SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 11: Changing the subject of a
formula
Authored by:
Margaret Ferguson
Reviewed by:
Jillian Hornby
Previously authored by:
Eddie Mullan
Heriot-Watt University
Edinburgh EH14 4AS, United Kingdom.
First published 2014 by Heriot-Watt University.
This edition published in 2016 by Heriot-Watt University SCHOLAR.
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SCHOLAR Study Guide Course Materials Topic 11: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Changing the subject of a formula
Contents
11.1 Changing the subject of a formula . . . . . . . . . . . . . . . . . . . . . . . . .
3
11.2 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.3 End of topic test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
8
2
TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
Learning objectives
By the end of this topic, you should be able to:
•
change the subject of a formula.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
1.1
3
Changing the subject of a formula
Rearranging formulae
Look at this formula: A = 3B
A is called the subject of the formula. The formula tells you how to calculate A when
you know B.
Reverse the process and B becomes the subject of the formula.
Look at this formula: A = B + 4
A is called the subject of the formula. The formula tells you how to calculate A when
you know B.
Reverse the process and B becomes the subject of the formula.
Look at this formula: C = aH + t
C is called the subject of the formula. The formula tells you how to calculate C when
you know a, H and t.
© H ERIOT-WATT U NIVERSITY
Go online
4
TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
Reverse the process and B becomes the subject of the formula.
..........................................
Examples
1.
Problem:
Make A the subject of the formula B = 3A + 2.
Solution:
Think what the formula does to A: A times 3 then add 2 gives B.
Reverse this: B minus 2 then divide by 3 gives A.
Write it as a formula: A = (B − 2) ÷ 3 or as A =
B−2
3
..........................................
2.
Problem:
The formula for the volume of a cylinder is V
Change the subject of the formula to r.
= π r 2 h.
Solution:
Think what the formula does to r: r squared times πh gives V .
Reverse this: V divided by πh then square root gives r.
V
Write it as a formula:r =
πh
..........................................
3.
Problem:
The formula to convert from Fahrenheit to Celsius is: C =
Change the subject of the formula to F .
5
9
(F − 32).
Solution:
Think what the formula does to F : F minus 32 multiplied by 5 /9 gives C.
Reverse this: C divided by 5 /9 then add 32 gives F .
© H ERIOT-WATT U NIVERSITY
TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
5
Remember when we divide by a fraction we change it to multiply by the reciprocal,
flipping the fraction over to become 9 /5 .
Therefore, ÷
5
9
becomes ×
Write it as a formula: F =
9
5.
9
5 C
+ 32
..........................................
4.
Problem:
a
b
=
1−g
1 + g,
change the subject of the formula to g.
Solution:
Sometimes it is easier to use the rules for rearranging an equation.
This is especially true when the subject we want appears more than once in the formula.
1 − g
a
=
multiply by (1 + g)
b
1 + g
a(1 + g)
= 1 − g
multiply by b
b
a(1 + g) = b(1 − g)
multiply out the brackets
a + ag = b − bg
a + ag + bg = b
add bg
subtract a
ag + bg = b − a
factorise the left hand side
g(a + b) = b − a
divide by (a + b)
b − a
g =
a + b
..........................................
Key point
Remember to re-arrange a step at a time and show all your working.
Changing the subject of a formula practice
Try the following questions on paper then check your answers.
Go online
Q1: Make F the subject of the formula E = 4F − 5
..........................................
Q2: Make h the subject of the formula V
=
1
2
bhl
..........................................
Q3: Make t the subject of the formula M =
1
2
k t2
..........................................
Q4: The formula to calculate your BMI (B) is B
and h is your height in metres.
=
w
h2
where w is your weight in kg
To find the height of someone when you know their weight and BMI, change the subject
of the formula to h.
..........................................
© H ERIOT-WATT U NIVERSITY
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TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
Changing the subject of a formula exercise
Key point
Go online
Remember to re-arrange a step at a time and show all your working.
Q5:
a) Make P the subject of the formula Q = 8P + 6
b) Make t the subject of the formula s = At + B
..........................................
Q6:
a) Make l the subject of the formula A =
b) Make a the subject of the formula C =
=
c) Make h the subject of the formula V
1
2 lh
a
m + b
1
2
3 πr h
..........................................
Q7:
a) Make H the subject of the formula Z =
=
b) Make r the subject of the formula V
c) Make b the subject of the formula d =
H2
R
4
2
3πr
√
ab + c
..........................................
