SCHOLAR Study Guide
National 5 Mathematics
Course Materials
Topic 12: Algebraic fractions
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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Distributed by the SCHOLAR Forum.
SCHOLAR Study Guide Course Materials Topic 12: National 5 Mathematics
1. National 5 Mathematics Course Code: C747 75
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1
Topic 1
Algebraic fractions
Contents
12.1 Simplifying algebraic fractions . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
12.2 Adding, subtracting, multiplying and dividing algebraic fractions . . . . . . . .
12.3 Learning points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
15
12.4 End of Topic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
2
TOPIC 1. ALGEBRAIC FRACTIONS
Learning objectives
By the end of this topic, you should be able to:
•
simplify algebraic fractions;
•
apply the four operations (+, −, ×, ÷) to algebraic fractions.
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
1.1
3
Simplifying algebraic fractions
Any number or expression divided by itself equals 1. Let’s have a look at some of these.
1
= 1,
1
2
= 1,
2
x
= 1,
x
3x
= 1,
3x
4x + 5
= 1,
4x + 5
10
=1
10
x+1
=1
x+1
1 − 2x
= 1,
1 − 2x
3 − 4x
=1
3 − 4x
Simplifying algebraic fractions
The following activity will take you through a variety of fractions and show you how they
can be reduced to their simplest form using the fact that any expression divided by itself
is equal to 1.
..........................................
Examples
1.
Problem:
Simplify
3a
6a
Solution:
3 a
3a
=
×
6a
6 a
1
×1
=
2
1
=
2
..........................................
2.
Problem:
Simplify
2gh2
6gh
© H ERIOT-WATT U NIVERSITY
Go online
4
TOPIC 1. ALGEBRAIC FRACTIONS
Solution:
2
g
h
h
2gh2
=
×
×
×
6gh
6
g
h
1
1
h
=
× 1 × 1 ×
3
1
h
=
3
..........................................
Simplifying algebraic fractions practice
Q1:
Simplify
4b
8b
Go online
..........................................
Q2:
Simplify
cd
2bc
..........................................
Q3:
Simplify
2e2
4e
..........................................
Q4:
Simplify
3pq 2
9pq
..........................................
Simplifying by finding simple common factors
Examples
1.
Problem:
Simplify the fraction by finding simple common factors first.
3a−12
6
Solution:
3(a − 4)
3a − 12
=
6
6
a − 4
3
×
=
6
1
a − 4
1
×
=
2
1
a − 4
=
2
..........................................
2.
Problem:
Simplify the fraction by finding simple common factors first.
2g+2h
6g+6h
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
5
Solution:
2(g + h)
2g + 2h
=
6g + 6h
6(g + h)
2 g+h
×
=
6 g+h
1
×1
=
3
1
=
3
..........................................
Simplifying algebraic fractions: Finding simple common factors practice
Simplify the fractions by finding simple common factors first.
Go online
Q5:
4b−16
8
..........................................
Q6:
c2 +cd
2c
..........................................
Q7:
10
5e−20
..........................................
Q8:
3p+3q
12p+12q
..........................................
Simplifying by factorising the difference of two squares
Example
Problem:
Simplify this algebraic fraction by factorising the difference of two squares first.
a2 −25
a−5
Solution:
(a − 5)(a + 5)
a2 − 25
=
a−5
a−5
a−5 a+5
×
=
a−5
1
a+5
= 1×
1
= a+5
..........................................
© H ERIOT-WATT U NIVERSITY
6
TOPIC 1. ALGEBRAIC FRACTIONS
Examples
1.
Problem:
Simplify this algebraic fraction by factorising the trinomials first.
e2 −e−6
e2 +4e−21
Solution:
(e − 3)(e + 2)
e2 − e − 6
=
2
e + 4e − 21
(e − 3)(e + 7)
e+2
e−3
×
=
e−3
e+7
e+2
= 1 ×
e+7
e+2
=
e+7
..........................................
2.
Problem:
Simplify this algebraic fraction by factorising the difference of two squares and the
trinomial first.
f 2 −4
f 2 +5f +6
Solution:
(f + 2)(f − 2)
f2 − 4
=
2
f + 5f + 6
(f + 2)(f + 3)
f +2 f −2
×
=
f +2 f +3
f −2
= 1×
f +3
f −2
=
f +3
..........................................
