Practice Maths

Terminating and Recurring Decimals

Key Ideas

Key Terms

terminating decimal
a decimal that ends after a finite number of digits (e.g. ¼ = 0.25, ⅜ = 0.375).
recurring decimal
a decimal with a digit or group of digits that repeats forever, shown using dot notation (e.g. ⅓ = 0.333… = 0.3̇).
dot notation
a way to write recurring decimals by placing a dot above the first and last repeating digit (e.g. 0.3̇ = 0.3333…, 0.1̇6̇ = 0.1666…).
rational number
a number that can be written as p⁄q where p and q are integers and q ≠ 0; every terminating and recurring decimal is rational.

Converting Fractions to Decimals

Divide the numerator by the denominator using short or long division. If the remainder reaches 0, the decimal terminates. If the remainder repeats, the decimal recurs.

Hot Tip To convert a recurring decimal to a fraction, use algebra. For 0.3̇: let x = 0.333…, then 10x = 3.333…. Subtract: 10x − x = 3, so 9x = 3, giving x = 1/3. For a two-digit repeat like 0.1̇2̇ = 0.121212…, multiply by 100 instead.

Worked Example

Question: Convert 3/8 and 5/6 to decimals. State whether each terminates or recurs.

3/8:
8 = 23 — only factor of 2, so it terminates.
3 ÷ 8 = 0.375    Terminating decimal.

5/6:
6 = 2 × 3 — has factor of 3, so it recurs.
5 ÷ 6 = 0.8333… = 0.8̇3̇    Recurring decimal.

Where does 1/3 sit on a number line from 0 to 1?

0 1 0.3 0.33 1/3 = 0.3̇ 0.34

Note: 0.3 < 1/3 = 0.3̇ < 0.34 < 1

Two Types of Decimal

When you convert a fraction to a decimal by dividing, one of two things happens: either the division eventually gives a zero remainder and the decimal ends, or the remainder repeats in a cycle and the decimal goes on forever. These give us two types of decimal:

  • Terminating decimal: The decimal has a finite number of digits after the decimal point and then stops. Examples: 14 = 0.25, 38 = 0.375, 720 = 0.35.
  • Recurring decimal: The decimal has a digit or group of digits that repeats forever. Examples: 13 = 0.3333... = 0.3̅, 16 = 0.1666... = 0.16̅, 17 = 0.142857142857... = 0.1̅4̅2̅8̅5̅7̅.

Dot Notation for Recurring Decimals

Writing dots over digits is the standard way to show recurring decimals without writing forever:

  • A dot over a single digit means that digit repeats: 0.3̅ = 0.3333...
  • Dots over the first and last digit of a repeating block means the whole block repeats: 0.1̅4̅ = 0.141414...
  • 0.1̅4̄2̄8̄5̄7̅ = 0.142857142857...

In this notation, the dots are placed directly above the digits. In printed text you will sometimes see an overline bar instead — both mean the same thing.

Converting Fractions to Decimals

To convert any fraction to a decimal, divide the numerator by the denominator. Work through the long division until you either get a zero remainder (terminating) or notice the same remainder appearing again (recurring).

Example — terminating: Convert 38 to a decimal.
3 ÷ 8: 3.000 ÷ 8 = 0.375. Remainder is 0, so it terminates. Answer: 0.375.

Example — recurring: Convert 23 to a decimal.
2 ÷ 3: 2.0000 ÷ 3 = 0.6666... The remainder 2 keeps repeating, so 23 = 0.6̅.

Which Fractions Terminate?

Here is the rule: a fraction in its simplest form terminates if and only if the denominator's only prime factors are 2 and/or 5. In other words, the denominator must be of the form 2a × 5b (where a and b are whole numbers, including zero).

  • 14 — denominator 4 = 22. Only factor is 2. Terminates.
  • 320 — denominator 20 = 22 × 5. Only factors are 2 and 5. Terminates.
  • 16 — denominator 6 = 2 × 3. Has a factor of 3. Recurring.
  • 512 — denominator 12 = 22 × 3. Has a factor of 3. Recurring.
  • 735 = 15 (simplified). Denominator 5. Terminates. (Always simplify first!)
Key tip: Always simplify the fraction to lowest terms before applying the rule. For example, 610 simplifies to 35, whose denominator is 5 (only factor 5), so it terminates: 35 = 0.6. If you forget to simplify, you might incorrectly think the fraction recurs because you see a factor of 2 and 5 in the denominator that isn't really there after simplification.

Why Do Fractions Recur?

