Shyama and Vyom walk up an escalator (moving stairway). The escalator moves at a constant speed. Shyama takes three steps for every two of Vyom's steps. Shyama gets to the top of the escalator after having taken 25 steps, while Vyom (because his slower pace lets the escalator do a little more of the work) takes only 20 steps to reach the top. If the escalator were turned off, how many steps would they have to take to walk up?

Options

A. 40
B. 50
C. 60
D. 80

Answer

To solve the problem step by step, we can break it down as follows:

### Step 1: Define Variables
Let \( L \) be the total number of steps on the escalator. Shyama takes 25 steps, and Vyom takes 20 steps.

### Step 2: Relate Shyama's and Vyom's Steps
We know that Shyama takes 3 steps for every 2 steps of Vyom. Therefore, the ratio of their steps can be expressed as:
\[ \frac{S}{V} = \frac{3}{2} \]
Where \( S \) is the number of steps taken by Shyama and \( V \) is the number of steps taken by Vyom.

### Step 3: Calculate Vyom's Equivalent Steps
Since Shyama takes 25 steps, we can find the equivalent number of steps Vyom would take using the ratio:
\[ V = \frac{2}{3} \times S = \frac{2}{3} \times 25 = \frac{50}{3} \text{ steps} \]

### Step 4: Calculate Total Steps for Vyom
The total distance covered by Vyom to reach the top of the escalator is:
\[ V + E = 20 + L \]
Where \( E \) is the distance covered by the escalator.

### Step 5: Relate the Steps of Vyom and the Escalator
The escalator moves a certain number of steps while Vyom walks. The distance covered by the escalator when Vyom takes 20 steps can be expressed as:
\[ E = \frac{L}{20} \times 20 = L \]

### Step 6: Set Up the Equation for Shyama
For Shyama, the total distance covered is:
\[ S + E = 25 + E \]
And we know from Vyom's equation that:
\[ E = \frac{5L}{6} \]

### Step 7: Equate the Two Expressions
Now we can equate the two expressions for the escalator:
\[ 25 + \frac{5L}{6} = 20 + L \]

### Step 8: Simplify the Equation
Rearranging gives:
\[ 25 - 20 = L - \frac{5L}{6} \]
This simplifies to:
\[ 5 = \frac{1L}{6} \]

### Step 9: Solve for L
Multiplying both sides by 6 gives:
\[ L = 30 \]

### Step 10: Calculate Total Steps Without the Escalator
If the escalator were turned off, the total number of steps they would have to take to walk up would be:
\[ L + 20 = 30 + 20 = 50 \]

Thus, the total number of steps they would have to take to walk up the escalator if it were turned off is **50 steps**.

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