Two men A and B travel from point P to Q, a distance of 84 km at 12 km/h and 16 km/h, respectively. B reachesQand returns immediately and meets AatR.Find the distance fromPtoR

Options

A. a. 72 km
B. b. 76 km
C. c. 78 km
D. d. 68 km

Answer

To solve the problem step by step, we need to analyze the situation involving two men, A and B, traveling from point P to Q and back.

### Step 1: Determine the time taken by B to reach Q
B travels from P to Q, which is a distance of 84 km at a speed of 16 km/h.

**Time taken by B = Distance / Speed**
\[ \text{Time taken by B} = \frac{84 \text{ km}}{16 \text{ km/h}} = 5.25 \text{ hours} \quad \text{(or } 5 \frac{1}{4} \text{ hours)} \]

**Hint:** Use the formula for time, which is distance divided by speed.

### Step 2: Determine the distance traveled by A in the same time
A travels at a speed of 12 km/h. We need to find out how far A has traveled in the same time that B takes to reach Q.

**Distance traveled by A = Speed × Time**
\[ \text{Distance traveled by A} = 12 \text{ km/h} \times 5.25 \text{ hours} = 63 \text{ km} \]

**Hint:** Remember that distance can be calculated by multiplying speed by time.

### Step 3: Calculate the distance between A and B when B reaches Q
When B reaches Q, A has traveled 63 km. The remaining distance from Q to A is:
\[ \text{Distance between A and Q} = 84 \text{ km} - 63 \text{ km} = 21 \text{ km} \]

**Hint:** Subtract the distance A has traveled from the total distance to find the remaining distance.

### Step 4: Determine the time taken by B to return to meet A
After reaching Q, B immediately returns towards A. The distance B needs to cover to meet A is 21 km. The combined speed of A and B when they are moving towards each other is:
\[ \text{Combined speed} = 12 \text{ km/h} + 16 \text{ km/h} = 28 \text{ km/h} \]

**Time taken to meet = Distance / Combined Speed**
\[ \text{Time taken to meet} = \frac{21 \text{ km}}{28 \text{ km/h}} = \frac{3}{4} \text{ hours} \]

**Hint:** When two objects move towards each other, their speeds add up.

### Step 5: Calculate the distance A travels in that time
Now, we need to find out how far A travels in the time it takes for B to return and meet him.

**Distance traveled by A in \(\frac{3}{4}\) hours:**
\[ \text{Distance} = 12 \text{ km/h} \times \frac{3}{4} \text{ hours} = 9 \text{ km} \]

**Hint:** Again, use the distance formula (Distance = Speed × Time).

### Step 6: Calculate the total distance from P to R
The total distance from P to R is the distance A traveled towards Q plus the distance A traveled while B was returning.

\[ \text{Distance from P to R} = 63 \text{ km} + 9 \text{ km} = 72 \text{ km} \]

**Hint:** Add the distances A traveled before and after B reached Q.

### Conclusion
The distance from P to R is **72 km**.

**Final Answer:** 72 km