Multiply three-digit whole numbers by decimal tenths

Ontario Grade 6 Mathematics

Number

A Grade 6 lesson for multiplying three-digit whole numbers by decimal tenths

A decimal tenth is one of ten equal parts of a whole. For example,

0.4

means four tenths. When we multiply a three-digit whole number by a decimal tenths amount, we find part of that whole number. We can first use whole-number multiplication, then use the meaning of tenths to place the decimal point. In this lesson, the decimal factor has one digit after the decimal point, such as

0.3

or

1.2

.

What you will learn

Explain what a decimal tenth means.
Multiply a three-digit whole number by a decimal tenths amount.
Use whole-number multiplication to help find the product.
Check that a product makes sense.

1. Build the meaning of tenths

First, review place value. In the number

246

, the digits show

2

hundreds,

4

tens, and

6

ones. A whole number does not have a decimal part written.

Tenths are parts made when one whole is split into ten equal parts. So

0.1

is one tenth, and

0.4

is four tenths. The digit after the decimal point tells how many tenths there are.

Imagine a strip that stands for

246

. If the strip is split into ten equal parts, each part is one tenth of

246

. Three parts are

0.3

of

246

. To find that amount, divide

246

into ten equal parts and take three of them.

There is also a useful multiplication way to think about this. First find

246

multiplied by

3

. That gives the amount for three groups of

246

. Since

0.3

is three tenths, the product must be one tenth of that whole-number product.

0.3 = \frac{3}{10}

$0.1$ means one tenth.
$0.3$ means three tenths, or three of ten equal parts.
A product by a number less than $1$ is less than the starting whole number.

2. Use whole-number multiplication

Multiply the three-digit number by the digit that names the tenths. Then use place value to find one tenth of that result.

For example, to find

246 \times 0.3

, first calculate

246 \times 3

. The answer is

738

. But

0.3

is not three whole groups; it is three tenths of a group. So the product is one tenth of

738

, which is

73.8

.

Moving from

738

to

73.8

makes each place value ten times smaller. The hundreds become tens, the tens become ones, and the ones become tenths. This is why the decimal point is placed one place from the right in the whole-number product.

If the whole-number product has fewer than two digits, write a zero first when needed. For example,

125 \times 0.4

starts with

125 \times 4 = 500

. One tenth of

500

is

50.0

, which is

50

.

N \times 0.d = \frac{N \times d}{10}

Multiply by the tenths digit as if it were a whole number.
The decimal factor has one digit after the decimal point, so the product has tenths when needed.
A final zero after the decimal point does not change the value.

3. Include tenths greater than one

A decimal tenths amount can be greater than one. For example,

1.2

means one whole and two tenths. That is twelve tenths altogether.

To multiply by

1.2

, multiply by

12

first, then find one tenth of the result. This works because

1.2

is twelve tenths. The answer will be a little greater than the starting number because

1.2

is greater than

1

.

You can use a quick size check. Multiplying by

0.6

should give less than the original number. Multiplying by

1.6

should give more. These checks do not give the exact answer, but they can show whether an answer is unreasonable.

1.2 = \frac{12}{10}

$1.2$ is twelve tenths, not twelve whole groups.
For a decimal factor below $1$ , expect a smaller product.
For a decimal factor above $1$ , expect a larger product.

4. Practise the method

Use the same steps for each question: multiply by the digits as whole numbers, then account for the tenths. Say what the decimal means before placing the decimal point.

Try these without looking at the examples:

318 \times 0.2

and

407 \times 1.3

. For the first, expect an answer less than

318

. For the second, expect an answer greater than

407

.

When checking, compare the product with the original number. Also check the whole-number multiplication before placing the decimal point. A small error in that first calculation will change the final answer.

Show the whole-number multiplication so it can be checked.
Use the size of the decimal factor to judge whether the product should be smaller or larger.

Worked example

Find three tenths of 246

Calculate

246 \times 0.3

.

Multiply by 3
The $3$ in $0.3$ means three tenths. First multiply $246$ by $3$ as whole numbers.
$246 \times 3 = 738$
Find one tenth
The product for three whole groups is $738$ . Three tenths is one tenth of that amount, so write the value in tenths.
$738 \div 10 = 73.8$

Answer:

246 \times 0.3 = 73.8

Check: Since

0.3

is less than

1

, the answer must be less than

246

. The result,

73.8

, fits.

Worked example

Find four tenths of 125

Calculate

125 \times 0.4

.

Multiply by 4
The $4$ in $0.4$ names four tenths. Multiply by $4$ first.
$125 \times 4 = 500$
Account for tenths
The factor is four tenths, so the whole-number product must be divided into ten equal parts. Here the result is a whole number.
$500 \div 10 = 50$

Answer:

125 \times 0.4 = 50

Check: Four tenths is less than one whole, so the answer should be less than

125

. The answer

50

is reasonable.

Worked example

Multiply by a decimal greater than one

Calculate

326 \times 1.4

.

Use fourteen tenths
$1.4$ means fourteen tenths. Multiply $326$ by $14$ as whole numbers.
$326 \times 14 = 4564$
Find one tenth
Fourteen tenths of $326$ is one tenth of $4564$ . Place the decimal point so the value is in tenths.
$4564 \div 10 = 456.4$

Answer:

326 \times 1.4 = 456.4

Check: Since

1.4

is greater than

1

, the product should be greater than

326

. The answer

456.4

fits.

Common mistakes and how to avoid them

Writing

738

for

246 \times 0.3

.

Correction:

738

is the answer to

246 \times 3

. Since

0.3

is three tenths, the product is one tenth of

738

, or

73.8

.

Putting the decimal point in a different place each time without checking place value.

Correction: Use the meaning of tenths. Divide the whole-number product by

10

, then check the size of the answer.

Thinking

1.4

means fourteen whole groups.

Correction:

1.4

means fourteen tenths. It is one whole and four tenths.

Lesson summary

A decimal tenths factor tells how many tenths of a number to find.
Multiply by the tenths digit or digits as whole numbers, then divide that product by $10$ .
Check whether the result should be less than or greater than the starting number.

Check your understanding

Question 1

What is

215 \times 0.4

?

$86$
$860$
$8.6$
$219$

Show answer and explanation

86

215 \times 4 = 860

, and one tenth of

860

is

86

. Since

0.4

is less than

1

, a result below

215

makes sense.

Question 2

What is

304 \times 1.2

?

$364.8$
$36.48$
$3648$
$305.2$

Show answer and explanation

364.8

304 \times 12 = 3648

. One tenth of

3648

is

364.8

. This is greater than

304

, as expected for a factor greater than

1

.

Key terms

Decimal tenths: Parts made when a whole is split into ten equal parts. The first digit after the decimal point counts tenths.
Product: The answer to a multiplication question.
Whole-number product: The answer found by multiplying with whole numbers before accounting for the tenths.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.7. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B2.7 · Multiply three-digit whole numbers by decimal tenths

Ontario Grade 6 Mathematics

A Grade 6 lesson for multiplying three-digit whole numbers by decimal tenths

What you will learn

1. Build the meaning of tenths

2. Use whole-number multiplication

3. Include tenths greater than one

4. Practise the method

Find three tenths of 246

Find four tenths of 125

Multiply by a decimal greater than one

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Key terms

Continue through Grade 6 Math

About this lesson and its review