Year 7 Maths: Compare rates: word problems

Introduction to Comparing Rates

Today, we’re going to learn about comparing rates through word problems. This means we will figure out which option is better based on different situations, like speed, price, or efficiency.

What is a Rate?

A rate is a way to compare two different quantities. For example, if we say “10 miles per hour,” we’re comparing distance to time.

Why Compare Rates?

Comparing rates helps us make better decisions. For example, if you’re choosing between two different mobile plans, knowing which plan gives you the best value for money helps you decide.

How to Compare Rates

Step-by-Step Guide

1. Identify the Rates: Read the problem carefully and pick out the rates you need to compare.
2. Convert Rates if Necessary: Make sure the rates are in the same units. For example, if one rate is in pence per item and another is in pounds per item, convert them to the same unit.
3. Calculate the Values: If needed, do some calculations to find out the actual values.
4. Compare the Values: Look at the numbers and decide which one is better.

Key Rules

• Always convert rates to the same unit before comparing.
• Pay attention to what the rates are describing (e.g., speed, cost, etc.).
• Double-check your calculations for accuracy.

Tips and Tricks

• Use a Table: Writing the rates side by side in a table can help you see which one is larger or better.
• Think About the Context: Consider what the rates mean in real life. Sometimes a slightly higher rate might still be the better choice.

Examples

Example 1: Comparing Prices

Question: A sandwich costs £3.50, and a drink costs £1.20. How much does one sandwich and one drink cost together?

Solution:

• Sandwich: £3.50
• Drink: £1.20
• Total Cost: $$3.50 + 1.20 = 4.70$$

So, together they cost £4.70.

Example 2: Comparing Speed

Question: Car A travels 150 miles in 3 hours. Car B travels 180 miles in 4 hours. Which car is faster?

Solution:

• Car A Speed: $$\frac{150 \text{ miles}}{3 \text{ hours}} = 50 \text{ miles per hour}$$
• Car B Speed: $$\frac{180 \text{ miles}}{4 \text{ hours}} = 45 \text{ miles per hour}$$

Car A is faster at 50 miles per hour.

Practice Questions

Easy Level Questions

1. A bike travels 20 miles in 1 hour. How far does it travel in 2 hours?
2. A candy costs 50p per piece. How much do 4 pieces cost?
3. A car uses 5 litres of petrol to travel 100 miles. How much petrol is used for 200 miles?
4. A jogger runs 30 minutes and covers 2 miles. What is the rate in miles per hour?
5. A book costs £10. How much do 3 books cost?
6. A train travels 60 miles in 1 hour. How far does it travel in 3 hours?
7. A ticket costs £15. How much do 2 tickets cost?
8. A gardener can plant 10 flowers in 2 hours. How many can he plant in 4 hours?
9. A car travels 120 miles in 2 hours. What is the speed in miles per hour?
10. A pack of biscuits costs £2.50. How much do 3 packs cost?
11. A runner completes a marathon (26 miles) in 4 hours. What is the speed in miles per hour?
12. A taxi charges £3 per mile. How much for a 5-mile ride?
13. A bottle of juice is £1.80. How much do 5 bottles cost?
14. A cyclist rides 15 miles in 30 minutes. What is the speed in miles per hour?
15. A pair of shoes costs £40. How much do 2 pairs cost?
16. A pizza costs £12. How much do 3 pizzas cost?
17. A swimmer swims 50 metres in 1 minute. What is the speed in metres per minute?
18. A fruit basket costs £15. How much do 2 baskets cost?
19. A delivery truck travels 200 miles in 4 hours. What is the speed in miles per hour?
20. A toy costs £8. How much do 4 toys cost?

Medium Level Questions

1. A car can travel 300 miles on 10 gallons of petrol. What is the miles per gallon?
2. A 500g pack of flour costs £1.50. How much would 2kg cost?
3. A bus travels 150 miles in 3 hours. How fast is the bus going?
4. A worker earns £240 for 40 hours of work. What is the hourly rate?
5. A 1-litre bottle of soda costs £2. How much do 5 bottles cost?
6. A factory produces 400 items in 8 hours. How many items are produced per hour?
7. A cyclist rides at a speed of 12 miles per hour. How far will he travel in 2.5 hours?
8. A pack of 6 pens costs £3. How much for 24 pens?
9. A car travels 180 miles in 3 hours. What is the speed in miles per hour?
10. A gym charges £30 for a monthly membership. How much does it cost for 6 months?
11. A train covers 120 miles in 2 hours. What is the speed in miles per hour?
12. A cookbook costs £15. How much do 4 cookbooks cost?
13. A restaurant bill is £50. If there are 5 people, how much does each person pay?
14. A plant grows 2 inches in a month. How tall will it be in a year?
15. A car can drive 240 miles on 8 gallons of fuel. What is the miles per gallon?
16. A concert ticket is £25. How much do 3 tickets cost?
17. A person walks 6 miles in 2 hours. What is the speed in miles per hour?
18. A box contains 12 chocolates for £4. How much for 48 chocolates?
19. A student reads 30 pages in 1 hour. How many pages can they read in 3 hours?
20. A printer uses 5 cartridges in 1000 pages. How many pages does one cartridge print?

Hard Level Questions

1. A car uses 50 litres of petrol to travel 450 miles. What is the fuel efficiency in miles per litre?
2. A 750g bag of rice costs £2.40. How much would 5kg cost?
3. A factory can produce 800 toys in 4 hours. How many toys can it produce in 6 hours?
4. A cyclist travels 36 miles in 2 hours and 30 minutes. What is the speed in miles per hour?
5. A store sells a pair of shoes for £60. If there is a 25% discount, what will the price be after the discount?
6. A train travels 300 miles in 5 hours. How fast is the train going in miles per hour?
7. A worker earns £400 for 50 hours of work. What is the hourly wage?
8. A recipe requires 3 cups of flour for 12 cookies. How much flour is needed for 30 cookies?
9. A car costs £15,000. If a person saves £500 a month, how many months will it take to save enough for the car?
10. A box of 24 chocolates costs £12. How much would 100 chocolates cost?
11. A jogger runs 6 miles in 45 minutes. What is their speed in miles per hour?
12. A bike rental costs £10 for the first hour and £5 for each additional hour. How much would it cost for 4 hours?
13. An athlete runs 100m in 12 seconds. What is their speed in metres per second?
14. A printer can print 20 pages in 1 minute. How long will it take to print 100 pages?
15. A person walks 8 miles in 2 hours. How far can they walk in 5 hours at the same pace?
16. A car can travel 200 miles on 12 gallons of petrol. What is the fuel efficiency in miles per gallon?
17. A store sells oranges at £0.80 each or £4.00 for a pack of 6. Which option is cheaper?
18. A runner completes a race in 1 hour and 30 minutes. What is the average speed in miles per hour if the race is 13 miles long?
19. A car travels 400 miles on 20 gallons. How many miles can it travel on 1 gallon?
20. A florist sells bouquets for £25 each. How much would 10 bouquets cost after a 10% discount?

Answers

Easy Level Answers

1. 40 miles
2. £2.00
3. 10 litres
4. 4 mph
5. £30.00
6. 180 miles
7. £30.00
8. 20 flowers
9. 60 mph
10. £7.50
11. 6.5 mph
12. £15.00
13. £9.00
14. 30 mph
15. £80.00
16. £36.00
17. 60 mph
18. £30.00
19. 200 miles
20. £32.00

Medium Level Answers

1. 30 miles per gallon
2. £7.50
3. 50 mph
4. £6.00
5. £10.00
6. 50 items per hour
7. 30 miles
8. £12.00
9. 60 mph
10. £180.00
11. 60 mph
12. £60.00
13. £10.00
14. 24 inches
15. 30 miles
16. £25.00
17. £1.00
18. 12 mph
19. 20 pages
20. £10.00

Hard Level Answers

1. 9 miles per litre
2. £8.00
3. 1200 toys
4. 14.4 mph
5. £45.00
6. 60 mph
7. £8.00
8. 7.5 cups
9. 30 months
10. £5.00
11. 10.67 mph
12. £25.00
13. 8.33 m/s
14. 5 minutes
15. 20 miles
16. 16.67 mpg
17. Pack of 6 is cheaper
18. 8.67 mph
19. 20 miles
20. £225.00

Conclusion

That’s it for today’s lesson on comparing rates! Remember to read the problems carefully and take your time when calculating. With practice, you’ll become great at comparing rates and making smart choices. Happy learning!