P, Q and R are three contestants in a 2 km race. If P can give Q a start of 100 m and P can give R a start of 138 m ,then how many metres start can Q give to R?

Options

A. (A.) 40m
B. (B.) 45m
C. (C.) 50m
D. (D.) 55m

Answer

To solve the problem step by step, let's analyze the information given and derive the answer systematically.

### Step 1: Understand the Problem
We have three contestants: P, Q, and R in a 2 km (2000 meters) race. We know:
- P can give Q a start of 100 meters.
- P can give R a start of 138 meters.

We need to find out how many meters start Q can give to R.

### Step 2: Calculate Distances for Q and R when P runs 2000 meters
1. When P runs 2000 meters, Q runs:
\[ \text{Distance Q runs} = 2000 - 100 = 1900 \text{ meters} \]
2. When P runs 2000 meters, R runs:
\[ \text{Distance R runs} = 2000 - 138 = 1862 \text{ meters} \]

### Step 3: Calculate the Ratio of Distances
Now, we need to find the ratio of the distances that Q and R run when P runs 2000 meters:
- Q runs 1900 meters.
- R runs 1862 meters.

### Step 4: Determine the Difference in Distances
To find out how much start Q can give R, we calculate the difference in the distances they run:
\[ \text{Difference} = 1900 - 1862 = 38 \text{ meters} \]
This means when Q runs 1900 meters, R runs 1862 meters, giving R a start of 38 meters.

### Step 5: Calculate the Start Q can give R in a 2000 Meter Race
Now, we need to find out how much start Q can give R in a full 2000 meter race. We set up a proportion based on the distances:
\[ \frac{\text{Start given by Q to R}}{\text{Distance Q runs}} = \frac{38 \text{ meters}}{1900 \text{ meters}} \]
Let \( x \) be the start that Q can give R in a 2000 meter race:
\[ \frac{x}{2000} = \frac{38}{1900} \]

### Step 6: Solve for x
Cross-multiplying gives:
\[ 1900x = 38 \times 2000 \]
Calculating the right side:
\[ 1900x = 76000 \]
Now, divide both sides by 1900:
\[ x = \frac{76000}{1900} = 40 \text{ meters} \]

### Final Answer
Thus, Q can give R a start of **40 meters** in a 2000 meter race.

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