High School

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A farmer ploughs his land in 12 days using 5 tractors. How long will it take if he uses only 3 tractors?

Let [tex]x[/tex] represent the number of days.

Calculation:
- Total tractor-days required: [tex]12 \times 5 = 60[/tex]
- Using 3 tractors: [tex]x = \frac{60}{3} = 20[/tex]

Therefore, it will take 20 days if he uses only 3 tractors.

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Karabo and John are at the same rest stop alongside a highway. Karabo starts driving along the highway at a constant speed of [tex]80 \text{ km/h}[/tex]. An hour later, John starts driving along the same highway in the same direction as Karabo at a constant speed of [tex]100 \text{ km/h}[/tex]. How long will it take John to catch up with Karabo?

Calculation:
- John will catch up with Karabo in 4 hours.

Answer :

Let's solve the given problem step by step.

Problem:
1. A farmer ploughs his land in 12 days using 5 tractors. How long will it take if he uses only 3 tractors?

2. Karabo and John are traveling along the highway. Karabo drives at 80 km/h and starts first. An hour later, John starts driving at 100 km/h in the same direction. How long will it take John to catch up with Karabo?

### Solution:

1. Farmer and Tractors:

- Initially, with 5 tractors, the farmer takes 12 days to plough the land.

- We can determine the total amount of work needed to plough the land in terms of "tractor-days," which is simply the number of tractors multiplied by the number of days they work:
[tex]\[
\text{Total work} = 5 \text{ tractors} \times 12 \text{ days} = 60 \text{ tractor-days}
\][/tex]

- If the farmer switches to using only 3 tractors, we want to find out how many days, denoted as [tex]\( x \)[/tex], it will take to complete the same amount of work:
[tex]\[
3 \text{ tractors} \times x \text{ days} = 60 \text{ tractor-days}
\][/tex]

- Solving for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{60 \text{ tractor-days}}{3 \text{ tractors}} = 20 \text{ days}
\][/tex]

So, it will take 20 days if the farmer uses 3 tractors.

2. Karabo and John on the Highway:

- Karabo’s movement:
- Speed = 80 km/h
- Starts at time = 0 hours

- John’s movement:
- Speed = 100 km/h
- Starts 1 hour later than Karabo

- When John starts driving, Karabo has already traveled:
[tex]\[
\text{Distance} = \text{Karabo's speed} \times \text{Time before John starts} = 80 \text{ km/h} \times 1 \text{ hour} = 80 \text{ km}
\][/tex]

- Relative speed (how fast John is catching up to Karabo):
[tex]\[
\text{Relative speed} = \text{John's speed} - \text{Karabo's speed} = 100 \text{ km/h} - 80 \text{ km/h} = 20 \text{ km/h}
\][/tex]

- Time for John to catch up:
[tex]\[
\text{Time to catch up} = \frac{\text{Initial distance between them}}{\text{Relative speed}} = \frac{80 \text{ km}}{20 \text{ km/h}} = 4 \text{ hours}
\][/tex]

- Total time from when Karabo starts until John catches up:
[tex]\[
= \text{1 hour (until John starts driving)} + \text{4 hours (to catch up)} = 5 \text{ hours}
\][/tex]

Thus, John will catch up to Karabo 5 hours after Karabo starts driving.

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