Direct Proportion: More Chocolate = More Happiness
Its chocolate. No words needed.
Proportion is linked to Ratio.
2 things are directly proportional if increasing one of them increases the other by the same amount or the same scale factor.
For example: When you buy fruit, increasing the weight of the fruit, increases the price.
To answer questions based on direct proportion, you usually work out the cost of one and then multiply up accordingly.
For example:
If you were to buy 6 cupcakes for £3.00, how much would 8 cupcakes cost?
Step 1: Find the price of 1 cupcake
£3 divided by 6 = 50p per cupcake
Step 2: Multiply by how many you need
8 x 50p = £4
So 8 of the same cupcakes would cost £4.00
Here’s another example:
If a car travels 20 miles and it takes 30 minutes, how long would it take the car to travel 35 miles - assume that the car’s speed stays the same.
First, work out how long it takes to travel 1 mile:
30/20 = 1.5, so every mile takes 1.5 minutes to complete.
We have 35 miles, multiplying 35 by 1.5 gives 52.5 minutes.
So it would take 52.5 minutes to travel 35 miles at the same speed.
Harder Direct Proportion
Sometimes you will come across a question that involves a task that needs to be completed. Something like painting a room or translating documents.
The question will state how long it takes a group of people to do and then ask you to work out how long it takes a different sized group of people.
Here is an example:
(this is from the cgp ks3 maths study book)
3 painters can paint 9 rooms per day.
How many rooms per day could 7 painters paint?
First, we need to know how many rooms 1 painter paints.
So divide 9 by 3.
9/3 = 3. This means: one painter paints 3 rooms per day.
The question asks how many rooms 7 painters can paint in a day.
We know 1 painter paints 3, so 7 painters will paint 7 x 3 = 21 rooms.
This example is similar to the buying food examples. e.g. I buy 6 cupcakes, how many would 8 cost.
But somtimes the way its worded can make it seem confusing.
Direct Proportion Questions: Divide to find 1 and then multiply to find all.
Or Divide for one, Times for all.
Free Quiz: Direct Proportion Word Problems
1. If 3 apples cost $6, how much would 5 apples cost?
Inverse Proportion
This is when things go opposite ways.
If you increase one thing, the other decreases.
E.g. When it gets hotter, we use less heating.
If we are decorating a room, the more helpers we have the less time it takes.
Inverse proportion questions can be a pain.
To solve them you do the opposite:
Multiply for one and then divide for all.
Let’s have a look at some examples.
It takes 6 workers 8 hours to complete a task.
How many hours will it take 12 workers to complete the task?
In this problem, we are dealing with inverse proportion, which means that when one number gets bigger, the other gets smaller. In this case, if you add more workers, the task will be finished faster, so the time needed to complete the task will be less.
We can use this simple rule for inverse proportion:
Multiply the original number of workers by the original number of hours. This gives us a constant number, which we use to find the missing information.
There are 2 ways to do this:
The fast way: We have double the workers, so the time will decrease by 1/2.
So it will take 12 workers 4 hours to complete the task.
Another way to look at it is like this:
If it takes 6 hours for 8 workers, one worker will take 6 x 8 hours = 48 hours.
If we have 12 workers, 48/12 = 4. So it will take 4 hours for 12 workers to complete the task.
Here is another example:
Question:
It takes 4 taps 12 minutes to fill a swimming pool. How long would it take 6 taps to fill the pool, assuming all taps run at the same speed?
In this question, we're dealing with inverse proportion because more taps will fill the pool faster, so the time will go down as the number of taps goes up.
The key idea is:
When you increase the number of taps, the time it takes to fill the pool will decrease.
Here’s how we solve it:
Step 1: Multiply the original number of taps by the original time:
4×12=484 \times 12 = 484×12=48
This gives us a constant number (48), this is how long it takes 1 tap to fill the pool.
Step 2: Now, divide the constant number by the new number of taps:
48÷6 = 8
So, with 6 taps, it will take 8 minutes to fill the swimming pool.
Inverse proportion works in situations where increasing one thing causes the other to decrease. Here, adding more taps makes the pool fill faster, so the time taken goes down.
Practice Questions: Inverse Proportion
1. It takes 3 people 12 days to paint a house. How long will it take 6 people to paint the same house?
