Question

$\frac{1}{4\frac{2}{7}}+\frac{1}{3\frac{11}{13}}+\frac{1}{\frac{5}{9}}=$

• A. 312/45
• B. 513/13
• C. 218/35
• D. 172/75

Answer 1

Answer:

D. 172/75

Step 1: Convert the mixed fractions to improper fractions

• For the first term: $4\frac{2}{7} = \frac{(4 \times 7) + 2}{7} = \frac{30}{7}$

• For the second term: $3\frac{11}{13} = \frac{(3 \times 13) + 11}{13} = \frac{50}{13}$

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Step 2: Reciprocate the fractions

Since each term is under a $\frac{1}{x}$ format, we invert the fractions:

• $\frac{1}{\frac{30}{7}} = \frac{7}{30}$

• $\frac{1}{\frac{50}{13}} = \frac{13}{50}$

• $\frac{1}{\frac{5}{9}} = \frac{9}{5}$

Now substitute these back into the original expression:
$\frac{7}{30} + \frac{13}{50} + \frac{9}{5}$

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Step 3: Find the Common Denominator

The Least Common Multiple (LCM) of the denominators $30$, $50$, and $5$ is $150$.

Convert each fraction to have a denominator of $150$:

• $\frac{7 \times 5}{30 \times 5} = \frac{35}{150}$

• $\frac{13 \times 3}{50 \times 3} = \frac{39}{150}$

• $\frac{9 \times 30}{5 \times 30} = \frac{270}{150}$

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Step 4: Add the numerators together

$\frac{35 + 39 + 270}{150}$
$= \frac{344}{150}$

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Step 5: Check and re-verify calculations

Let's double-check the calculations carefully.

• First term: $1 / (30/7) = 7/30$

• Second term: $1 / (50/13) = 13/50$

• Third term: $1 / (5/9) = 9/5$

• Sum: $\frac{35}{150} + \frac{39}{150} + \frac{270}{150} = \frac{344}{150}$

• Simplifying $\frac{344}{150}$ by dividing numerator and denominator by 2 gives $\frac{172}{75}$.