Question

Two people are climbing up two different moving escalators, each of which has 120 visible steps. The ratio of the first person's stepping speed to the speed of the first escalator is 2:3. The ratio of the second person's stepping speed to the speed of the second escalator is 3:5. Find the total number of steps the two people actually step on, added together.

• A. 85
• B. 93
• C. 80
• D. 75

Hint:

Steps actually climbed = total steps \( \times \frac{\text{person's speed}}{\text{person's speed} + \text{escalator's speed}}\). Apply this to both ratios and add the results.

Answer 1

The Correct Option is B

Step 1: Set up the idea behind escalator problems.
When a person walks up a moving escalator, their speed relative to the ground is the sum of their own stepping speed and the escalator's speed, since both act in the same upward direction.
The number of steps a person actually plants their feet on is not the full 120 visible steps; it is only the fraction of the escalator's length that corresponds to their own contribution to the total speed.
Steps actually climbed \( = 120 \times \dfrac{\text{person's speed}}{\text{person's speed} + \text{escalator's speed}} \).

Step 2: Apply this to the first person.
The ratio of person to escalator is \(2:3\), so if person's speed \(= 2k\) and escalator's speed \(= 3k\), the combined speed is \(5k\).
Fraction of steps climbed by the person \( = \dfrac{2k}{5k} = \dfrac{2}{5} \).
Steps climbed \( = 120 \times \dfrac{2}{5} = 48 \).

Step 3: Apply the same idea to the second person.
The ratio here is \(3:5\), so person's speed \(= 3m\) and escalator's speed \(= 5m\), giving a combined speed of \(8m\).
Fraction of steps climbed \( = \dfrac{3m}{8m} = \dfrac{3}{8}\).
Steps climbed \( = 120 \times \dfrac{3}{8} = 45 \).

Step 4: Add the two amounts.
\[ 48 + 45 = 93 \]

Step 5: Why the other options are wrong.
85, 80 and 75 would only appear if the ratios were applied directly to 120 without first finding person's speed as a fraction of the combined speed (a common error is to use \(\frac{2}{3}\) and \(\frac{3}{5}\) of 120 instead of \(\frac{2}{5}\) and \(\frac{3}{8}\)). Only 93 comes from the correct combined-speed fractions.

Final Answer:
Together, the two people step on 93 steps.
\[ \boxed{93} \]