Q8:
Make y the subject of the formula j y + k y = q
..........................................
Q9:
Make p the subject of the formula
a
p
=
b
p−1
..........................................
Q10: Make x the subject of the formula
x+5
2−x
=
y
z
..........................................
Q11: Height (h) of a rocket can be calculated using the formula h
where t is the time taken and v is the speed.
=
−16t 2 + vt,
To find the speed of the rocket, change the subject of the formula to v.
..........................................
Q12: In Physics, acceleration (a) is calculated using the formula a = v −t u , where v
is the final velocity (speed), u is the initial velocity (speed) and t is time taken (seconds).
To find the initial velocity, change the subject of the formula to u.
..........................................
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TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
1.2
Learning points
•
By reversing the operations performed in the original formula you can change the
subject.
•
The rules for solving equations can be applied.
•
Re-arrange a step at a time and show all working.
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TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
1.3
End of topic test
End of topic test
Key point
Go online
Remember to re-arrange a step at a time and show all your working.
Change the subject of these equations to the variable stated.
Q13:
a) x + a = b to x.
x
d
b)
= e to x.
c) g =
1
2
h x to x.
..........................................
Change the subject of these equations to the variable stated.
Q14:
a) A = π r 2 to r.
b) t =
c)
h2
=
d) k =
r2
s
to r.
m
r3
to r.
p2
+ r 2 to r.
..........................................
Change the subject of these equations to the variable stated.
Q15:
a) G = 2 (y − z) to y.
b) Q =
3
4
(y + v) to y.
..........................................
Q16: Make g the subject of the formula a g − b h = c g
..........................................
Q17: In Physics the formula for the magnetic force of a particle (F ) is
F =
mv 2
r
Change the subject of the formula to v.
..........................................
Q18: Make R the subject of the formula L =
k R2
3j
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. CHANGING THE SUBJECT OF A FORMULA
Q19: The number of rolls (R) of wallpaper required to paper a room can be calculated
ph
using the formula R =
0·53 (10 − r) + 1, where p is the perimeter of the room in
metres, h is the height of the room in metres and r is the length of the pattern repeat in
metres.
To find the length of the pattern repeat, change the subject of the formula to r.
..........................................
..........................................
© H ERIOT-WATT U NIVERSITY
9
10
ANSWERS: TOPIC 11
Answers to questions and activities
11 Changing the subject of a formula
Changing the subject of a formula practice (page 5)
Q1:
Think what the formula does to F : F times 4 minus 5 gives E.
Reverse this: E plus 5 all divided by 4 gives F .
Write it as a formula: F =
E+5
4
Q2:
Think what the formula does to h and reverse this.
Write it as a formula: h =
2V
bl
remember ÷
1
2
becomes ×
2
1
Q3:
Think what the formula does to t and reverse this.
2M
Write it as a formula: t =
k
Q4:
Think what the formula does to h and reverse this.
w
Write it as a formula: h =
B
Changing the subject of a formula exercise (page 6)
Q5:
a) P =
b) t =
Q−6
8
s−B
A
Q6:
a) l =
2A
h
b) a = m (C − b)
c) h =
3V
π r2
Q7:
a) H =
b) r =
c) b =
Q8:
y =
√
ZR
3V
4π
d2 − c
a
q
j+k
Q9:
Steps:
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ANSWERS: TOPIC 11
•
•
11
What do you get when you cross-multiply? a(p − 1) = bp
What do you get when you expand the brackets and collect all terms on the left
hand side of the equation? ap − a − bp = 0
Answer: p =
a
a−b
Q10: x =
2y − 5z
y+z
Q11: v =
h + 16t2
t
Q12: u = v − at
End of topic test (page 8)
Q13:
a) x = b − a
b) x = d e
c) x = 2hg
Q14:
a) r =
A
π
√
b) r =
st
h2 − p2
c) r =
d) r = 3 m
k
Q15:
a) y =
b) y =
Q16: g =
Q17: v =
Q18: R =
G
2 + z
4
3Q −
v
bh
(a − c)
Fr
m
3j L
k
Q19:
Hint:
ph
0 · 53 (10 − r)
ph
(10 − r) =
0 · 53 (R − 1)
ph
− 10
−r =
0 · 53 (R − 1)
R − 1 =
•
Answer: r = 10 −
ph
0·53 (R − 1)
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