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Out of the following rational numbers [tex]$\frac{3}{5}$[/tex], [tex]$\frac{3}{4}$[/tex], [tex][tex]$\frac{14}{5}$[/tex][/tex], and [tex]$\frac{2}{3}$[/tex], the fraction that cannot be expressed as a recurring decimal is:

a) [tex]$\frac{2}{3}$[/tex]

b) [tex][tex]$\frac{3}{5}$[/tex][/tex]

c) [tex]$\frac{3}{4}$[/tex]

d) [tex]$\frac{14}{5}$[/tex]

Sagot :

GinnyAnswer
Let's examine each of the given rational numbers to determine which one cannot be expressed as a recurring decimal.

### Step-by-Step Solution:

1. Fraction [tex]\( \frac{3}{5} \)[/tex]:

- The numerator is 3, and the denominator is 5.
- A fraction can be expressed as a terminating decimal if the denominator in its simplest form has only the prime factors 2 and/or 5.
- The denominator 5 is already a prime number and it is a factor of only 5, which is allowed.
- Therefore, [tex]\( \frac{3}{5} \)[/tex] is a terminating decimal.

2. Fraction [tex]\( \frac{3}{4} \)[/tex]:

- The numerator is 3, and the denominator is 4.
- The prime factorization of 4 is [tex]\( 2^2 \)[/tex], which is a factor of only 2.
- Since the denominator has only prime factors of 2, [tex]\( \frac{3}{4} \)[/tex] is a terminating decimal.

3. Fraction [tex]\( \frac{14}{5} \)[/tex]:

- The numerator is 14, and the denominator is 5.
- The denominator 5 is already a prime number and it is a factor of only 5, which is allowed.
- Therefore, [tex]\( \frac{14}{5} \)[/tex] is a terminating decimal.

4. Fraction [tex]\( \frac{2}{3} \)[/tex]:

- The numerator is 2, and the denominator is 3.
- The prime factorization of 3 is simply 3, which is not 2 or 5.
- Since the denominator has a prime factor that is not 2 or 5, [tex]\( \frac{2}{3} \)[/tex] is a recurring decimal.

### Conclusion:

Among the given fractions [tex]\( \frac{3}{5}, \frac{3}{4}, \frac{14}{5} \)[/tex], and [tex]\( \frac{2}{3} \)[/tex], the fraction that cannot be expressed as a recurring decimal is:

[tex]\[ \frac{3}{5}, \frac{3}{4}, \][/tex] and [tex]\[ \frac{14}{5} \][/tex] are terminating decimals, whereas [tex]\[ \frac{2}{3} \][/tex] is a recurring decimal.

However, since the question is specifically asking for the fraction that can be expressed as a terminating decimal:
- It is clear that this is actually [tex]\( \frac{3}{5} \)[/tex] among these options.

Hence, the fraction that cannot be expressed as a recurring decimal is [tex]\( \frac{3}{5} \)[/tex].

- The answer is [tex]\( \boxed{1} \)[/tex] as per the choices (a), (b), (c), (d), where [tex]\( \frac{3}{5} \)[/tex] is the correct non-recurring decimal.

Therefore:

The correct choice is:

b) [tex]\( \frac{3}{5} \)[/tex]
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