Question

2-13. which of the fractions listed below will have repeating decimals and which will have terminating decimals? explain how you know. a. $\frac{2}{5}$ b. $\frac{7}{45}$ c. $\frac{5}{99}$ d. $\frac{27}{250}$ 2-14. here are the lengths (in inches) of snakes in a reptile display at the zoo: 10, 31, 36, 36, 38, 42, 47, 48, 49

Explanation:

Step1: Recall fraction - decimal conversion rule

A fraction $\frac{a}{b}$ (where $a,b$ are integers, $b
eq0$) has a terminating decimal if the denominator $b = 2^m\times5^n$, where $m$ and $n$ are non - negative integers. If the denominator has prime factors other than 2 and 5, it has a repeating decimal.

Step2: Analyze option a

For $\frac{2}{5}$, the denominator $b = 5=5^1$. Since the denominator is of the form $2^m\times5^n$ (here $m = 0,n = 1$), it has a terminating decimal. $\frac{2}{5}=0.4$.

Step3: Analyze option b

For $\frac{7}{45}$, factorize the denominator $45=3^2\times5$. Since it has a prime factor of 3 other than 2 and 5, it has a repeating decimal.

Step4: Analyze option c

For $\frac{5}{99}$, factorize the denominator $99 = 3^2\times11$. Since it has prime factors of 3 and 11 other than 2 and 5, it has a repeating decimal.

Step5: Analyze option d

For $\frac{27}{250}$, factorize the denominator $250=2\times5^3$. Since it is of the form $2^m\times5^n$ (here $m = 1,n = 3$), it has a terminating decimal. $\frac{27}{250}=0.108$.

Answer:

Fractions with repeating decimals: b. $\frac{7}{45}$, c. $\frac{5}{99}$
Fractions with terminating decimals: a. $\frac{2}{5}$, d. $\frac{27}{250}$