +0.03; 2nd +0.03 changes the sum by $0.06$ because both numbers increase. The result is $4.00 + 4.59 = 8.59$ , not $8.53$ .
The idea underneath. Adding an amount to one number and subtracting the same amount from the other keeps the sum the same. A 0 in the hundredths place lets you work with tenths.