Answer :
Sure, let's break down each part of the question step-by-step:
### 3.1.1 How many machines will you need to produce 500 items per hour?
1. One machine produces 100 items in one hour.
2. To find out how many machines are needed to produce 500 items in one hour, divide 500 by 100.
3. [tex]\(\frac{500}{100} = 5\)[/tex]
You will need 5 machines to produce 500 items per hour.
### 3.1.2 How long will it take 4 machines to produce 100 items?
1. One machine produces 100 items in one hour.
2. Four machines working together will produce 400 items in one hour (4 machines × 100 items).
3. To produce 100 items with 4 machines: [tex]\( \frac{100}{400} = 0.25 \)[/tex] hours.
4. Converting 0.25 hours into minutes: [tex]\(0.25 \times 60 = 15\)[/tex] minutes.
It will take 4 machines 15 minutes to produce 100 items.
### 3.1.3 How many items do 8 machines produce in 45 minutes?
1. One machine produces 100 items in one hour, or [tex]\(\frac{100}{60} \approx 1.67\)[/tex] items per minute.
2. Therefore, 8 machines produce [tex]\(8 \times 1.67 = 13.36\)[/tex] items per minute.
3. In 45 minutes, 8 machines will produce [tex]\(13.36 \times 45 = 600\)[/tex] items.
8 machines produce 600 items in 45 minutes.
### 3.2 If a rescue vessel has enough provisions for 10 people to survive for 6 days, how long will 12 people survive on the vessel?
1. Original total provisions amount to [tex]\(10 \times 6 = 60\)[/tex] people-days.
2. Distributing these provisions among 12 people: [tex]\(\frac{60}{12} = 5\)[/tex] days.
12 people will survive for 5 days on the vessel.
### 3.3.1 If a motorist wants to complete his journey in 4 hours, at what speed must he travel?
1. The distance covered at 80 km/h in 4.5 hours is [tex]\(80 \times 4.5 = 360\)[/tex] km.
2. To complete the same journey in 4 hours, the speed required is [tex]\(\frac{360}{4} = 90\)[/tex] km/h.
The motorist must travel at 90 km/h to complete the journey in 4 hours.
### 3.3.2 If he drives at 110 km/h, how long will the journey be? Provide the answer in hours and minutes.
1. Using the previously calculated distance of 360 km, if driving at 110 km/h, the time taken would be [tex]\(\frac{360}{110} \approx 3.27\)[/tex] hours.
2. Converting the decimal into hours and minutes:
- [tex]\(3.27\)[/tex] hours equates to 3 hours and [tex]\(0.27 \times 60 \approx 16\)[/tex] minutes.
The journey will take approximately 3 hours and 16 minutes.
### 3.1.1 How many machines will you need to produce 500 items per hour?
1. One machine produces 100 items in one hour.
2. To find out how many machines are needed to produce 500 items in one hour, divide 500 by 100.
3. [tex]\(\frac{500}{100} = 5\)[/tex]
You will need 5 machines to produce 500 items per hour.
### 3.1.2 How long will it take 4 machines to produce 100 items?
1. One machine produces 100 items in one hour.
2. Four machines working together will produce 400 items in one hour (4 machines × 100 items).
3. To produce 100 items with 4 machines: [tex]\( \frac{100}{400} = 0.25 \)[/tex] hours.
4. Converting 0.25 hours into minutes: [tex]\(0.25 \times 60 = 15\)[/tex] minutes.
It will take 4 machines 15 minutes to produce 100 items.
### 3.1.3 How many items do 8 machines produce in 45 minutes?
1. One machine produces 100 items in one hour, or [tex]\(\frac{100}{60} \approx 1.67\)[/tex] items per minute.
2. Therefore, 8 machines produce [tex]\(8 \times 1.67 = 13.36\)[/tex] items per minute.
3. In 45 minutes, 8 machines will produce [tex]\(13.36 \times 45 = 600\)[/tex] items.
8 machines produce 600 items in 45 minutes.
### 3.2 If a rescue vessel has enough provisions for 10 people to survive for 6 days, how long will 12 people survive on the vessel?
1. Original total provisions amount to [tex]\(10 \times 6 = 60\)[/tex] people-days.
2. Distributing these provisions among 12 people: [tex]\(\frac{60}{12} = 5\)[/tex] days.
12 people will survive for 5 days on the vessel.
### 3.3.1 If a motorist wants to complete his journey in 4 hours, at what speed must he travel?
1. The distance covered at 80 km/h in 4.5 hours is [tex]\(80 \times 4.5 = 360\)[/tex] km.
2. To complete the same journey in 4 hours, the speed required is [tex]\(\frac{360}{4} = 90\)[/tex] km/h.
The motorist must travel at 90 km/h to complete the journey in 4 hours.
### 3.3.2 If he drives at 110 km/h, how long will the journey be? Provide the answer in hours and minutes.
1. Using the previously calculated distance of 360 km, if driving at 110 km/h, the time taken would be [tex]\(\frac{360}{110} \approx 3.27\)[/tex] hours.
2. Converting the decimal into hours and minutes:
- [tex]\(3.27\)[/tex] hours equates to 3 hours and [tex]\(0.27 \times 60 \approx 16\)[/tex] minutes.
The journey will take approximately 3 hours and 16 minutes.
