Multiply Decimals: Problems, Comparisons and Mistakes | Grade 5 Math | LevelKite

You already know how to multiply decimals and where to put the decimal point. In this lesson you will use that skill in three ways. You will solve story problems, predict how big an answer will be before you multiply, and find and fix mistakes.

Two words will come up often. The numbers you multiply are called factors. The answer to a multiplication is called the product. In $3 \times 0.5 = 1.5$ , the factors are $3$ and $0.5$ , and the product is $1.5$ .

In a story problem, you decide which numbers to multiply, find the product, and then make sure the product makes sense. An estimate helps you with that last part.

An estimate is a number close to the exact answer that you find with easy numbers. To estimate a product, round one or both factors. To round means to change a number to a nearby number that is easier to use. Then multiply the easier numbers.

Take $4.8 \times 6.1$ . The factor $4.8$ is close to $5$ , and $6.1$ is close to $6$ , so the product is close to $5 \times 6 = 30$ . The exact product is $29.28$ , which is close to $30$ , so its size makes sense. An answer of $292.8$ or $2.928$ would be far from $30$ . That would tell you to look for a mistake, such as a decimal point in the wrong place.

Worked example

A story problem with an estimate check

One lap of a walking trail is $2.3$ kilometers long. Maya walks $3.5$ laps. How many kilometers does she walk?

Decide what to multiply. Every lap has the same length, 2.3 kilometers, and Maya walks that length 3.5 times. So multiply the number of laps by the length of one lap.
$3.5 \times 2.3$
Estimate first. Round 3.5 up to 4 and 2.3 down to 2, because whole numbers are easy to multiply. The exact answer should be close to 8 kilometers.
$4 \times 2 = 8$
Multiply as if the factors were the whole numbers 35 and 23. Split 23 into 20 and 3, because each part is easy to multiply by 35. Then add the two results.
$35 \times 23 = 700 + 105 = 805$
Place the decimal point. The factor 3.5 is 35 tenths, and 2.3 is 23 tenths. Tenths times tenths gives hundredths, so the product is 805 hundredths, which is 8.05.
$3.5 \times 2.3 = 8.05$
Check against the estimate. The product 8.05 is close to 8, so the size of the answer makes sense. An answer of 80.5 or 0.805 would be far from 8.

Answer

Maya walks $8.05$ kilometers, which is close to the estimate of $8$ kilometers.

An estimate takes a little multiplying. Sometimes you can tell how a product compares with one of its factors with no multiplying at all.

Pick one factor to be your starting number, and check that it is greater than 0. The other factor tells you how many copies of the starting number you get: exactly one copy for a factor of $1$ , more than one copy for a factor greater than $1$ , and only part of one copy for a factor less than $1$ . Because the starting number is greater than 0, part of one copy is less than the starting number, and more than one copy is greater than it.

The table shows this with two starting numbers that are greater than 0: $8$ and $0.6$ . Each row shows the same amount two ways: as a number sentence, which uses numbers and symbols, and in words.

Starting numbers greater than 0: what the other factor does to the product

Starting number	Number sentence	In words	Other factor compared with 1	Product compared with starting number
$8$	$0.5 \times 8 = 4$	half of one 8	$0.5 < 1$	$4 < 8$
$8$	$1 \times 8 = 8$	exactly one 8	$1 = 1$	$8 = 8$
$8$	$1.5 \times 8 = 12$	one 8 and half of another 8	$1.5 > 1$	$12 > 8$
$0.6$	$0.5 \times 0.6 = 0.3$	half of one 0.6	$0.5 < 1$	$0.3 < 0.6$
$0.6$	$1.5 \times 0.6 = 0.9$	one 0.6 and half of another 0.6	$1.5 > 1$	$0.9 > 0.6$

Key concept

Know how big a product should be before you trust it

You can compare a product with one of its factors without multiplying. Call that factor the starting number, and make sure it is greater than 0. Then look at the other factor:

Less than 1: the product is less than the starting number.
Equal to 1: the product equals the starting number.
Greater than 1: the product is greater than the starting number.

This rule is only for starting numbers greater than 0. Multiplying 0 always gives 0, so a starting number of 0 stays 0: $0.5 \times 0 = 0$ and $1.5 \times 0 = 0$ .

An estimate with easy numbers then shows roughly where the product should land.

Worked example

Order three numbers without multiplying

Without multiplying, put these three numbers in order from least to greatest: $0.9 \times 4.6$ , $4.6$ and $1.1 \times 4.6$ .

