If 4a + 7b/4c + 7d=4a - 7b/4c - 7d, then which one of the following is correct?

Correct Answer is : ab=cda/b=c/dba=dc

Correct Answer is : ab=cda/b=c/dba=dc

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UPSC CDS I 2019 Math Paper

Question : 38 of 100

Marks: +1, -0

\frac{4a + 7b}{4c + 7d}=\frac{4a - 7b}{4c - 7d}

, then which one of the following is correct?

Solution:

Concept:

Use the property of equal ratios and apply componendo and dividendo to simplify.

Explanation:

Start with the given equation:

\frac{4a + 7b}{4c + 7d} = \frac{4a - 7b}{4c - 7d}

Rearrange to group the terms:

\frac{4a + 7b}{4a - 7b} = \frac{4c + 7d}{4c - 7d}

Apply componendo and dividendo:

\frac{(4a+7b)+(4a-7b)}{(4a+7b)-(4a-7b)} = \frac{(4c+7d)+(4c-7d)}{(4c+7d)-(4c-7d)}

Simplify the numerators and denominators: left side becomes

\frac{8a}{14b}

, right side becomes

\frac{8c}{14d}

So,

\frac{8a}{14b} = \frac{8c}{14d}

Cancel the common factor

\frac{8}{14}

to obtain

\frac{a}{b} = \frac{c}{d}

Answer:

Option A:

\frac{a}{b} = \frac{c}{d}

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