The third proportional between (a2-b2) and (a+b)2 is:

Option c: (a+b)3/a-b. Solution: Let the third proportional between a2-b2 and (a+b)2 be x . By definition of third proportional, we have: (a2-b2) : (a+b)2 = (a+b)2 : x This can be written in fractional form as: a2-b2/(a+b)2 = (a+b)2/x Multiplying both sides by x , we get: (a2-b2) × x = (a+b)2 × (a+b)2 (a2-b2) × x = (a+b)4 Now, solving for x : x = (a+b)4/a2-b2 Recall the algebraic identity: a2-b2 = (a-b)(a+b) . Substitute this into the denominator: x = (a+b)4/(a-b)(a+b) Cancel out one term of (a+b) from both the numerator and the denominator: x = (a+b)3/a-b This matches the expression in Option C. Note: The textbook answer key incorrectly specifies Option D as the correct choice. However, as mathematically proven above, the correct mathematical derivation leads to Option C. Hence, Option C is the correct answer.

Ratio, Proportion, Indices, LogarithmMCQPYQ May 18Question 847 of 305

All Questions

The third proportional between $\displaystyle (a^2-b^2)$ and $\displaystyle (a+b)^2$ is:

Options

\displaystyle \frac{a+b}{a-b}

\displaystyle \frac{a-b}{a+b}

\displaystyle \frac{(a+b)^3}{a-b}

\displaystyle \frac{(a-b)^2}{a+b}

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Correct Answer

✅ Option c — $\displaystyle \frac{(a+b)^3}{a-b}$

All Options:

A $\displaystyle \frac{a+b}{a-b}$
B $\displaystyle \frac{a-b}{a+b}$
C $\displaystyle \frac{(a+b)^3}{a-b}$
D $\displaystyle \frac{(a-b)^2}{a+b}$

Detailed Solution & Explanation

Let the third proportional between

\displaystyle a^2-b^2

and

\displaystyle (a+b)^2

\displaystyle x

.
By definition of third proportional, we have:

(a^2-b^2) : (a+b)^2 = (a+b)^2 : x

This can be written in fractional form as:

\frac{a^2-b^2}{(a+b)^2} = \frac{(a+b)^2}{x}

Multiplying both sides by

\displaystyle x

, we get:

(a^2-b^2) \cdot x = (a+b)^2 \cdot (a+b)^2

(a^2-b^2) \cdot x = (a+b)^4

Now, solving for

\displaystyle x

x = \frac{(a+b)^4}{a^2-b^2}

Recall the algebraic identity:

\displaystyle a^2-b^2 = (a-b)(a+b)

. Substitute this into the denominator:

x = \frac{(a+b)^4}{(a-b)(a+b)}

Cancel out one term of

\displaystyle (a+b)

from both the numerator and the denominator:

x = \frac{(a+b)^3}{a-b}

This matches the expression in **Option C**.
Note: The textbook answer key incorrectly specifies **Option D** as the correct choice. However, as mathematically proven above, the correct mathematical derivation leads to **Option C**.
Hence, **Option C** is the correct answer.

About This Chapter: Ratio, Proportion, Indices, Logarithm

Paper

Paper 3: Quantitative Aptitude

Weightage

5-7 Marks

Key Topics

Ratio, Proportion, Indices, Logarithms

This chapter covers Ratio, Proportion, Indices, Logarithms and is part of Paper 3: Quantitative Aptitude in the CA Foundation exam.

View Official ICAI Syllabus

Exam Strategy Tip

This topic carries 5-7 Marks weightage. Focus on understanding core concepts rather than memorizing.

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The third proportional between (a2−b2)\displaystyle (a^2-b^2)(a2−b2) and (a+b)2\displaystyle (a+b)^2(a+b)2 is: