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Solve Inverse Proportion Problems Algebraically

In this worksheet, students will find the values of unknown variables which are inversely proportional to each other, using appropriate notation to express such relationships algebraically.

'Solve Inverse Proportion Problems Algebraically' worksheet

Key stage:  KS 4

GCSE Subjects:   Maths

GCSE Boards:   AQA, Eduqas, Pearson Edexcel, OCR

Curriculum topic:   Ratio, Proportion and Rates of Change

Curriculum subtopic:   Ratio, Proportion and Rates of Change, Direct and Inverse Proportion

Difficulty level:  

Worksheet Overview

QUESTION 1 of 10

Just when you have are hopefully getting to grips with direct proportion using algebra, this activity will throw inverse proportion at you too!

 

key 

 

Key Information:

Two quantities are said to be inversely proportional if, as one quantity increases, the other quantity decreases at the same rate.

They are doing the opposite to each other.

 

For example, the faster you run over a given distance, the less time it takes.

 

 

When two variables are proportional (so one increases as the other does, or vice versa), we would write this as: y ∝ x

We use the letter k to represent the constant, and we would find the multiplier using the formula: y = k × x

 

 

However, for inverse proportion, the opposite is happening, so we have to rearrange our formula to become:

y ∝ k/x

 

 

 

pencil 

 

It's time to practise now...

 

 

e.g. x is inversely proportional to y, and when x = 4 then y = 9.

a) Find y when x = 2.

 

Firstly, we need to find our multiplier by adding in the numbers we know:

y ∝ k/x

9 = k ÷ 4

k = 9 × 4

So k = 36

 

 

Now we can substitute this value into our formula with the new value for x:

y = 36 ÷ 2

y = 18

So when x = 2, y = 18. 

 

 

b) Find x when y = 3.

All we need to do here is to take the same multiplier and divide by the value given for y:

3 = 36 ÷ x

3x = 36

x = 12

 

 

 

Okay - let's dive in!

 

skydiver 

 

In this activity, you will find the values of unknown variables which are inversely proportional to each other, using appropriate notation to express these relationships algebraically. 

The variable t is inversely proportional to m.

 

If t = 6 when m = 2, find the missing values in the statements below. 

The variable w is inversely proportional to x.

 

If w = 5 when x = 12, find the missing values in the statements below. 

The variable y is inversely proportional to x.

 

If y = 5 when x = 2, find the missing values in the statements below. 

y is inversely proportional to x.  

 

Consider the values in the table below:

 

x 12 ?
y 15 5

 

Use what you know to find the values of the missing number, represented asin the table. 

y is inversely proportional to x.  

 

Consider the values in the table below:

 

x 1 3 6
y A B 15

 

Use what you know to find the values of the missing numbers, represented as A and in the table. 

Underwater view of two whales and a diver

 

The temperature t of the water in the sea (in °c) is inversely proportional to the depth d (in km).

 

The temperature of the sea was 6°c at a depth of 4 km.

 

What would the temperature of the sea be at a depth of 8 km?

q is inversely proportional to the square of t.

 

If q = 8.5 when t = 4, find the value of q when t = 5

 

Show your answer to 2 decimal places.

x and y are positive quantities.  

 

y is inversely proportional to x².

 

If y = 40 when x = 10, find the value of y when x = 1.

y is inversely proportional to √x.

 

If y = 6 when x = 4, find:

 

a) y when x = 9;

 

b) x when y = 12.

Match each inversely proportional relationship on the left to its correct algebraic representation on the right. 

Column A

Column B

y is inversely proportional to the square root of ...
y ∝ k / x³
y is inversely proportional to the square of x
y ∝ k / √x
y varies inversely with x
y ∝ k / x
y is inversely proportional to the cube of x
y ∝ k / x²
  • Question 1

The variable t is inversely proportional to m.

 

If t = 6 when m = 2, find the missing values in the statements below. 

CORRECT ANSWER
EDDIE SAYS
We just need to remember how to write the relationships out, then we can substitute the information we have been given into this formula to find our multiplier (k). t ∝ k/m When t = 6, m = 2, so we know that: 6 = k ÷ 2 6 × 2 = 12 So our multiplier (k) is 12. Now let's apply this information in the two scenarios we have been given. t when m = 4: t = k ÷ 4 t = 12 ÷ 4 = 3 m when t = 4.8: 4.8 = k ÷ m 4.8 = 12 ÷ m 4.8m = 12 m = 12 ÷ 4.8 = 2.5 How was that first challenge?
  • Question 2

The variable w is inversely proportional to x.

 

If w = 5 when x = 12, find the missing values in the statements below. 

CORRECT ANSWER
EDDIE SAYS
So let's start with creating our formulae: w ∝ k/x Then let's find our multiplier (k): 5 = k ÷ 12 5 × 12 = 60 Now we can apply this multiplier to the new scenarios. w when x = 3: w = k ÷ 3 w = 60 ÷ 3 w = 20 x when w = 10: 10 = k ÷ x 10 = 60 ÷ x 10x = 60 x = 60 ÷ 10 = 6
  • Question 3

The variable y is inversely proportional to x.

 

If y = 5 when x = 2, find the missing values in the statements below. 