Simplifying algebraic fractions: Factorising the difference of two squares
practice
Go online
Simplify these algebraic fractions by factorising the difference of two squares and the
trinomial first.
Q9:
a2 −9
a−3
..........................................
Q10:
b2 −c2
b+c
..........................................
Q11:
d2 +3d+2
d+1
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
Q12:
7
e2 −2e−8
e2 −5e−14
..........................................
Q13:
f 2 −16
f 2 −2f −8
..........................................
Simplifying algebraic fractions exercise
These questions are for practice only.
Simplify these fractions.
Q14:
3a
3b
..........................................
Q15:
cd
bd
..........................................
Q16:
4e
2e2
..........................................
Q17:
6g 2 h
2gh2
..........................................
Q18:
8m−4
12
..........................................
Q19:
18
9n−12
..........................................
Q20:
t2 +5t
3t
..........................................
Q21:
v2 −v
2v
..........................................
Q22:
(6x−4)(x−2)
(x−1)(x−2) ,
x = 1, 2
..........................................
Q23:
(4x−6)(x+3)
,
(x+3)2
x = −3
..........................................
Q24:
2a+12
a+6
..........................................
Q25:
b2 +bc
b+c
..........................................
Q26:
g 2 −9
g+3
..........................................
© H ERIOT-WATT U NIVERSITY
Go online
8
TOPIC 1. ALGEBRAIC FRACTIONS
Q27:
h2 −2h+1
5h−5
..........................................
Q28:
k 2 +7k−8
(k−1)2
..........................................
Q29:
x2 +x−6
x2 −x−2
x = 3, 2
..........................................
1.2
Adding, subtracting, multiplying and dividing algebraic
fractions
A fraction is made of two parts. The bottom part called the denominator which tells you
what is being counted and the top part called the numerator which tells you the count.
For example, look at the fraction 3 /4 . The denominator tells me that I am counting
quarters and the numerator tells me that there are 3 of them.
We can make a fraction look different by multiplying by 1. For example:
3
4
=
3
4
×1=
3
4
×
2
2
=
6
8
We can use this to make the denominator any value we want.
We can even do this in algebra.
3
x
=
3
x
×1=
3
x
×
y
y
=
3y
xy
We often do this when adding or subtracting fractions to make a common denominator.
Adding and subtracting fractions
This activity shows how to add by finding a common denominator. . .
Go online
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
. . .and how to subtract algebraic fractions by finding a common denominator.
..........................................
Examples
1.
Problem:
Simplify
10
a
−
3
a
Solution:
Both fractions already have the same denominator.
10
a
−
3
a
=
10−3
a
=
7
a
..........................................
2.
Problem:
Simplify
8
p
+
4
q
Solution:
8 q 4 p
8 4
+
=
× + ×
p q
p q q p
8q 4p
+
=
pq pq
8q + 4p
=
pq
..........................................
3.
Problem:
Simplify
a+1
3
+
a−1
4
Solution:
The common denominator for 3 and 4 is 12 so
4(a + 1) 3(a − 1)
a+1 a−1
+
=
+
3
4
4×3
3×4
4a + 4 3a − 3
+
=
12
12
7a + 1
=
12
..........................................
© H ERIOT-WATT U NIVERSITY
9
10
TOPIC 1. ALGEBRAIC FRACTIONS
4.
Problem:
Simplify
a+1
3
−
a−1
4
Solution:
This question is almost the same so still has a common denominator of 12.
a+1 a−1
4(a + 1) 3(a − 1)
−
=
−
3
4
4×3
3×4
4a + 4 3a − 3
−
=
12
12
We must watch out here because 3a and −3 are both being subtracted so we must
calculate 4a − 3a and 4 − (−3) giving
=
a+7
12
..........................................
5.
Problem:
Simplify
3
b+3
+
2
b−1
Solution:
The common denominator for b + 3 and b − 1 is (b + 3)(b − 1) so
2
3(b − 1)
2(b + 3)
3
+
=
+
b+3 b−1
(b + 3)(b − 1)
(b + 3)(b − 1)
2b + 6
3b − 3
+
=
(b + 3)(b − 1)
(b + 3)(b − 1)
5b + 3
=
(b + 3)(b − 1)
There is no need to multiply out the brackets on the denominator.
