Fractions and Linear Equations
Fraction Operations
While you can perform operations on fractions using the calculator, for this worksheet you must perform the
operations by hand. You must show all steps for full credit, but you are encouraged to check your answers
using the calculator.
 Reducing:
A fraction is reduced if the numerator and denominator have no common factors. You can reduce a fraction by
dividing both numerator and denominator by the same value until they have no factors in common.
Example:
Consider the fraction 18/12. Dividing both 18 and 12 by the common factor 2 results in the fraction 9/6.
Dividing both 9 and 6 by the common factor 3 results in the fraction 3/2. Because the values 3 and 2 have no
factors in common, the fraction has been reduced to lowest terms:
18 9 2
= =
12 6 3
 Addition/Subtraction:
To add or subtract fractions, they must have the same denominator. If two fractions do not have a common
denominator, this is achieved by multiplying both numerator and denominator by the same number, which does
not change the value of the fraction.
Example:
To add the fractions 1/2 and 1/5, we must first find a common multiple of the denominators 2 and 5. The
smallest, or least, common multiple is the value 10. To obtain a common denominator, both numerator and
denominator of 1/2 are multiplied by 5 and both numerator and denominator of 1/5 are multiplied by 2. The
fractions are then added by combining the numerators. A pictorial representation of the sum is also provided
below.
1 5 1 2
5
2
7
∗ + ∗ =
+
=
2 5 5 2 10 10 10
Subtraction is similar:
1 5 1 2
5
2
3
∗ − ∗ =
−
=
2 5 5 2 10 10 10
Note:
To show your work in computing the sum 1/2 + 1/5 online, type 1/2 + 1/5 = 5/10 + 2/10 = 7/10.
If you include an additional step, keep in mind the proper use of parentheses:
2
26
1/2 + 1/5 = 5/10 + 2/10 = (5 + 2)/10 = 7/10. The expression 5 + 2/10 would be 5 + 10 = 5 .
 Multiplication:
To multiply two fractions, the numerators are multiplied together and the denominators are multiplied together.
Keep in mind that the denominator of an integer is an understood 1.
Examples:
2 4 2∗4
8
∗ =
=
3 5 3 ∗ 5 15
3∗
4
3 4
3∗4
12 4
= ∗
=
=
=
27 1 27 1 ∗ 27 27 9
 Division:
In order to divide by a fraction, multiply by its reciprocal. That is, reverse the numerator and denominator of
the fraction by which you are dividing and change the operation from division to multiplication.
Examples:
2 4 2 5 10 5
÷ = ∗ =
=
3 5 3 4 12 6
3÷
4
3 27 81
= ∗
=
27 1 4
4
Linear Equations
At this time only equations in one variable will be addressed. A linear equation is one in which the variable is
raised only to the first power. To solve a linear equation, isolate the variable on one side of the equation by
performing the same operation (addition, subtraction, multiplication, or division) to both sides of the equation.
Note:
 You have likely been solving linear equations since grade school, where you were asked to write into the
box the missing number: 3 +  = 5. This is no different than solving the linear equation 3 + x = 5.
 When dealing with variable expressions, remember that you can only combine like terms, which contain the
same variable(s) raised to the same powers. For example, 2x + 3x = 5x, but 2 + 3x cannot be simplified
further.
Example 1:
To solve the equation 2 = 9x – 12x for x, first combine the like terms on the right-hand side of the equation and
then divide both sides by -3 to isolate the x.
2 = 9𝑥 − 12𝑥
2 = −3𝑥
2
−3𝑥
=
−3
−3
2
− =𝑥
3
Note:
 When dealing with negative fractions, the negative sign can be placed in front of the fraction or either in
2
−2
2
the numerator or denominator: − 3 = 3 = −3.
 You can verify that your answer is correct by replacing the variable in the original equation with the
value found: 2 = 9(-2/3) – 12(-2/3) → 2 = -6 + 8. Because this is a true equation, the solution x = -2/3 is
correct.
Example 2:
To solve the equation below, we can start by multiplying all terms on both sides by the denominator 2 to
eliminate the fraction.
9𝑥
= 4 − 11𝑥
2
9𝑥
2 ∗ ( ) = 2 ∗ 4 − 2 ∗ 11𝑥
2
9𝑥 = 8 − 22𝑥
9𝑥 + 22𝑥 = 8 − 22𝑥 + 22𝑥
31𝑥 = 8
31𝑥
8
=
31
31
8
𝑥=
31
Example 3:
If two fractions are equal to each other, then you can “cross multiply” to eliminate the fractions. That is,
multiply the denominator of each fraction by the numerator of the other and set the products equal. In the
solution of the equation below, this is equivalent to multiplying both sides by the common denominator, 18.
Note: To show your work in solving the above equation online, you could type:
3x(9) = 2(4x – 5)
27x = 8x – 10
19x = -10
x = -10/19
Example 4:
If either side of the equation contains another term, cross multiplication cannot be used. In the following
example, the fractions are eliminated by multiplying through by the common denominator, 7.
𝑏 + 10
2𝑏
+1=
7
7
𝑏 + 10
2𝑏
7∗(
)+7∗1= 7∗( )
7
7
𝑏 + 10 + 7 = 2𝑏
17 = 𝑏
Fractions and Linear Equations
Examples
(Solutions provided at bottom of page)
Reduce each fraction to lowest terms
1.
99
2.
108
21
3.
28
18
33
Perform each fraction operation by hand. Give the answer in fractional form reduced to lowest terms.
4.
5.
6.
7.
3
5
+2
4
3
5
−2
4
3
∗
4
3
5
2
5
÷2
4
Solve each equation. Give non-integer solutions in fractional form reduced to lowest terms.
8. 4 + 10 = 6 + x
9. 12(x + 4) + 1 = 5x - 2
10. 12 =
11.
12.
13.
4𝑥
5
𝑥−1
9
+ 2 = 4(𝑥 − 10)
𝑥+8
5
=
8(𝑥+7)
12
7𝑥
8
=
− 10
3(𝑥−11)
8
14. 6(x + 4) + 12 = 8x - 4
15.
4𝑥
10
+ 3 = 10(𝑥 − 6)
Solutions:
1.
4.
7.
10.
13.
11/12
3/4 + 10/4 = 13/4
3/4 * 2/5 = 6/20 = 3/10
x = 109
x = -211/7
2.
5.
8.
11.
14.
3/4
3/4 – 10/4 = -7/4
x=8
x = 105/8
x = 20
3.
6.
9.
12.
15.
6/11
15/8
x = -51/7
x = 464/27
x = 105/16
Worksheet 5
Fractions and Linear Equations
1. Reduce each fraction to lowest terms.
16
36
a. 24
b. 90
c.
4
d.
24
45
63
e.
5
40
Perform each fraction operation by hand. You must show your work for full credit. Give the answer in
fractional form reduced to lowest terms.
2.
4.
4
3
2
+5
4
3.
2
∗
3 5
5.
4
2
3
−5
4
2
÷5
3
Solve each equation. Give non-integer solutions in fractional form reduced to lowest terms.
6.
𝑥−9
4
7. 8x – 5 = 9 + 3(x + 7)
=3
8. 8(x + 1) + 10x = 9
10.
5(𝑥+3)
2
=
8(𝑥−4)
3
9.
𝑥+9
2
−3 =
11𝑥
5