First find the cost of one
4 notebooks cost $20. How much do 7 notebooks cost? 7 is not 2 times or 3 times 4, so doubling or tripling the 4 notebooks will not reach 7. Instead, go through one notebook.
The $20 is shared equally by the 4 notebooks, so one notebook costs 20 ÷ 4 = $5. The cost of one is called the unit value.
The 4 notebooks share the $20 equally, so one of them is $5.
Then multiply up
Once you know that one notebook costs $5, any number of them is easy: 7 notebooks cost 7 × $5 = $35. You can check it against the first fact: 4 × $5 = $20.
Finding the value of one and then multiplying is the unitary method, used here for two quantities in direct proportion. It works for any two numbers, not only when one is a multiple of the other.
7 notebooks at $5 each: 7 × 5 = $35.
When one is not a whole number
The unit value does not have to be a whole number. 2 kg of apples cost $5, so 1 kg costs 5 ÷ 2 = $2.50, and 3 kg cost 3 × $2.50 = $7.50.
Reading the price of one from a graph
In the drawing below, the line shows the price of any weight of pears. 2 kg of pears cost $6. Every point on the line has the same price per kilogram, so the price of one kilogram is the height of the line at 1 kg.
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
Drag the weight from 2 kg to 1 kg. 2 kg cost $6, and 1 kg costs $3, half as much: 6 ÷ 2 = 3.
The usual mistakes
Adding the new number onto the old total. If 4 notebooks cost $20, 7 notebooks do not cost $20 + 7 = $27. They cost 7 × $5 = $35.
Adding the number of items to the price of one. 7 notebooks at $5 each is 7 × 5 = $35, not 7 + 5 = $12.
Dividing the wrong way round. One notebook costs 20 ÷ 4 = $5. 4 ÷ 20 = 0.2 is how many notebooks one dollar buys.
One worker for one hour
Sometimes the "one" is one worker for one hour. In the next problem, 6 pickers work for 3 hours, which is 6 × 3 = 18 picker-hours. Finding what one picker fills in one hour is the unit value, and every other question follows from it.
Worked example: Apple Pickers in an Orchard: A Bigger Team, Then a Bigger Order
Question In an orchard, 6 pickers fill 90 crates of apples in 3 hours. Every picker works at the same steady rate. (a) How many crates do 9 pickers fill in 3 hours? (b) The orchard has an order for 400 crates, which must be filled in 5 hours. How many pickers are needed?
1.Draw one box for each picker in each hour: 6 pickers for 3 hours make 6 × 3 = 18 boxes, and together they hold 90 crates.
One box for each picker in each hour: 6 × 3 = 18 boxes, holding 90 crates between them. 2.One box, one picker for one hour, holds 90 ÷ 18 = 5 crates.
One box, one picker for one hour, holds 90 ÷ 18 = 5 crates. 3.(a) 9 pickers for 3 hours make 9 × 3 = 27 boxes, so they fill 27 × 5 = 135 crates.
(a) 9 × 3 = 27 boxes, and 27 × 5 = 135 crates. 4.(b) 400 crates need 400 ÷ 5 = 80 boxes.
(b) 400 ÷ 5 = 80 boxes, laid out in 5 rows, one row for each hour. 5.The 80 boxes are spread over 5 hours, so each hour needs 80 ÷ 5 = 16 boxes, one for each picker: 16 pickers are needed. Check: 16 × 5 × 5 = 400 crates.
Each hour has 80 ÷ 5 = 16 boxes, one for each picker: 16 pickers.
Answer: (a) 135 crates; (b) 16 pickers
Common mistakes
- Scaling (a) by the time as well as by the team. The time is still 3 hours; only the number of pickers changes, from 6 to 9, so the crates change in the same way: 90 × 96 = 135.
- Treating a longer time as needing more pickers. Each picker fills more crates in 5 hours than in 3, so a longer time needs fewer pickers; dividing the 80 picker-hours by the 5 hours is what gives the team.