Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: , and . Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

Understand the Question A rational number in simplest form has a terminating decimal expansion if the prime factorisation of the denominator contains only prime factors and/or (i.e. of the form , where ). If contains any prime factor other than or , its decimal expansion is non-terminating repeating. Part (i) Step 1 · Analyse and Verify by Division The fraction is in simplest form. Prime factorisation of the denominator : Since the prime factors of the denominator are only and , the decimal expansion is terminating. Checking by long division: Answer > (i) Terminating; Part (ii) Step 1 · Analyse and Verify by Division The fraction is in simplest form. Prime factorisation of the denominator : Since the denominator contains a prime factor (other than or ), the decimal expansion is repeating. Checking by long division: Answer > (ii) Repeating; Part (iii) Step 1 · Analyse and Verify by Division The fraction is in simplest form. Prime factorisation of the denominator : Since the prime factors of the denominator are only and , the decimal expansion is terminating. Checking by long division: Answer > (iii) Terminating; Common Mistakes Not Checking Simplest Form First: Always ensure is reduced to lowest terms (i.e., and are co-prime) before checking denominator factors, or extra common factors might mislead the test. Bar Placement on Repeating Decimals: Placing the repeating bar over non-repeating digits (e.g. writing instead of ).

The World of Numbers | Exercise 3.5

Question 1

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: $\dfrac{7}{20}$ , $\dfrac{4}{15}$ and $\dfrac{13}{250}$ . Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

Solution

Understand the Question

A rational number $\dfrac{p}{q}$ in simplest form has a terminating decimal expansion if the prime factorisation of the denominator $q$ contains only prime factors $2$ and/or $5$ (i.e. of the form $2^n 5^m$ , where $n, m \ge 0$ ).
If $q$ contains any prime factor other than $2$ or $5$ , its decimal expansion is non-terminating repeating.

(i) $\dfrac{7}{20}$

Step 1 · Analyse $\dfrac{7}{20}$ and Verify by Division

The fraction $\dfrac{7}{20}$ is in simplest form.

Prime factorisation of the denominator $20$ :

\begin{aligned} 20 &= 2 \times 10 \\ &= 2 \times 2 \times 5 \\ &= 2^2 \times 5^1 \end{aligned}

Since the prime factors of the denominator are only $2$ and $5$ , the decimal expansion is terminating. Diagram 1

Checking by long division: $7 \div 20 = 0.35$

Answer

(i) Terminating; $0.35$

(ii) $\dfrac{4}{15}$

Step 1 · Analyse $\dfrac{4}{15}$ and Verify by Division

The fraction $\dfrac{4}{15}$ is in simplest form.

Prime factorisation of the denominator $15$ : $15 = 3 \times 5$

Since the denominator contains a prime factor $3$ (other than $2$ or $5$ ), the decimal expansion is repeating. Diagram 2

Checking by long division: $4 \div 15 = 0.2666\dots = 0.2\overline{6}$

Answer

(ii) Repeating; $0.2\overline{6}$

(iii) $\dfrac{13}{250}$

Step 1 · Analyse $\dfrac{13}{250}$ and Verify by Division

The fraction $\dfrac{13}{250}$ is in simplest form.

Prime factorisation of the denominator $250$ :

\begin{aligned} 250 &= 2 \times 125 \\ &= 2 \times 5 \times 25 \\ &= 2 \times 5 \times 5 \times 5 \\ &= 2^1 \times 5^3 \end{aligned}

Since the prime factors of the denominator are only $2$ and $5$ , the decimal expansion is terminating. Diagram 3

Checking by long division: $13 \div 250 = 0.052$

Answer

(iii) Terminating; $0.052$

Common Mistakes

Not Checking Simplest Form First: Always ensure $\dfrac{p}{q}$ is reduced to lowest terms (i.e., $p$ and $q$ are co-prime) before checking denominator factors, or extra common factors might mislead the test.
Bar Placement on Repeating Decimals: Placing the repeating bar over non-repeating digits (e.g. writing $\overline{0.26}$ instead of $0.2\overline{6}$ ).

Question 1

More questions in Exercise 3.5