The fraction (7)/(20) is already in its simplest form since 7 and 20 share no common factors other than 1. Factorising the denominator, we get 20 = 2² × 5^1, which contains only the prime factors 2 and 5. Therefore, (7)/(20) has a terminating decimal expansion.

The fraction (7)/(20) is already in its simplest form since 7 and 20 share no common factors other than 1. Factorising the denominator, we get 20 = 2² × 5^1, which contains only the prime factors 2 and 5. Therefore, (7)/(20) has a terminating decimal expansion.

The fraction (4)/(15) is in simplest form because the numerator 4 and denominator 15 are co-prime. The denominator factors as 15 = 3 × 5. Because it contains the prime factor 3 (which is neither 2 nor 5), the fraction (4)/(15) has a non-terminating repeating decimal expansion.

The fraction (4)/(15) is in simplest form because the numerator 4 and denominator 15 are co-prime. The denominator factors as 15 = 3 × 5. Because it contains the prime factor 3 (which is neither 2 nor 5), the fraction (4)/(15) has a non-terminating repeating decimal expansion.

The fraction (13)/(250) is in simplest form since 13 is a prime number and does not divide 250. Factorising the denominator gives 250 = 2 × 125 = 2^1 × 5³. Since the only prime factors are 2 and 5, (13)/(250) has a terminating decimal expansion.

The fraction (13)/(250) is in simplest form since 13 is a prime number and does not divide 250. Factorising the denominator gives 250 = 2 × 125 = 2^1 × 5³. Since the only prime factors are 2 and 5, (13)/(250) has a terminating decimal expansion.

We verify our findings by computing the exact decimal value of each fraction. For the first fraction, (7)/(20) = (7 × 5)/(20 × 5) = (35)/(100) = 0.35 (terminating). For the second, (4)/(15) = 0.2666 = 0.2 6 (repeating). For the third, (13)/(250) = (13 × 4)/(250 × 4) = (52)/(1000) = 0.052 (terminating). All predictions are confirmed.

We verify our findings by computing the exact decimal value of each fraction. For the first fraction, (7)/(20) = (7 × 5)/(20 × 5) = (35)/(100) = 0.35 (terminating). For the second, (4)/(15) = 0.2666 = 0.2 6 (repeating). For the third, (13)/(250) = (13 × 4)/(250 × 4) = (52)/(1000) = 0.052 (terminating). All predictions are confirmed.

Exercise Set 3.5 1. Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: $\frac{7}{20}, \frac{4}{15} \text{ and } \frac{13}{250}$ . Then check your answers

Answer: Terminating: $\frac{7}{20}$ and $\frac{13}{250}$ ; Repeating: $\frac{4}{15}$ . Checking gives $\frac{7}{20} = 0.35$ , $\frac{4}{15} = 0.2\bar{6}$ , and $\frac{13}{250} = 0.052$ .

Step-by-step solution

Step 1: State condition for terminating decimals

A simplified rational number $\frac{p}{q}$ , where $p$ and $q$ are co-prime integers, has a terminating decimal expansion if and only if the prime factorisation of the denominator $q$ contains only 2 and 5 as prime factors (that is, of the form $2^m \times 5^n$ for non-negative integers $m$ and $n$ ). If any other prime factor is present in $q$ , the decimal expansion is non-terminating repeating.

Step 2: Analyze the fraction 7/20

The fraction $\frac{7}{20}$ is already in its simplest form since 7 and 20 share no common factors other than 1. Factorising the denominator, we get $20 = 2^2 \times 5^1$ , which contains only the prime factors 2 and 5. Therefore, $\frac{7}{20}$ has a terminating decimal expansion.

Step 3: Analyze the fraction 4/15

The fraction $\frac{4}{15}$ is in simplest form because the numerator 4 and denominator 15 are co-prime. The denominator factors as $15 = 3 \times 5$ . Because it contains the prime factor 3 (which is neither 2 nor 5), the fraction $\frac{4}{15}$ has a non-terminating repeating decimal expansion.

Step 4: Analyze the fraction 13/250

The fraction $\frac{13}{250}$ is in simplest form since 13 is a prime number and does not divide 250. Factorising the denominator gives $250 = 2 \times 125 = 2^1 \times 5^3$ . Since the only prime factors are 2 and 5, $\frac{13}{250}$ has a terminating decimal expansion.

Step 5: Check answers by conversion to decimals

We verify our findings by computing the exact decimal value of each fraction. For the first fraction, $\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 0.35$ (terminating). For the second, $\frac{4}{15} = 0.2666\dots = 0.2\bar{6}$ (repeating). For the third, $\frac{13}{250} = \frac{13 \times 4}{250 \times 4} = \frac{52}{1000} = 0.052$ (terminating). All predictions are confirmed.

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Exercise Set 3.5 1. Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 720,415 and 13250\frac{7}{20}, \frac{4}{15} \text{ and } \frac{13}{250}207​,154​ and 25013​. Then check your answers