How much for one
A tap is filling a tank, and every hour it adds the same 12 liters of water. The amount that goes with 1 hour, 12 liters, is called the rate. We say the tap fills at 12 liters per hour. "Per" means "for each", so this means 12 liters for each hour.
A rate always joins two different quantities, so it has two units. Here they are liters and hours. A car might travel at 50 kilometers per hour, rice might cost $3 per kilogram, and a typist might type 40 words per minute. Each one tells you how much of the first quantity goes with one of the second.
The top line counts hours and the bottom line counts liters. The mark above 1 hour is 12 liters, which is the rate.
Every hour is one more step
The tap adds the same amount every hour, so each hour is one more step of 12 liters. After 1 hour there are 12 liters, after 2 hours there are 2 × 12 = 24 liters, and after 3 hours there are 3 × 12 = 36 liters.
So to find the amount for any number of hours, multiply the rate by the number of hours. In 5 hours the tap adds 5 × 12 = 60 liters.
3 hours is 3 steps of 12 liters, so the mark under 3 hours is 3 × 12 = 36 liters.
Finding the rate
Sometimes a question gives a total and asks for the rate. A car travels 150 km in 3 hours at a steady speed. The 150 km is shared equally between the 3 hours, so the car travels 150 ÷ 3 = 50 km in each hour. Its rate is 50 km per hour.
Once you know the rate, multiply as before. In 5 hours the car travels 5 × 50 = 250 km.
150 km sits under 3 hours. Sharing it between the 3 hours puts 50 km under 1 hour, and 5 steps of 50 reach 250 km.
y = kx is a straight line through the origin, and y/x is k at every point on it: the price per kilogram is the gradient
Drag the weight to 1 kg and read the price
The line joins each mass of rice to its price. Drag the kg handle along the line to 1 kg: the price there, $2.50, is the rate. Every other kilogram adds the same $2.50.
The usual mistakes
Adding the rate and the time. At 12 liters per hour for 3 hours, 12 + 3 = 15 is not the amount of water. Each of the 3 hours adds 12 liters, so multiply: 3 × 12 = 36.
Counting one step too many. 3 hours is 3 steps of 12 liters, not 4. The line starts at 0 liters at 0 hours, before any water has run.
Rates for a job
A rate can also measure work. If a painter can paint a hall in 12 hours, the painter paints of the hall in each hour. When two painters work at the same time, the parts of the hall they paint in the same hour add together. In the next problem the hall is cut into 60 equal units, because 60 is the lowest common multiple of the three painters' times, so every rate is a whole number of units per hour.
Worked example: Cooperative Shared Task Completion
Question Worker A can paint a school hall in 12 hours working alone. Worker B can paint the same hall in 15 hours working alone, and Worker C can paint it in 20 hours working alone. (a) What fraction of the hall can Worker A and Worker B paint together in 1 hour? (b) How many hours will it take for all three workers to paint the entire hall if they work together simultaneously?
1.Total work = 60 units.
Call the hall 60 units, a number 12, 15 and 20 all divide. 2.Worker A: 60 ÷ 12 = 5 units/hr.
Each worker’s hourly share: 5, 4 and 3 units. 3.Worker B: 60 ÷ 15 = 4 units/hr.
Each worker’s hourly share: 5, 4 and 3 units. 4.Worker C: 60 ÷ 20 = 3 units/hr.
Each worker’s hourly share: 5, 4 and 3 units. 5.Workers A and B together do 5 + 4 = 9 units/hr.
A and B together: 9 units an hour. 6.(a) Fraction in 1 hour = 960 = 320.
(a) 960 = 320 of the hall. 7.All three workers together do 5 + 4 + 3 = 12 units/hr.
All three: 12 units an hour. 8.Total time to complete 60 units: 60 ÷ 12 = 5 hours.
All three: 12 units an hour. 9.(b) 5 hours.
(b) 60 ÷ 12 = 5 hours.
Answer: (a) 320 of the hall; (b) 5 hours
Common mistakes
- Adding the hours directly (12 + 15 + 20 = 47 hours) or averaging them, concluding it takes longer working together.
- Taking the common denominator incorrectly when adding the three fractions.