While travelling from his home to the office in a car, if Amit increases his speed by 5km/hr, he will reach the office 40mins earlier than usual, and if he decreases his speed by 5km/hr, he will get the office 2hours late. Amit decides to walk to his office on a day, how much earlier should he leave his home if his walking speed is 40% of his car's speed

Let's break this problem down step by step.

### Step 1: Define Variables
- Let \( d \) represent the distance between Amit's home and office in kilometers.
- Let \( v \) represent the usual speed of Amit's car in kilometers per hour.
- Let \( t \) represent the usual time taken by Amit to reach the office in hours.

From the basic relation of distance, time, and speed:
\[
d = v \cdot t
\]

### Step 2: Formulate Equations
#### Case 1: Amit increases his speed by 5 km/h
When Amit increases his speed by 5 km/h, he reaches 40 minutes (or \(\frac{2}{3}\) hour) earlier:
\[
d = (v + 5)(t - \frac{2}{3})
\]
Substitute \( d = v \cdot t \) into this equation:
\[
v \cdot t = (v + 5)(t - \frac{2}{3})
\]

#### Case 2: Amit decreases his speed by 5 km/h
When Amit decreases his speed by 5 km/h, he reaches 2 hours later:
\[
d = (v - 5)(t + 2)
\]
Substitute \( d = v \cdot t \) again:
\[
v \cdot t = (v - 5)(t + 2)
\]

Now, we have two equations:
1. \( v \cdot t = (v + 5)(t - \frac{2}{3}) \)
2. \( v \cdot t = (v - 5)(t + 2) \)

### Step 3: Solve the System of Equations
We will solve these two equations to find \( v \) and \( t \).

#### First equation:
\[
v \cdot t = (v + 5)(t - \frac{2}{3})
\]
Expanding the right-hand side:
\[
v \cdot t = v \cdot t - \frac{2}{3}v + 5t - \frac{10}{3}
\]
Simplify:
\[
0 = -\frac{2}{3}v + 5t - \frac{10}{3}
\]
Multiply through by 3 to eliminate the fractions:
\[
0 = -2v + 15t - 10
\]
Rearrange:
\[
2v = 15t - 10 \quad \text{(Equation 1)}
\]

#### Second equation:
\[
v \cdot t = (v - 5)(t + 2)
\]
Expanding the right-hand side:
\[
v \cdot t = v \cdot t + 2v - 5t - 10
\]
Simplify:
\[
0 = 2v - 5t - 10
\]
Rearrange:
\[
2v = 5t + 10 \quad \text{(Equation 2)}
\]

### Step 4: Solve for \( v \) and \( t \)
Now, we have two equations:
1. \( 2v = 15t - 10 \)
2. \( 2v = 5t + 10 \)

Equate the right-hand sides of both equations:
\[
15t - 10 = 5t + 10
\]
Solve for \( t \):
\[
15t - 5t = 10 + 10
\]
\[
10t = 20
\]
\[
t = 2 \quad \text{(hours)}
\]

Substitute \( t = 2 \) into one of the original equations to find \( v \):
\[
2v = 5t + 10
\]
\[
2v = 5(2) + 10
\]
\[
2v = 20
\]
\[
v = 10 \quad \text{km/h}
\]

### Step 5: Amit's Walking Speed
Amit's walking speed is 40% of his car's speed, so:
\[
\text{Walking speed} = 0.4 \times 10 = 4 \quad \text{km/h}
\]

### Step 6: Time Taken by Walking
The distance \( d \) between home and office is:
\[
d = v \cdot t = 10 \cdot 2 = 20 \quad \text{km}
\]
The time taken by walking is:
\[
\text{Walking time} = \frac{d}{\text{Walking speed}} = \frac{20}{4} = 5 \quad \text{hours}
\]

### Step 7: How Much Earlier Amit Should Leave
If Amit usually takes 2 hours by car and takes 5 hours by walking, then he needs to leave:
\[
5 - 2 = 3 \quad \text{hours earlier}
\]

Thus, Amit should leave **3 hours earlier** if he decides to walk.

---

Would you like further details or explanations on any part?

Here are 5 questions to further explore:
1. What would be the result if Amit's walking speed was 50% of his car's speed?
2. How would the solution change if Amit increases his speed by 10 km/h instead?
3. What other methods could be used to solve this type of distance-time-speed problem?
4. How would delays in traffic affect Amit's travel time if he walks?
5. How can this problem be modeled graphically?

**Tip:** Always express distances and speeds in consistent units when solving distance-time-speed problems.

While travelling from his home to the office in a car, if Amit increases his speed by 5km/hr, he will reach the office 40mins earlier than usual, and if he decreases his speed by 5km/hr, he will get the office 2hours late. Amit decides to walk to his office on a day, how much earlier should he leave his home if his walking speed is 40% of his car's speed?

Solve a real-world distance-speed-time problem where Amit's car speed changes impact his travel time. Learn how to calculate walking time if his walking speed is 40% of his car's speed. Suitable for Grades 9-12.

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