Question

Find the reciprocal of the sum of the reciprocals of the given numbers.
$\frac{2}{5}, \frac{5}{4}, \frac{6}{8}, \frac{6}{5}, \frac{9}{6}$

• A. 82/15
• B. 62/15
• C. 15/92
• D. 72/15

Answer 1

Answer:

C. 15/92

The final value is $\frac{15}{92}$.

1: Find the reciprocal of each given number

To find the reciprocal of a fraction, simply flip the numerator and denominator:

• Reciprocal of $\frac{2}{5} = \frac{5}{2}$

• Reciprocal of $\frac{5}{4} = \frac{4}{5}$

• Reciprocal of $\frac{6}{8}$ (which is $\frac{3}{4}$) $= \frac{4}{3}$

• Reciprocal of $\frac{6}{5} = \frac{5}{6}$

• Reciprocal of $\frac{9}{6}$ (which is $\frac{3}{2}$) $= \frac{2}{3}$

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2: Find the sum of these reciprocals

$\text{Sum} = \frac{5}{2} + \frac{4}{5} + \frac{4}{3} + \frac{5}{6} + \frac{2}{3}$

The Least Common Multiple (LCM) of the denominators (2, 5, 3, 6) is 30. Convert each fraction:

• $\frac{5}{2} = \frac{75}{30}$

• $\frac{4}{5} = \frac{24}{30}$

• $\frac{4}{3} = \frac{40}{30}$

• $\frac{5}{6} = \frac{25}{30}$

• $\frac{2}{3} = \frac{20}{30}$

Now add the numerators together:
$\text{Sum} = \frac{75 + 24 + 40 + 25 + 20}{30} = \frac{184}{30}$

Simplify the fraction by dividing by 2:
$\text{Sum} = \frac{92}{15}$

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3: Take the reciprocal of the sum

The question asks for the reciprocal of this total sum:
$\text{Final Answer} = \frac{15}{92}$