Direct proportion
More workers, more bricks: both change by the same factor and their quotient stays the same.
How many workers for 18000 bricks?
At a building site, 5 workers move 4500 bricks in a day. The site now needs 18000 bricks moved in a day.
How many workers will it take?
What this lesson covers
The idea
When a : b :: c : d, both quantities change by the same factor and their quotient stays the same; such proportions are direct proportions, and the fourth term is d = bc/a.
How many workers for 18000 bricks?
At a building site, 5 workers move 4500 bricks in a day. The site now needs 18000 bricks moved in a day.
How many workers will it take?
- 10 workers
- 20 workers
- 25 workers
Change the team
Change the number of workers and watch the bricks. Find the team for each of the three jobs. Look at the Factor column and at Bricks ÷ workers.
Same factor, same quotient
When a : b :: c : d, both quantities change by the same factor and their quotient stays the same; such proportions are direct proportions, and the fourth term is d = bc/a.
The site needs 18000 bricks, not 4500. We write 4500 : 18000 :: 5 : x Bricks became 18000 ÷ 4500 = 4 times as many, so the workers become 4 times as many: 5 × 4 = 20 workers.
With the rule: a = 4500, b = 18000, c = 5, so x = 18000 × 5 ÷ 4500 = 20
The quotients are the same: 4500 ÷ 5 = 900, 18000 ÷ 20 = 900. Each worker moves 900 bricks. More workers, more bricks: both go up together.
Notes
When a : b :: c : d, both quantities change by the same factor and their quotient stays the same; such proportions are direct proportions, and the fourth term is d = bc/a.
Check yourself
12 notebooks cost ₹180. At the same price for each notebook, what do 20 notebooks cost? (in ₹)
Answer: 300
12 : 20 :: 180 : x, so x = 20 × 180 ÷ 12 = 300. One notebook costs 180 ÷ 12 = ₹15, and 20 notebooks cost 20 × 15 = ₹300.
5 workers move 4500 bricks, 10 workers move 9000 and 20 workers move 18000. Which is the same for all three pairs?
A tap fills 15 litres in 6 minutes. How many litres does it fill in 10 minutes?
Answer: 25 litres
6 : 10 :: 15 : x, so x = 10 × 15 ÷ 6 = 25 litres. The tap fills 15 ÷ 6 = 2.5 litres every minute: 10 × 2.5 = 25.
6 cows give 30 litres of milk a day. How many cows give 80 litres a day, if every cow gives the same?
Answer: 16 cows
30 : 80 :: 6 : x, so x = 80 × 6 ÷ 30 = 16 cows. Each cow gives 30 ÷ 6 = 5 litres, and 80 ÷ 5 = 16.
- bricks − workers. It is 4495, 8990 and 17980. These are all different.
- bricks + workers. It is 4505, 9010 and 18020. These are all different.
- bricks ÷ workers — correct. Yes! 4500 ÷ 5 = 900, 9000 ÷ 10 = 900, 18000 ÷ 20 = 900. The quotient stays the same.
- bricks × workers. It is 22500, 90000 and 360000. These are all different.