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1. **Factory Machine Production**

- A machine produces 100 items in one hour. Answer the following questions, assuming that the machines work at the same tempo.

1.1. How many machines will you need to produce 500 items per hour?

1.2. How long will it take 4 machines to produce 100 items?

1.3.
- How many items do 8 machines produce in 45 minutes?
- If a rescue vessel has enough provisions for 10 people to survive for 6 days, how long will 12 people survive on the vessel?

2. **Motorist Travel Time**

- A motorist takes 4.5 hours to reach his destination at [tex]$80 \text{ km/h}$[/tex].

2.1. If he wants to complete his journey in 4 hours, at what speed must he travel?

2.2. If he drives at [tex]$110 \text{ km/h}$[/tex], how long will the journey be? Give your answer in hours and minutes.

[Total: 10 points]

Answer :

Sure! Let's go through the solution step-by-step for each part of the question.

Part 3.1.1: How many machines are needed to produce 500 items per hour?

- Each machine produces 100 items in one hour.
- To produce 500 items in one hour, you need to divide the total items needed by the number of items one machine can produce in an hour.
- Calculation: 500 items ÷ 100 items per machine per hour = 5 machines.

So, you need 5 machines.

Part 3.1.2: How long will it take 4 machines to produce 100 items?

- 4 machines each produce 100 items per hour.
- Together, these 4 machines produce 4 × 100 = 400 items in one hour.
- To find out how long it will take to produce 100 items, divide the total items needed by the total rate of production.
- Calculation: 100 items ÷ 400 items per hour = 0.25 hours.

It will take 0.25 hours (or 15 minutes).

Part 3.1.3: How many items do 8 machines produce in 45 minutes?

- 8 machines produce 8 × 100 = 800 items in a full hour.
- Since we want to find the number of items produced in 45 minutes, first convert 45 minutes into hours: 45 minutes ÷ 60 minutes/hour = 0.75 hours.
- Multiply the rate of production by the time: 800 items per hour × 0.75 hours = 600 items.

8 machines will produce 600 items in 45 minutes.

If a rescue vessel has enough provisions for 10 people to survive for 6 days, how long will 12 people survive on the vessel?

- The total provisions can be calculated as 10 people × 6 days = 60 person-days of provisions.
- To find how long 12 people will survive, divide the total person-days by the number of people.
- Calculation: 60 person-days ÷ 12 people = 5 days.

12 people will survive for 5 days.

Part 3.3: If the motorist wants to complete their journey in 4 hours, at what speed must they travel?

- The initial journey duration is 4.5 hours at 80 km/h.
- Distance traveled = Speed × Time = 80 km/h × 4.5 hours = 360 km.
- To complete this journey of 360 km in 4 hours, use the formula: Speed = Distance ÷ Time.
- Calculation: 360 km ÷ 4 hours = 90 km/h.

The motorist must travel at 90 km/h.

Part 3.3.2: If the motorist drives at 110 km/h, how long will the journey be?

- Use the journey distance calculated before, which is 360 km.
- Time = Distance ÷ Speed.
- Calculation: 360 km ÷ 110 km/h ≈ 3.27 hours.
- To convert 0.27 hours into minutes: 0.27 hours × 60 minutes/hour ≈ 16.36 minutes.

The journey will take approximately 3 hours and 16 minutes.

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