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Answer :
Sure! Let's solve each problem step-by-step using the unitary or ratio method.
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Problem 1: Electricity Usage of Home Theater System
A home theater system uses 2 units of electricity in 3 hours. We need to find how many units it will use in 9 hours.
1. First, calculate the electricity used per hour:
[tex]\( \text{Units per hour} = \frac{2 \ \text{units}}{3 \ \text{hours}} \)[/tex].
2. Then, calculate the electricity used in 9 hours:
[tex]\( \text{Units for 9 hours} = \text{Units per hour} \times 9 \)[/tex].
3. Round the result to the nearest unit.
Final answer: 6 units.
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Problem 2: Car Production
a) Cars Produced in 5 Hours
The car production line produces 24 cars in 9 hours. We need to find out how many cars are produced in 5 hours.
1. Calculate the number of cars produced per hour:
[tex]\( \text{Cars per hour} = \frac{24 \ \text{cars}}{9 \ \text{hours}} \)[/tex].
2. Then, calculate the number of cars produced in 5 hours:
[tex]\( \text{Cars in 5 hours} = \text{Cars per hour} \times 5 \)[/tex].
Final answer: Approximately 13 cars.
b) Time to Produce 2000 Cars
1. Use the cars per hour from the previous step.
2. Calculate how many hours are needed for 2000 cars:
[tex]\( \text{Hours for 2000 cars} = \frac{2000 \ \text{cars}}{\text{Cars per hour}} \)[/tex].
Final answer: 750 hours.
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Problem 3: Road Painting
a) Paint Needed for 100 km
The machine uses 180 liters of paint for 6 km of road. We need to find out the paint needed for 100 km.
1. Calculate the amount of paint used per kilometer:
[tex]\( \text{Liters per km} = \frac{180 \ \text{liters}}{6 \ \text{km}} \)[/tex].
2. Then, calculate the paint needed for 100 km:
[tex]\( \text{Paint for 100 km} = \text{Liters per km} \times 100 \)[/tex].
Final answer: 3000 liters.
b) Length of Road for 500 Litres of Paint
1. Use the liters per kilometer from the previous step.
2. Calculate the length of road that can be painted with 500 liters:
[tex]\( \text{Length for 500 liters} = \frac{500 \ \text{liters}}{\text{Liters per km}} \)[/tex].
Final answer: Approximately 17 km.
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Problem 4: Chocolate Wrapping
a) Chocolate Bars Wrapped in 8-Hour Day
The machine wraps 300 chocolate bars in 2 minutes. We need to find out how many bars can be wrapped in an 8-hour day.
1. Calculate the wrapping rate per minute:
[tex]\( \text{Bars per minute} = \frac{300 \ \text{bars}}{2 \ \text{minutes}} \)[/tex].
2. Convert the 8-hour day into minutes and calculate:
[tex]\( 8 \ \text{hours} = 8 \times 60 \ \text{minutes} = 480 \ \text{minutes} \)[/tex].
3. Calculate the total bars wrapped in 480 minutes:
[tex]\( \text{Bars in 8 hours} = \text{Bars per minute} \times 480 \)[/tex].
Final answer: 72,000 bars.
b) Time to Wrap 1000 Chocolate Bars
1. Use the bars per minute from the previous step.
2. Calculate how many minutes are needed for 1000 bars:
[tex]\( \text{Time for 1000 bars} = \frac{1000 \ \text{bars}}{\text{Bars per minute}} \)[/tex].
Final answer: Approximately 7 minutes.
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Problem 5: Cricket Balls
a) Balls Bowled in 30 Minutes
The machine bowls 3 balls every 2 minutes. We need to calculate how many balls in 30 minutes.
1. Calculate balls per minute:
[tex]\( \text{Balls per minute} = \frac{3 \ \text{balls}}{2 \ \text{minutes}} \)[/tex].
2. Then, calculate the balls bowled in 30 minutes:
[tex]\( \text{Balls in 30 minutes} = \text{Balls per minute} \times 30 \)[/tex].
Final answer: 45 balls.
b) Time to Bowl 48 Balls
1. Use the balls per minute from the previous step.
2. Calculate how many minutes are required for 48 balls:
[tex]\( \text{Time for 48 balls} = \frac{48 \ \text{balls}}{\text{Balls per minute}} \)[/tex].
Final answer: 32 minutes.
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Problem 6: Fruit Juice Mix
The required ratio of orange juice to apple juice is 7:5. If 168 liters of orange juice are used, calculate the liters of apple juice needed.
1. Set up the ratio equation:
[tex]\( \frac{168 \ \text{liters (orange)}}{\text{Apple juice}} = \frac{7}{5} \)[/tex].
2. Calculate the quantity for apple juice:
[tex]\( \text{Apple juice} = \frac{168 \times 5}{7} \)[/tex].
Final answer: 120 liters.
