 # Solve Inverse Proportion Problems Algebraically

In this worksheet, students will be able to solve inverse proportion problems. 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:   ### QUESTION 1 of 10 Just when you have got to grips with direct proportion using algebra, your wonderful maths teacher throws inverse proportion at you. 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.

Normally you would write  y x

You use the letter k for the constant and would find the multiplier  y = k x (x).

However the inverse is happening so we have to rearrange our formula to become y k/x (opposite of x is ÷) Time to practice

X is inversely proportional to Y       X = 4  and Y = 9

Find y when x = 2

Find the multiplier

y  k/x

9 = k/4  Do the opposite to divide  9 x 4 = 36.

Now substitute

y = 36 ÷2 = 18

Now find x when y = 3

All you do here is take the multiplier and divide by the value given for y.

36 ÷ 3 = 12 Time to strap yourself back into your parachute.

T is inversely proportional to M.  If T = 6 when M = 2  find:

a) T when M = 4

b) M when T = 4.8

T = 4

T= 3

M= 3.5

M= 2.5

W is inversely proportional to x.  If W = 5 when X = 12  find:

a) W when X = 3

b) X when W = 10

W = 20

W= 30

X= 6

X= 5

Y is inversely proportional to x.  When y = 5,  x = 2

W = 20

W= 30

X= 6

X= 5

y is inversely proportional to x.  Use the information to find the missing gaps in the table.

 X 12 y 15 12

y is inversely proportional to x.

Which of the following 2 values are correct for the missing values of y

 x 1 3 6 y 15

30

40

20

90 The temperature  t of the water in the sea, is C, is inversely proportional to the depth d in km's.

The temperature was 6 C at a depth of 4 km.

What would the temperature be at a depth of 8 km.

30

40

20

90

Q is inversely proportional to the square of  T.

When T = 4,  Q = 8.5

Calculate the value of Q when T= 5

x and y are positive quantities.

y is inversely proportional to x²

When y = 40, x = 10

Find the value of y when x = 10

Y is inversely proportional to √ x when y = 6 and x = 4.

Find the value of Y when x = 9

Find the value of x when y = 12

Mathe up the following

## Column B

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

T is inversely proportional to M.  If T = 6 when M = 2  find:

a) T when M = 4

b) M when T = 4.8

T= 3
M= 2.5
EDDIE SAYS
The most important thing to do is find the multiplier. Without this we will be stuck. T = k ÷ m Substitute 6 = k & divide; 2 Do the opposite 6 x 2 Multiplier is 12 T = 12÷4 = 3 To find M Take the multiplier and divide by the value for T 12 ÷ 4.8 = 2.5 It looks more scary than it actually is
• Question 2

W is inversely proportional to x.  If W = 5 when X = 12  find:

a) W when X = 3

b) X when W = 10

W = 20
X= 6
EDDIE SAYS
T = k ÷ m Substitute 5 = k & divide; 12 Do the opposite5 x 12 Multiplier is 60 w = 60 ÷ 3 = 20 To find x take the multiplier and divide it by the value for w. 60 ÷ 10 = 6 It is just a case of keep practising.
• Question 3

Y is inversely proportional to x.  When y = 5,  x = 2

EDDIE SAYS
y = k ÷ x 5 = K ÷ 2 Do the opposite 2 x 5 = 10 and here we have the multiplier. How valuable is the multiplier? It is our true friend in these questions. y = 10 ÷ 20 = 0.5 To find y use our multiplier 10 ÷ 4 = 2.5
• Question 4

y is inversely proportional to x.  Use the information to find the missing gaps in the table.

 X 12 y 15 12

x = 15
EDDIE SAYS
y = k÷ x 15 = k ÷ 12 12 x 15 = 180 180 ÷ 12 =15 Are you getting the hang of this now?
• Question 5

y is inversely proportional to x.

Which of the following 2 values are correct for the missing values of y

 x 1 3 6 y 15

30
90
EDDIE SAYS
Phone a friend (the multiplier) y = k ÷ x 15 = k ÷ 6 Do the opposite 15 x 6 = 90 when x = 1 90 & divide; 1 = 90 when x = 3 90 & divide; 3 = 30
• Question 6 The temperature  t of the water in the sea, is C, is inversely proportional to the depth d in km's.

The temperature was 6 C at a depth of 4 km.

What would the temperature be at a depth of 8 km.

EDDIE SAYS
Who do we call..... the multiplier. T = k ÷ d 6= K ÷: 4 Do the opposite 6 x 4 = 24 T = 24 ÷ 8 = 3 This is a sneaky way of asking about inverse proportion. That water is too cold for me.
• Question 7

Q is inversely proportional to the square of  T.

When T = 4,  Q = 8.5

Calculate the value of Q when T= 5

5.44
EDDIE SAYS
Oh that curved ball, don't you just love them. Multiplier Q = k ÷ T² 8.5 = k ÷ 16 8.5 x 16 = 136 136 ÷ 25 = 5.44 Did you remember to square your numbers?
• Question 8

x and y are positive quantities.

y is inversely proportional to x²

When y = 40, x = 10

Find the value of y when x = 10

10
EDDIE SAYS
y = k ÷ x² 40 = k ÷ 25 40 x 25 = 1000 y = k ÷ 10² 1000 ÷ 100 = 10
• Question 9

Y is inversely proportional to √ x when y = 6 and x = 4.

Find the value of Y when x = 9

Find the value of x when y = 12

EDDIE SAYS
You know by now that the multiplier is key. y = k/√x 6 = k ÷: 2 2 x 6 = 12 y = 12 ÷ √9 = 4 And on to the final push To find x 12 = 12 ÷√x 12 ÷12 = 1
• Question 10

Mathe up the following

## 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
Makes life a lot easier seeing these all written down like this. I hope your stress levels have now gone down and your expertise levels gone up. Inverse proportion at its best.
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