Question

Given below are two statements :
Statement I: The ratio of the son's age to the father's age is 1 : 5. The product of their ages is 245. The ratio of the son's age to the father's age after 9 years will be 3 : 5.
Statement II: A milkman adds 30 litres of water to 70 litres of milk. After selling 1/5th of the total quantity, he adds water equal to the quantity he sold. The proportion of water to milk he sells now would be 28 : 72. In the light of the above statements, choose the correct answer from the options given below.

• A. Both Statement I and Statement II are true
• B. Both Statement I and Statement II are false
• C. Statement I is true but Statement II is false
• D. Statement I is false but Statement II is true

Answer 1

The Correct Option is B

Let's evaluate both Statement I and Statement II to determine their validity step-by-step:

Evaluating Statement I:

1. **Given Information:** - The ratio of the son's age to the father's age is \(1:5\). - The product of their ages is 245. - The ratio of the son's age to the father's age after 9 years will be \(3:5\). 2. **Let the son's age be \(x\) and the father's age be \(5x\):** - \(x \times 5x = 245\) - Simplifying, we have \(5x^2 = 245\). - Therefore, \(x^2 = 49\). - Solving, \(x = 7\). 3. **Current Ages:** - Son's age = \(7\). - Father's age = \(5 \times 7 = 35\). 4. **Calculating ages after 9 years:** - Son's age = \(7 + 9 = 16\). - Father's age = \(35 + 9 = 44\). 5. **Ratio after 9 years:** - The ratio of son's age to father's age = \(\frac{16}{44} = \frac{4}{11}\), not \(3:5\). 6. **Conclusion:** - The ratio provided in the statement after 9 years is incorrect. - Hence, Statement I is false.

Evaluating Statement II:

1. **Initial Mixture Composition:** - Water: \(30\) liters - Milk: \(70\) liters - Total = \(30 + 70 = 100\) liters 2. **After selling 1/5th of the mixture:** - Sold quantity = \( \frac{1}{5} \times 100 = 20\) liters - New quantities after selling: - Water: \(30 - \frac{30}{100} \times 20 = 24\) liters - Milk: \(70 - \frac{70}{100} \times 20 = 56\) liters 3. **Adding water equal to quantity sold:** - Added water = \(20\) liters - New water quantity = \(24 + 20 = 44\) liters - Milk remains \(56\) liters 4. **New ratio of water to milk:** - Water to milk ratio = \(\frac{44}{56} = \frac{11}{14}\). 5. **Conclusion:** - The given proportion after adjustments (28:72) is incorrect; actual ratio is not equivalent to \(28:72 = 7:18\). - Hence, Statement II is false. **Final Conclusion: Both Statement I and Statement II are false.**