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: Assign concrete speeds instead of using letters.
For the first escalator, pick numbers in the ratio $2:3$: let the person's stepping speed be $2$ steps per second and the escalator's speed be $3$ steps per second. Their speeds add up on the way up, since escalator and person both move upward together.

Step 2: Find how long the first person takes to reach the top.
The escalator's visible length is 120 steps, and ground speed is $2 + 3 = 5$ steps per second, so time taken $= \frac{120}{5} = 24$ seconds.
In those 24 seconds, the person's own feet move at 2 steps per second, so steps actually stepped on $= 2 \times 24 = 48$.

Step 3: Repeat for the second person with the $3:5$ ratio.
Let the person's speed be 3 steps per second and the escalator's speed be 5 steps per second, so ground speed $= 3 + 5 = 8$ steps per second.
Time taken $= \frac{120}{8} = 15$ seconds.
Steps stepped on $= 3 \times 15 = 45$.

Step 4: Add the steps from both people.
$48 + 45 = 93$.

Step 5: Sanity check with different concrete numbers.
Using speeds of 4 and 6 steps per second (still ratio $2:3$) for the first escalator gives time $= 120/10 = 12$ seconds and steps $= 4 \times 12 = 48$, the same answer as before, confirming the actual chosen numbers do not matter, only the ratio.

Final Answer:
The two people together step on 93 steps.
\[ \boxed{93} \]