Question

In a race \(A\), \(B\) and \(C\) take part. \(A\) beats \(B\) by \(30\) meters, \(B\) beats \(C\) by \(20\) meters and \(A\) beats \(C\) by \(48\) meters.
Given below are two statements:
Statement I: The length of the race is \(300\) meters.
Statement II: The speed of \(A\), \(B\) and \(C\) are in the ratio \(50:45:40\).
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

Hint:

In such race problems, always remember that the ratio of distances covered in the same time is equal to the ratio of speeds. If \(L\) is the length, then \(V_A:V_B = L:L-d_1\) and \(V_B:V_C = L:L-d_2\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When \(A\) beats \(B\) by \(x\) meters in a race of length \(L\), it means when \(A\) covers \(L\), \(B\) covers \((L-x)\). The ratio of their speeds is \(\frac{V_A}{V_B} = \frac{L}{L-x}\).
Step 2: Key Formula or Approach:
Use the relationship \(\frac{V_A}{V_C} = \frac{V_A}{V_B} \times \frac{V_B}{V_C}\).
Step 3: Detailed Explanation:
Let the race length be \(L\).
1. \(A\) beats \(B\) by \(30\)m: \(\frac{V_A}{V_B} = \frac{L}{L-30}\).
2. \(B\) beats \(C\) by \(20\)m: \(\frac{V_B}{V_C} = \frac{L}{L-20}\).
3. \(A\) beats \(C\) by \(48\)m: \(\frac{V_A}{V_C} = \frac{L}{L-48}\).
Combining these:
\[ \frac{L}{L-48} = \frac{L}{L-30} \times \frac{L}{L-20} \implies \frac{1}{L-48} = \frac{L}{(L-30)(L-20)} \]
\[ (L-30)(L-20) = L(L-48) \]
\[ L^2 - 50L + 600 = L^2 - 48L \]
\[ 2L = 600 \implies L = 300 \text{ meters.} \]
So, Statement I is true.
4. Checking speed ratios:
Speed of \(A = V_A = \text{constant} \times 300\).
Speed of \(B = V_B = \text{constant} \times (300-30) = 270\).
Speed of \(C\): Since \(\frac{V_B}{V_C} = \frac{300}{280}\), then \(V_C = V_B \times \frac{280}{300} = 270 \times \frac{28}{30} = 9 \times 28 = 252\).
Ratio \(V_A : V_B : V_C = 300 : 270 : 252\).
Dividing by \(6\): \(50 : 45 : 42\).
Since the ratio in Statement II is \(50:45:40\), Statement II is false.
Step 4: Final Answer:
Statement I is true, but Statement II is false.