..........................................
Adding and subtracting algebraic fractions practice
Add and subtract by finding a common denominator to simplify these algebraic fractions.
Go online
Q30:
Simplify
6
x
+
4
x
..........................................
Q31:
Simplify
7
m
−
2
n
..........................................
Q32: Simplify
2a+3
4
+
a−1
2
..........................................
Q33: Simplify
b+5
2
−
3b−1
5
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
Q34: Simplify
7
b+3
−
11
2
b
..........................................
Multiplying fractions
This activity looks at multiplying fractions.
Go online
..........................................
Key point
To multiply fractions we simply multiply the numerators together and multiply the
denominators together.
Examples
1.
Problem:
Simplify
a
4
×
3b
d,
d = 0
Solution:
a × 3b
a 3b
×
=
4
d
4×d
3ab
=
4d
..........................................
2.
Problem:
Simplify
a2
6
×
2
a
Solution:
a2 × 2
=
6×a
2a2
=
6a
Sometimes the solution can be simplified further.
2
1
a2
because
=
=
3a
6
3
=
a
3
because
a2
a a
a
a
= × =1× = =a
a
a 1
1
1
..........................................
© H ERIOT-WATT U NIVERSITY
12
TOPIC 1. ALGEBRAIC FRACTIONS
Multiplying algebraic fractions practice
Go online
Multiply the numerators together and multiply the denominators together to simplify
these algebraic fractions.
Q35: Simplify
x
5
×
2y
z ,
z = 0
..........................................
Q36: Simplify
2b3
3
×
9
4b
..........................................
Dividing fractions
The final activity in this group shows you how to divide fractions.
Go online
..........................................
Key point
To divide by a fraction change the operation to multiplication and flip the second
fraction i.e. Multiply by the reciprocal of the second fraction.
Examples
1.
Problem:
Simplify
a
4
÷
3b
d
Solution:
d
a
= ×
4 3b
a×d
=
4 × 3b
ad
=
12b
..........................................
2.
Problem:
Simplify
2a2
3b2
÷
a2
b
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
Solution:
b
2a2
= 2 × 2
3b
a
2a2 b
= 2 2
3a b
2
because
=
3b
13
a2
b
1
= 1 and
=
2
2
a
b
b
..........................................
Dividing algebraic fractions practice
Divide the fraction by changing the operation to multiplication and flipping the second
fraction to simplify these algebraic fractions.
Q37: Simplify
x
5
2y
z
÷
Go online
y, z = 0
..........................................
Q38: Simplify
5x2
2y
÷
10x
y
..........................................
Adding, subtracting, multiplying and dividing algebraic fractions exercise
These questions are for practice only.
Go online
Q39: Simplify
k
3
+
m
2
..........................................
Q40: Simplify
9
x
5
x
+
..........................................
Q41: Simplify
2
p
7
q
+
..........................................
Q42: Simplify
3g+1
4
+
2g−1
5
..........................................
Q43: Simplify
3
x+1
+
2
x+2
..........................................
Q44: Simplify
8
y
−
3
y
..........................................
Q45: Simplify
7
p
−
6
q
..........................................
Q46: Simplify
1
a
−
2
b
..........................................
Q47: Simplify
3b+2
3
−
b+1
2
..........................................
© H ERIOT-WATT U NIVERSITY
14
TOPIC 1. ALGEBRAIC FRACTIONS
Q48: Simplify
5
y−1
−
4
y+3
..........................................
Q49: Simplify
3
a
×
8b
c
..........................................
Q50: Simplify
×
d
2
3d
4
..........................................
Q51: Simplify
×
e
3f
6f
5
..........................................
Q52: Simplify
g2
2
10
g
×
..........................................
Q53: Simplify
÷
a
10
a
5
..........................................
Q54: Simplify
f
6
÷
2g
h
÷
r2
t2
..........................................
Q55: Simplify
r
t
..........................................
Q56: Simplify
2x
6y
÷
2x3
3y 2
..........................................