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These solutions illustrate the use of the unitary method and ratio method to find the desired results.
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Problem 1: Electricity Usage of Home Theater System
A home theater system uses 2 units of electricity in 3 hours. We need to find how many units it will use in 9 hours.
1. First, calculate the electricity used per hour:
[tex]\( \text{Units per hour} = \frac{2 \ \text{units}}{3 \ \text{hours}} \)[/tex].
2. Then, calculate the electricity used in 9 hours:
[tex]\( \text{Units for 9 hours} = \text{Units per hour} \times 9 \)[/tex].
3. Round the result to the nearest unit.
Final answer: 6 units.
---
Problem 2: Car Production
a) Cars Produced in 5 Hours
The car production line produces 24 cars in 9 hours. We need to find out how many cars are produced in 5 hours.
1. Calculate the number of cars produced per hour:
[tex]\( \text{Cars per hour} = \frac{24 \ \text{cars}}{9 \ \text{hours}} \)[/tex].
2. Then, calculate the number of cars produced in 5 hours:
[tex]\( \text{Cars in 5 hours} = \text{Cars per hour} \times 5 \)[/tex].
Final answer: Approximately 13 cars.
b) Time to Produce 2000 Cars
1. Use the cars per hour from the previous step.
2. Calculate how many hours are needed for 2000 cars:
[tex]\( \text{Hours for 2000 cars} = \frac{2000 \ \text{cars}}{\text{Cars per hour}} \)[/tex].
Final answer: 750 hours.
---
Problem 3: Road Painting
a) Paint Needed for 100 km
The machine uses 180 liters of paint for 6 km of road. We need to find out the paint needed for 100 km.
1. Calculate the amount of paint used per kilometer:
[tex]\( \text{Liters per km} = \frac{180 \ \text{liters}}{6 \ \text{km}} \)[/tex].
2. Then, calculate the paint needed for 100 km:
[tex]\( \text{Paint for 100 km} = \text{Liters per km} \times 100 \)[/tex].
Final answer: 3000 liters.
b) Length of Road for 500 Litres of Paint
1. Use the liters per kilometer from the previous step.
2. Calculate the length of road that can be painted with 500 liters:
[tex]\( \text{Length for 500 liters} = \frac{500 \ \text{liters}}{\text{Liters per km}} \)[/tex].
Final answer: Approximately 17 km.
---
Problem 4: Chocolate Wrapping
a) Chocolate Bars Wrapped in 8-Hour Day
The machine wraps 300 chocolate bars in 2 minutes. We need to find out how many bars can be wrapped in an 8-hour day.
1. Calculate the wrapping rate per minute:
[tex]\( \text{Bars per minute} = \frac{300 \ \text{bars}}{2 \ \text{minutes}} \)[/tex].
2. Convert the 8-hour day into minutes and calculate:
[tex]\( 8 \ \text{hours} = 8 \times 60 \ \text{minutes} = 480 \ \text{minutes} \)[/tex].
3. Calculate the total bars wrapped in 480 minutes:
[tex]\( \text{Bars in 8 hours} = \text{Bars per minute} \times 480 \)[/tex].
Final answer: 72,000 bars.
b) Time to Wrap 1000 Chocolate Bars
1. Use the bars per minute from the previous step.
2. Calculate how many minutes are needed for 1000 bars:
[tex]\( \text{Time for 1000 bars} = \frac{1000 \ \text{bars}}{\text{Bars per minute}} \)[/tex].
Final answer: Approximately 7 minutes.
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Problem 5: Cricket Balls
a) Balls Bowled in 30 Minutes
The machine bowls 3 balls every 2 minutes. We need to calculate how many balls in 30 minutes.
1. Calculate balls per minute:
[tex]\( \text{Balls per minute} = \frac{3 \ \text{balls}}{2 \ \text{minutes}} \)[/tex].
2. Then, calculate the balls bowled in 30 minutes:
[tex]\( \text{Balls in 30 minutes} = \text{Balls per minute} \times 30 \)[/tex].
Final answer: 45 balls.
b) Time to Bowl 48 Balls
1. Use the balls per minute from the previous step.
2. Calculate how many minutes are required for 48 balls:
[tex]\( \text{Time for 48 balls} = \frac{48 \ \text{balls}}{\text{Balls per minute}} \)[/tex].
Final answer: 32 minutes.
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Problem 6: Fruit Juice Mix
The required ratio of orange juice to apple juice is 7:5. If 168 liters of orange juice are used, calculate the liters of apple juice needed.
1. Set up the ratio equation:
[tex]\( \frac{168 \ \text{liters (orange)}}{\text{Apple juice}} = \frac{7}{5} \)[/tex].
2. Calculate the quantity for apple juice:
[tex]\( \text{Apple juice} = \frac{168 \times 5}{7} \)[/tex].
Final answer: 120 liters.
---
These solutions illustrate the use of the unitary method and ratio method to find the desired results.
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