CAT 2018Slot 1QAQuestion & Solution
Question
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?
Options
$\frac{1}{5}$
$\frac{6}{19}$
$\frac{1}{4}$
$\frac{7}{33}$
Solution
1. Concept Used
- Topic: Ratio and Proportion — Working with changing ratios after transfer
- Formula: $$ \frac{4x + a}{9x - a} = \frac{5}{6} $$
2. Calculation
Let the original number of marbles with Raju and Lalitha be (4x) and (9x) respectively, consistent with the given ratio (4:9). Let (a) be the number of marbles Lalitha transfers to Raju.
After the transfer, Raju has (4x + a) marbles and Lalitha has (9x - a) marbles. The new ratio is given as (5:6), so we set up the equation:
$$ \frac{4x + a}{9x - a} = \frac{5}{6} $$
Cross-multiplying both sides:
$$ 6(4x + a) = 5(9x - a) $$
$$ 24x + 6a = 45x - 5a $$
Bringing like terms together:
$$ 6a + 5a = 45x - 24x $$
$$ 11a = 21x $$
$$ a = \frac{21x}{11} $$
Now, the fraction of Lalitha's original marbles that she gave away is:
$$ \frac{a}{9x} = \frac{\frac{21x}{11}}{9x} = \frac{21x}{11 \times 9x} = \frac{21}{99} = \frac{7}{33} $$
3. Solution
Answer = Option D ✅
The fraction of Lalitha's original marbles given to Raju is (\dfrac{7}{33}).