Arithmetic

Ratio, Proportion & Variation

Medium

Raju and Lalitha originally had marbles in the ratio $4:9$ . Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became $5:6$ . What fraction of her original number of marbles was given by Lalitha to Raju?

A) $6/19$

B) $7/33$

C) $1/4$

D) $1/5$

✅ Correct Option: 2

Raju and Lalitha start with marbles in the ratio 4:9. After Lalitha gives some marbles to Raju, their new ratio becomes 5:6. We need to find what fraction of her original marbles Lalitha gave away.

Since the initial ratio is 4:9, we can write:

Raju's initial marbles = 4x

Lalitha's initial marbles = 9x

When we have a ratio like 4:9, we use a common factor 'x' to represent the actual quantities. This ensures the ratio remains 4:9 regardless of the actual numbers.

Let's say Lalitha gives y marbles to Raju.

After the transfer:

Raju's marbles = 4x + y (he receives y marbles)

Lalitha's marbles = 9x - y (she gives away y marbles)

The new ratio is 5:6, so:

$\dfrac{4x + y}{9x - y} = \dfrac{5}{6}$

Cross-multiplying:

$6(4x + y) = 5(9x - y)$

$24x + 6y = 45x - 5y$

This eliminates fractions and gives us a linear equation that's easier to solve.

Moving all terms with y to one side and all terms with x to the other:

$6y + 5y = 45x - 24x$

$11y = 21x$

Therefore: $y = \dfrac{21x}{11}$

The fraction of original marbles that Lalitha gave to Raju is:

$\dfrac{y}{9x} = \dfrac{\frac{21x}{11}}{9x}$

$\frac{21x}{11} \times \dfrac{1}{9x} = \dfrac{21x}{11 \times 9x} = \dfrac{21}{99}$

Reducing to lowest terms:

$\dfrac{21}{99} = \dfrac{7}{33}$

Answer: $\dfrac{7}{33}$

When solving ratio problems with transfers, always:

Use variables that maintain the original ratio

Account for what each person gains/loses

Set up the new ratio equation

Verify your answer with concrete numbers

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CAT Past Year Questions (Topic-Wise)

Raju and Lalitha originally had marbles in the ratio $4:9$ . Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became $5:6$ . What fraction of her original number of marbles was given by Lalitha to Raju?