Find the starting number. Both products multiply 4.6, so use 4.6 as the starting number. It is greater than 0, so you can compare each product with 4.6 by looking at the other factor.
Look at the first product. The starting number 4.6 is greater than 0 and the other factor, 0.9, is less than 1. The product is only part of one 4.6, so it is less than 4.6.
$0.9 \times 4.6 < 4.6$
Look at the other product. The starting number 4.6 is greater than 0 and the other factor, 1.1, is greater than 1. The product is more than one 4.6, so it is greater than 4.6.
$1.1 \times 4.6 > 4.6$
Put the three numbers in order. One product is less than 4.6 and the other is greater than 4.6, so 4.6 goes in the middle.
$0.9 \times 4.6 < 4.6 < 1.1 \times 4.6$
Multiply only to check the reasoning. As whole numbers, 9 times 46 is 414 and 11 times 46 is 506. Tenths times tenths gives hundredths, so the products are 4.14 and 5.06.
$4.14 < 4.6 < 5.06$

Answer

From least to greatest, the order is $0.9 \times 4.6$ , then $4.6$ , then $1.1 \times 4.6$ .

You now have ways to check a product, whether the work is yours or someone else's. Ask three questions:

Is it close to an estimate? If it is far away, look at the decimal point.
Does it compare correctly with a factor? For a starting number greater than 0, a factor less than 1 gives a smaller product, a factor greater than 1 gives a larger product, and a factor of 1 keeps it the same.
Do the place values fit? When you add, you line up the decimal points so that tenths are added to tenths, and the sum is still in tenths. When you multiply, tenths times tenths gives hundredths, so the decimal point does not come straight down.

The table shows three common mistakes and a check that catches each one.

Three common mistakes and a check that catches each one

Mistake	Work with the mistake	Check that catches it	Corrected work
Decimal point in the wrong place	$4.1 \times 1.9 = 0.779$	Estimate: $4 \times 2 = 8$ . The answer $0.779$ is far from $8$ .	$41 \times 19 = 779$ . Tenths times tenths gives hundredths: $7.79$ .
Decimal points lined up as in addition	$1.2 \times 0.3 = 3.6$ , with the point brought straight down	Since $1.2$ is greater than 0 and $0.3 < 1$ , the product must be less than $1.2$ . But $3.6 > 1.2$ .	$12 \times 3 = 36$ . Tenths times tenths gives hundredths: $0.36$ .
Expecting every product to be greater than its factors	$0.6 \times 0.7 = 0.42$ is changed to $4.2$ because $0.42$ is smaller than both factors	Both factors are greater than 0 and less than 1, so the product is less than each factor.	Keep $0.42$ . It is less than $0.6$ and less than $0.7$ .

Worked example

Find and fix a mistake

A ribbon is $4.2$ meters long. Lena uses $0.6$ of the ribbon. A classmate writes $4.2 \times 0.6 = 25.2$ and says Lena uses $25.2$ meters. Find the mistake and correct it.

Compare the product with a factor. The starting number 4.2 is greater than 0 and the other factor, 0.6, is less than 1, so the product must be less than 4.2. That fits the story, because part of a ribbon is shorter than the whole ribbon. The classmate's answer is greater than 4.2, so it cannot be right.
$25.2 > 4.2$
Estimate. Round 4.2 down to 4. Keep 0.6, because rounding it to a whole number would change it a lot, and 4 times 6 tenths is an easy 24 tenths. The exact answer should be close to 2.4 meters, and 25.2 is far from that.
$4 \times 0.6 = 2.4$
Find the mistake. The digits are right, because 42 times 6 is 252. The classmate brought the decimal point straight down, as in addition, which leaves 252 tenths. But 4.2 is 42 tenths and 0.6 is 6 tenths, and tenths times tenths gives hundredths.
$42 \times 6 = 252$
Correct the work. The product is 252 hundredths, which is 2.52. It is less than 4.2 and close to the estimate of 2.4, so it passes both checks.
$4.2 \times 0.6 = 2.52$

Answer

The classmate lined up the decimal point as in addition, which put it in the wrong place. Lena uses $2.52$ meters of ribbon.

Takeaway: Know how big a decimal product should be before you trust it. An estimate with easy numbers shows roughly where the product should land. Comparing with a factor tells you more: for a starting number greater than 0, a factor less than 1 gives a smaller product, a factor greater than 1 gives a larger product, and a factor of 1 keeps it the same. Multiplying 0 always gives 0, so this rule does not fit a starting number of 0. When a product fails a check, look at the decimal point first and think about place value.