B. Number · Number Sense: Big Numbers, Square Roots and Rational Numbers

Converting Between Fractions and Decimals

B1.7About 40 minutes

Learning goals

I can change a fraction to a decimal using equivalent fractions or division.
I can change a decimal to a fraction in simplest form.
I can tell whether a fraction makes a terminating or a repeating decimal.
I can convert mixed numbers and negative fractions and decimals.

Words to know

convert
Change a number to a different form without changing its value.

terminating decimal
A decimal that ends, like 0.625.

repeating decimal
A decimal with a digit or group of digits that repeats forever, like 0.454545…. It can be written with a bar over the repeating part.

mixed number
A whole number and a fraction together, like 2 3/4.

Remember from Grade 6

In Grade 6 you changed fractions with denominators like 10, 100 and 1000 into decimals, used equivalent fractions (3/8 = 375/1000 = 0.375), and saw that 1 ÷ 3 = 0.333…. This year you will convert any fraction, including mixed numbers and negative numbers.

Which is more?

Owen's recipe for bannock uses 3/4 cup of water. His measuring jug is marked in decimals of a cup: 0.25, 0.5, 0.75, 1. Where should he fill to? 3/4 = 0.75, so he fills to the 0.75 mark.

We often need to switch between fractions and decimals. Here are the methods.

Fraction → decimal

Method 1: Equivalent fraction with 10, 100 or 1000. If you can multiply the denominator to make 10, 100 or 1000, do that.

7/20 = 35/100 = 0.35 (multiply by 5)

Method 2: Divide. The fraction bar means divide. a/b = a ÷ b. This works for every fraction.

5/8 = 5 ÷ 8 = 0.625
5/6 = 5 ÷ 6 = 0.8333… (the 3 repeats)

Decimal → fraction

Read the decimal's place value, write it as a fraction, then simplify.

0.48 = 48 hundredths = 48/100 = 12/25
0.375 = 375/1000 = 3/8

0.35 on a hundred grid: 35 hundredths. The shaded part can also be split into 7 groups of 5 squares out of 20 groups, so 0.35 = 7/20.

Worked example

Example 1: Equivalent fraction

Write 7/20 as a decimal.

20 × 5 = 100, so multiply the numerator and denominator by 5.
7/20 = 35/100.

7/20 = 0.35

Worked example

Example 2: Divide

Write 5/8 as a decimal.

5 ÷ 8: 8 doesn't go into 5, so write 5 as 5.000.
50 ÷ 8 = 6 R2 → 0.6. 20 ÷ 8 = 2 R4 → 0.62. 40 ÷ 8 = 5 R0 → 0.625. The remainder is 0, so it stops.

5/8 = 0.625 (a terminating decimal)

Worked example

Example 3: A mixed number that repeats

Write 2 5/6 as a decimal.

The whole number part stays: 2.
5/6 = 5 ÷ 6 = 0.8333… (the remainder 2 keeps coming back, so the 3 repeats).
So 2 5/6 = 2.8333…, which we can also write with a bar over the repeating 3.

2 5/6 = 2.8333…, about 2.83 rounded to the nearest hundredth.

Worked example

Example 4: Negative decimal to fraction

Write −1.375 as a mixed number in simplest form.

The whole number part is 1. The decimal part 0.375 = 375/1000.
Simplify: the GCF of 375 and 1000 is 125. 375 ÷ 125 = 3 and 1000 ÷ 125 = 8.
Keep the negative sign.

−1.375 = −1 3/8

Terminating or repeating?

Write the fraction in simplest form and look at the denominator. If its only prime factors are 2s and 5s (like 2, 4, 5, 8, 10, 16, 20, 25, 40, 50), the decimal terminates. If it has any other prime factor (like 3, 7 or 11), the decimal repeats. 3/8 terminates (8 = 2 × 2 × 2); 5/6 repeats (6 = 2 × 3).

The denominator decides: only 2s and 5s means the decimal stops; any other prime factor means it repeats forever.

Hands-on activity

Calculator pattern detective

You need: a calculator, paper and a pencil.

Use the calculator to change these fractions to decimals. Write each answer.

1/9, 2/9, 4/9
1/11, 2/11, 3/11
1/8, 3/8, 5/8

Which ones terminate? Which repeat? What repeats in each?
Predict 5/9 and 4/11 before you check.
Bonus: look at your measuring cups and spoons at home. Write each fraction label as a decimal (1/4 cup = 0.25 cup, 1/3 cup = 0.333… cup).

What to look for: ninths repeat one digit (2/9 = 0.222…). Elevenths repeat a pair (1/11 = 0.0909…, 2/11 = 0.1818…, 3/11 = 0.2727…). Eighths terminate (1/8 = 0.125). So 5/9 = 0.555… and 4/11 = 0.3636…. A calculator may round the last digit it shows, so 2/3 might show as 0.6666666667.

Just for fun

1/7 as a decimal is 0.142857142857… The six digits 142857 repeat forever, like a song that's stuck in your head.

Your turn

Write 3/5 as a decimal.

Write 9/20 as a decimal.

Write 3/8 as a decimal.

Write 0.85 as a fraction in simplest form.

Write 1/6 as a decimal.

Write −2/5 as a decimal.

Write 3.25 as a mixed number.

More practice: warm-up

Write 1/2 as a decimal.

Write 0.7 as a fraction.

Write 1/4 as a decimal.

Write 0.03 as a fraction.

Write 1/10 as a decimal.

More practice: on your own

Write 11/25 as a decimal.

Write 7/8 as a decimal.

Write 0.625 as a fraction in simplest form.

Write 4/9 as a decimal.

Write −1 3/5 as a decimal.

Challenge

Write 5/12 as a decimal. Round it to the nearest thousandth.

Which of these make terminating decimals: 1/6, 2/6, 3/6, 4/6, 5/6? Explain.

Is 0.3 equal to 1/3? If not, which is greater, and by how much (as a fraction)?

Check your understanding

Quiz

What is 3/4 as a decimal?

What is 0.4 as a fraction in simplest form?

What is 2/3 as a decimal?

What is 1 1/8 as a decimal?

What is 0.35 as a fraction in simplest form?

Which fraction makes a repeating decimal?

For parents and teachers

Big idea: Grade 7 expectation B1.7 asks students to convert between fractions, decimal numbers, and percents, in various contexts. This lesson does fractions and decimals; the next lesson adds percents. Recalling the most common equivalents by heart (like 1/8 = 0.125 = 12.5%) is B2.2 in the Operations unit.

Common difficulties:

Writing 3/4 as 0.34. The fraction bar means divide.
Not simplifying (0.48 = 48/100 instead of 12/25).
Thinking 0.3 = 1/3. 1/3 is 0.333…, slightly more.
Losing the whole number or the sign in mixed numbers (2 5/6 → 0.833).

How to help: Do the calculator pattern activity. In the kitchen, ask your child to say measuring-cup fractions as decimals.