Question

Two cars A and B start from the same point and travel the same distance. The ratio of their speeds is 3: 4. Car A takes 30 minutes more than car B to complete the journey. If car B takes 't' minutes, then what is the value of 't' (in minutes)?

• A. (60)
• B. (90)
• C. (120)
• D. (150)

Hint:

If speed ratio is (a:b), the corresponding travel time ratio is flipped to (b:a). The units of difference between these ratios is (|b - a|). Here, the difference between parts is (4 - 3 = 1 part). Since 1 part represents 30 minutes, Car B's time ((3 parts)) is simply calculated as: (3 30 = 90 minutes). This mental calculation eliminates algebraic setup steps completely!

Answer 1

The Correct Option is B

Concept: The motion of objects traveling over spaces is governed by the core relation: \[ \text{Distance} = \text{Speed} \times \text{Time} \] When multiple moving objects traverse the exact same total distance ((D = constant)), their speed and elapsed travel time become inversely proportional to each other. That means: \[ \text{Speed} \propto \frac{1}{\text{Time}} \quad \Rightarrow \quad \frac{S_A}{S_B} = \frac{T_B}{T_A} \]

Step 1: Construct the time equations using the inverse proportionality principle. We are given the ratio of the traveling speeds of Car A and Car B: \[ S_A : S_B = 3 : 4 \quad \Rightarrow \quad \frac{S_A}{S_B} = \frac{3}{4} \] Since the travel distance is identical for both entities, the inverse time relationship holds: \[ \frac{T_B}{T_A} = \frac{S_A}{S_B} = \frac{3}{4} \] This can be rewritten linearly to model their travel times via a common proportionality constant factor, say (x): \[ T_B = 3x \] \[ T_A = 4x \]

Step 2: Substitute the specific variables given inside the text. The question establishes that Car B takes exactly (t) minutes to complete the tracking journey. \[ T_B = t = 3x \quad \Rightarrow \quad x = \frac{t}{3} \] Therefore, the travel time for Car A can be expressed dynamically in terms of (t): \[ T_A = 4x = 4\left(\frac{t}{3}\right) = \frac{4t}{3} \]

Step 3: Introduce the absolute time difference given in the problem statement. Car A takes 30 minutes longer than Car B to finalize the course. Therefore: \[ T_A - T_B = 30 \text{ minutes} \] Substitute the expressions for (T_A) and (T_B) in terms of (t) into this relationship: \[ \frac{4t}{3} - t = 30 \]

Step 4: Perform linear isolation to evaluate the target parameter (t). Find a common denominator on the left-hand side of the linear equation: \[ \frac{4t - 3t}{3} = 30 \] \[ \frac{t}{3} = 30 \] Multiply both sides of the expression by 3 to compute the final answer: \[ t = 30 \times 3 = 90 \text{ minutes} \] Thus, the value of (t) is 90 minutes, which perfectly matches Option (B).