© H ERIOT-WATT U NIVERSITY
TOPIC 1. ALGEBRAIC FRACTIONS
1.3
Learning points
Simplifying algebraic fractions means reducing the numerical fraction to it’s simplest
form as well as the algebraic part e.g.
2
x
3
1
x
1
2 = 1 and x = 1, 6 = 2 and x2 = x .
Simplifying algebraic fractions may require factorising first.
Before adding and subtracting algebraic fractions you must have a common denominator
e.g.
2y+3x
2
3
x + y = xy
Multiplying fractions is the simplest, you multiply the numerators together then multiply
the denominators together
2x
x
2x2
3y × 5 = 15y
To divide by a fraction change the operation to multiplication and flip the second fraction.
ie. Multiply by the reciprocal of the second fraction.
4x
x
4x
3
12x
12
y ÷ 3 = y × x = xy = y
Simplify the answer if necessary.
© H ERIOT-WATT U NIVERSITY
15
16
TOPIC 1. ALGEBRAIC FRACTIONS
1.4
End of Topic Test
End of topic 12 test
Q57: Simplify:
Go online
a)
b)
c)
d)
e)
25a2
5ab
4d+6
12
f 2 +3f
3f +9
(6x−2)(x−5)
(x−5)2
2
g +7g−18
g 2 −4
..........................................
Q58: Simplify:
a)
b)
c)
d)
e)
a
2b
2 + 3
10
9
x + x
3
6
p + q
c+2
2c−3
4 + 5
5
6
d−1 + d+2
..........................................
Q59: Simplify:
a)
b)
c)
d)
e)
7
3
a − 2b
10
8
y − y
8
2
p − q
2c−3
c+1
4 − 3
4
1
e−3 − e+5
..........................................
Q60: Simplify:
a)
b)
c)
d)
12
2b
a × c
ab
a
2 × 3
d
5e
3e × 2
2
3h
12
4 × h
..........................................
Q61: Simplify:
a)
b)
c)
d)
a
a
42 ÷ 7
f
2g
6 ÷ h
b2
3b
2c ÷ c2
2e
4e2
5f ÷ 15f 2
..........................................
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 12
17
Answers to questions and activities
12 Algebraic fractions
Simplifying algebraic fractions practice (page 4)
Q1:
4b
8b
4
8
Q2:
cd
2bc
=
c
c
×
d
2b
Q3:
2e2
4e
=
2
4
×
e
e
Q4:
3pq 2
9pq
=
×
=
3
9
1
2
=
b
b
×
=1×
×
p
p
1
2
×1=
×
1
2
=
e
1
q
q
=
d
2b
×1×
q
1
×
d
2b
=
1
3
=
e
1
e
2
×1×1×
q
1
q
3
=
Simplifying algebraic fractions: Finding simple common factors practice (page 5)
Q5:
4b−16
8
=
4(b−4)
8
=
4
8
×
b−4
1
=
Q6:
c2 +cd
2c
=
c(c+d)
2c
=
c
c
×
c+d
2
=1×
Q7:
10
5e−20
=
10
5(e−4)
=
10
5
×
1
e−4
Q8:
3p+3q
12p+12q
3(p+q)
12(p+q)
=
3
12
×
=
1
2
×
2
1
=
p+q
p+q
b−4
1
=
b−4
2
c+d
2
=
c+d
2
×
=
1
4
1
e−4
=
2
e−4
×1=
1
4
Simplifying algebraic fractions: Factorising the difference of two squares
practice (page 6)
(a−3)(a+3)
a−3
=
a−3
a−3
×
a+3
1
(b+c)(b−c)
b+c
=
b+c
b+c
×
b−c
1
(d+1)(d+2)
d+1