CORRECT ANSWER
EDDIE SAYS
It is just a matter of practice with these questions. So let's start with creating our formula: y ∝ k/x Then let's find our multiplier (k): 5 = k ÷ 2 5 × 2 = 10 Now we can apply this multiplier to the new scenarios. y when x = 20: y = k ÷ 20 y = 10 ÷ 20 = 0.5 x when y = 4: 4 = 10 ÷ x 4x = 10 x = 10 ÷ 4 = 2.5
  • Question 4

y is inversely proportional to x.  

 

Consider the values in the table below:

 

x 12 ?
y 15 5

 

Use what you know to find the values of the missing number, represented asin the table. 

CORRECT ANSWER
EDDIE SAYS
It is best to write out what we want to know, as visualising this in the same way we have been practising will make things easier. We want to find out x, when y = 5 (?). Now we can just solve this in the same way that we have been practising. y ∝ k/x We have both variables in the first column, so we need to use this info to find our multiplier: 15 = k ÷ 12 15 × 12 = 180 Now let's find our missing value (?): y = 180 ÷ x 5 = 180 ÷ x 5x = 180 x = 180 ÷ 5 = 36
  • Question 5

y is inversely proportional to x.  

 

Consider the values in the table below:

 

x 1 3 6
y A B 15

 

Use what you know to find the values of the missing numbers, represented as A and in the table. 

CORRECT ANSWER
EDDIE SAYS
This time, we want to find out y, when x = 1 (A) and when x = 3 (B). y ∝ k/x We have both variables in the end column, so we need to use this info to find our multiplier: 15 = k ÷ 6 15 × 6 = 90 Now let's find our missing values. A) y when x = 1: y = k ÷ 1 y = 90 ÷ 1 = 90 B) y when x = 3: y = k ÷ 3 y = 90 ÷ 3 = 30
  • Question 6

Underwater view of two whales and a diver

 

The temperature t of the water in the sea (in °c) is inversely proportional to the depth d (in km).

 

The temperature of the sea was 6°c at a depth of 4 km.

 

What would the temperature of the sea be at a depth of 8 km?

CORRECT ANSWER
EDDIE SAYS
This time, our variables are hidden within the words of a problem, so we need to break it down. Then we can just solve this in the same way that we have been practising. t ∝ k/d 6 = k ÷ 4 6 × 4 = 24 (our multiplier) t = 24 ÷ 8 So when d= 24, t = 3. That water sounds pretty freezing!
  • Question 7

q is inversely proportional to the square of t.

 

If q = 8.5 when t = 4, find the value of q when t = 5

 

Show your answer to 2 decimal places.

CORRECT ANSWER
5.44
EDDIE SAYS
Oops, this is a bit of a curve-ball! This time, we do not have a simple inverse relationship between our variables, it is slightly more complex. The question tells us that 'q is inversely proportional to the square of t', we write this as: q ∝ k/t2 Now we just proceed as normal by finding our multiplier: 8.5 = k ÷ (4)2 8.5 × 16 = 136 Now we can apply this multiplier as usual: q = k/t2 q = 136 ÷ (5)2 q = 136 ÷ 25 = 5.44
  • Question 8

x and y are positive quantities.  

 

y is inversely proportional to x².

 

If y = 40 when x = 10, find the value of y when x = 1.

CORRECT ANSWER
EDDIE SAYS
Similarly to the last question, we have to tackle a slightly more complex relationship here. The question tells us that 'y is inversely proportional to x2', we write this in the same way as the previous question: y ∝ k/x2 Now we just proceed as normal by finding our multiplier: 40 = k ÷ (10)2 40 × 100 = 400 Now we can apply this multiplier as usual: y = k/x2 y = 400 ÷ (1)2 = 400
  • Question 9

y is inversely proportional to √x.

 

If y = 6 when x = 4, find:

 

a) y when x = 9;

 

b) x when y = 12.

CORRECT ANSWER
EDDIE SAYS
y ∝ k/√x Find our multiplier: 6 = k ÷ √4 6 × 2 = 12 Apply this multiplier: a) y = k/√x y = 12 ÷ √9 y = 12 ÷ 3 = 4 b) y = k/√x 12 = 12 ÷ √x 12 × √x = 12 √x = 12 ÷ 12 x = 12 = 1 Did you remember that the inverse of a √ is a 2?
  • Question 10

Match each inversely proportional relationship on the left to its correct algebraic representation on the right. 

CORRECT ANSWER

Column A

Column B

y is inversely proportional to th...
y ∝ k / √x
y is inversely proportional to th...
y ∝ k / x²
y varies inversely with x
y ∝ k / x
y is inversely proportional to th...
y ∝ k / x³
EDDIE SAYS
A good one to finish with here - it makes life a lot easier seeing these complex relationships expressed with algebra, doesn't it? Remember, all we need to do is apply the regular inverse expression of y and x (y ∝ k/x) then change this as required. Hopefully, your stress levels have now gone down and your expertise levels have gone up during this activity - an example of inverse proportion at its best! You can now find the values of unknown variables which are inversely proportional to each other, using appropriate notation to express these relationships algebraically.
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