When you divide n by d, the remainder at each step must be one of the values 0, 1, 2, ..., d−1. If the remainder is never 0, by the time you have done d steps you must get a remainder you have seen before — and the pattern repeats from that point. This is why every fraction is either terminating or recurring. There is no third option. Decimals that neither terminate nor recur (like π = 3.14159...) cannot be written as fractions — these are called irrational numbers, which you will study in the next lesson.

Mastery Practice

  1. Convert each fraction to a decimal. State whether it is terminating (T) or recurring (R). Fluency

    1. 1/4
    2. 1/3
    3. 3/5
    4. 2/9
    5. 7/8
    6. 5/11
    7. 3/4
    8. 4/15

  2. Use dot notation to write each recurring decimal. Fluency

    1. 0.4444…
    2. 0.727272…
    3. 0.1666…
    4. 0.583333…
    5. 0.142857142857…
    6. 0.090909…
    7. 0.3181818…
    8. 0.6666…

  3. Without dividing, state whether each fraction produces a terminating or recurring decimal. Explain your reasoning using prime factors of the denominator. Fluency

    1. 3/16
    2. 7/15
    3. 9/25
    4. 5/14
    5. 11/40
    6. 3/7
    7. 13/50
    8. 4/21

  4. Use the algebraic method to convert each recurring decimal to a fraction in its simplest form. Show all steps. Understanding

    1. 0.4̇ (i.e. 0.4444…)
    2. 0.7̇ (i.e. 0.7777…)
    3. 0.1̇2̇ (i.e. 0.121212…)
    4. 0.3̇6̇ (i.e. 0.363636…)
    5. 0.8̇ (i.e. 0.8888…)
    6. 0.2̇7̇ (i.e. 0.272727…)
    7. 0.5̇ (i.e. 0.5555…)
    8. 0.1̇4̇ (i.e. 0.141414…)

  5. Order each set of numbers from smallest to largest. Convert recurring decimals to enough decimal places to compare. Understanding

    1. 0.3̇,   0.34,   1/3,   0.3
    2. 2/3,   0.6̇6̇,   0.67,   0.666
    3. 5/8,   0.625,   0.6̇2̇,   0.63
    4. 0.1̇2̇,   0.12,   2/17,   0.125
    5. 7/9,   0.77,   0.7̇,   0.778
    6. 1/6,   0.16,   0.1̇6̇,   0.17

  6. Apply your knowledge to these real-world problems. Problem Solving

    1. A ribbon 1 metre long is cut equally among 3 students. Express each student’s share as a decimal. Is this terminating or recurring? How might you deal with this in practice?
    2. A circular pie chart has 9 equal sectors. What fraction of the full circle is each sector? Convert this to a decimal using dot notation.
    3. A recipe calls for dividing 2 cups of flour equally among 6 people.
      1. Express each person’s share as a fraction in simplest form.
      2. Convert to a decimal. Is it terminating or recurring?
      3. A measuring cup only shows two decimal places. What value would you use, and by how much does this differ from the exact amount?
    4. Priya says “0.9̇ is less than 1 because it never quite reaches 1.” Use the algebraic method to show that 0.9̇ = 1 exactly, and explain what this means.

  7. Complete the table. Fill in every “?” cell. Use dot notation for recurring decimals and simplify fractions fully. Understanding

    Fraction Decimal Percentage T or R?
    1/4???
    ?0.6̇?R
    3/8?37.5%?
    ??44%T
    5/9???
    ?0.35??

  8. Sort the numbers below into the correct column of the table. Understanding

    Numbers:   0.75,   1/3,   7/8,   0.1̇,   11/25,   5/6,   0.625,   4/15,   0.4̇5̇,   9/40

    Terminating decimal Recurring decimal
      

    Explain your reasoning: how did you decide which column each number belongs in?

  9. Mark each of the following decimals on the number line by estimating their position. Write the letter label at the correct location. Understanding

    A = 0.25,   B = 0.3̇,   C = 0.5,   D = 0.6̇6̇,   E = 0.75

    0 0.25 0.5 0.75 1

    Hint: convert each decimal to enough places to compare, then estimate the position between the labelled tick marks.

  10. A student made errors in the working below. Find each mistake and write the correct solution. Problem Solving

    1. Student writes: “3/6 = 0.5, which is terminating because 6 = 2 × 3, and 3 is a prime factor so it recurs.” What is the error?
    2. Student writes: “Let x = 0.4̇5̇. Then 10x = 4.5̇4̇, so 10x − x = 4.1, giving x = 4.1/9.” Find and fix the error.
    3. Student writes: “0.7̇ = 7/10 because the 7 is in the tenths place.” Is this correct? If not, find the correct fraction.
    4. A student says: “All recurring decimals are less than 1.” Give a counter-example to show this is false.
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