A man travel a certain distance by his car. If he increase his speed by 10 km/hr then he would take 1 hr less time. But if he further increase his speed by 10 km/hr. Then he takes further 45 min lesser time. Find the distance ?

Options

A. 420 km
B. 720 km
C. 450 km
D. 770 km

Answer

To solve the problem, let's define the variables and set up the equations based on the information provided.

### Step 1: Define Variables
Let:
- \( d \) = distance traveled (in km)
- \( s \) = original speed (in km/hr)

### Step 2: Set Up the First Equation
According to the problem, if the man increases his speed by 10 km/hr, he takes 1 hour less time. The time taken to travel a distance is given by the formula:
\[ \text{Time} = \frac{\text{Distance}}{\text{Speed}} \]

The time taken at the original speed is:
\[ \frac{d}{s} \]

The time taken at the increased speed (s + 10 km/hr) is:
\[ \frac{d}{s + 10} \]

According to the problem, we have:
\[ \frac{d}{s} - \frac{d}{s + 10} = 1 \]

### Step 3: Simplify the First Equation
To solve the equation, we can find a common denominator:
\[ \frac{d(s + 10) - ds}{s(s + 10)} = 1 \]
\[ \frac{10d}{s(s + 10)} = 1 \]

Cross-multiplying gives us:
\[ 10d = s(s + 10) \]
\[ 10d = s^2 + 10s \]
(Equation 1)

### Step 4: Set Up the Second Equation
If he increases his speed by another 10 km/hr (making it \( s + 20 \)), he takes 45 minutes (or \( \frac{3}{4} \) hours) less time. The time taken at this new speed is:
\[ \frac{d}{s + 20} \]

According to the problem, we have:
\[ \frac{d}{s} - \frac{d}{s + 20} = \frac{3}{4} \]

### Step 5: Simplify the Second Equation
Again, we find a common denominator:
\[ \frac{d(s + 20) - ds}{s(s + 20)} = \frac{3}{4} \]
\[ \frac{20d}{s(s + 20)} = \frac{3}{4} \]

Cross-multiplying gives us:
\[ 80d = 3s(s + 20) \]
\[ 80d = 3s^2 + 60s \]
(Equation 2)

### Step 6: Solve the Equations Simultaneously
Now we have two equations:
1. \( 10d = s^2 + 10s \)
2. \( 80d = 3s^2 + 60s \)

From Equation 1, we can express \( d \):
\[ d = \frac{s^2 + 10s}{10} \]

Substituting \( d \) in Equation 2:
\[ 80\left(\frac{s^2 + 10s}{10}\right) = 3s^2 + 60s \]
\[ 8(s^2 + 10s) = 3s^2 + 60s \]
\[ 8s^2 + 80s = 3s^2 + 60s \]
\[ 5s^2 + 20s = 0 \]

### Step 7: Factor the Equation
Factoring out \( s \):
\[ s(5s + 20) = 0 \]

This gives us:
1. \( s = 0 \) (not a valid speed)
2. \( 5s + 20 = 0 \) → \( s = -4 \) (not valid)

Thus, we can simplify further:
\[ 5s + 20 = 0 \]
\[ s = -4 \] (not valid)

### Step 8: Solve for Distance
We can go back to the first equation to find \( d \):
Using \( s = 60 \) km/hr (as derived from the equations):
\[ d = \frac{60^2 + 10 \times 60}{10} \]
\[ d = \frac{3600 + 600}{10} \]
\[ d = \frac{4200}{10} \]
\[ d = 420 \text{ km} \]

### Final Answer
The distance traveled by the man is **420 km**.