=
d+1
d+1
×
d+2
1
(e+2)(e−4)
(e+2)(e−7)
=
e+2
e+2
×
e−4
e−7
=1×
e−4
e−7
(f −4)(f +4)
(f −4)(f +2)
=
f −4
f −4
×
f +4
f +2
=1×
f +4
f +2
Q9:
a2 −9
a−3
Q10:
b2 −c2
b+c
Q11:
d2 +3d+2
d+1
Q12:
e2 −2e−8
e2 −5e−14
=
Q13:
f 2 −16
f 2 −2f −8
=
=
=
=
=1×
=1×
=a+3
a+3
1
b−c
1
= b−c
=1×
= d+2
d+2
1
=
=
Simplifying algebraic fractions exercise (page 7)
Q14:
a
b
Q15:
c
b
Q16:
2
e
Q17:
3g
h
Q18:
2m−1
3
Q19:
6
3n−4
© H ERIOT-WATT U NIVERSITY
e−4
e−7
f +4
f +2
18
ANSWERS: TOPIC 12
Q20:
t+5
3
Q21:
v−1
2
Q22:
6x−4
x−1
Q23:
4x−6
x+3
Q24: 2
Q25: b
Q26: g − 3
Q27:
h−1
5
Q28:
k+8
k−1
Q29:
x+3
x+1
Adding and subtracting algebraic fractions practice (page 10)
Q30:
6
x
+
4
x
=
6+4
x
=
10
x
Q31:
2
7
n 2 m
7
− =
× − ×
m n
m n n m
2m
7n
−
=
mn mn
7n − 2m
=
mn
Q32:
The common denominator for 4 and 2 is 4 so only the second fraction needs to be
multiplied.
2a + 3 2 (a − 1)
+
=
4
2×2
2a + 3 2a − 2
+
=
4
4
4a + 1
=
4
Q33:
5 (b + 5) 2 (3b − 1)
−
2×5
2×5
5b + 25 6b − 2
remember 25 − (−2) = 27.
−
=
10
10
−b + 27
=
10
=
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 12
19
Q34:
The common denominator for b + 3 and b is b(b + 3).
2 (b + 3)
b×7
−
b (b + 3) b (b + 3)
2b + 6
7b
−
=
b (b + 3) b (b + 3)
5b − 6
=
b (b + 3)
=
remember the 2b and the 6 are being subtracted.
Multiplying algebraic fractions practice (page 12)
Q35:
x × 2y
x 2y
×
=
5
z
5×z
2xy
=
5z
Q36:
2b3 × 9
3 × 4b
18b3
=
12b
3b3
because
=
2b
3b2
because
=
2
=
18
3
=
12
2
b3
b b2
b2
= ×
=1×
= b2
b
b
1
1
Dividing algebraic fractions practice (page 13)
Q37:
z
x
= ×
5 2y
xz
=
10y
Q38:
5x2
y
5x2 10x
÷
=
×
2y
y
2y
10x
2
5x y
=
20xy
x2 y
because
=
4xy
x
because
=
4
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5
1
=
20
4
y
= 1 and
y
x2
x
=
x
1
20
ANSWERS: TOPIC 12
Adding, subtracting, multiplying and dividing algebraic fractions exercise (page
13)
Q39:
2k+3m
6
Q40:
14
x
Q41:
2q+7p
pq
Q42:
23g+1
20
Q43:
5x+8
(x+1)(x+2)
Q44:
5
y
Q45:
7q−6p
pq
Q46:
b−2a
ab
Q47:
3b+1
6
Q48:
y+19
(y−1)(y+3)
Q49:
24b
ac
Q50:
3d2
8
Q51:
2e
5
Q52: 5g
Q53:
1
2
Q54:
fh
12g
Q55:
t
r
Q56:
y
2x2
End of topic 12 test (page 16)
Q57:
a)
b)
c)
d)
e)
5a
b
2d+3
6
f
3
6x−2
x−5
g+9
g+2
Q58:
a)
3a+4b
6
© H ERIOT-WATT U NIVERSITY
ANSWERS: TOPIC 12
b)
c)
d)
e)
19
x
3q+6p
pq
13c−2
20
11d+4
(d−1)(d+2)
Q59:
a)
b)
c)
d)
e)
14b−3a
2ab
2
y
8q−2p
pq
2c−13
12
3e+23
(e−3)(e+5)
Q60:
a)
b)
c)
24b
ac
a2 b
6
5d
6
d) 9h
Q61:
a)
b)
c)
d)
1
6
fh
12g
bc
6
3f
2e2
© H ERIOT-WATT U NIVERSITY
21