Basic Mathematics for College Students

Find the LCD to add and subtract fractions.

When we add or subtract fractions that have different denominators, the least
common denominator is not always obvious. We can use a concept studied earlier to
determine the LCD for more difficult problems that involve larger denominators. To
illustrate this, let’s find the least common denominator of and (Note, the LCD is
not80.)
We have learned that both 8 and 10 must divide the LCD exactly. This divisibility
requirement should sound familiar. Recall the following fact from Section 1.8.

The Least Common Multiple (LCM)

The least common multiple (LCM)of two whole numbers is the smallest

whole number that is divisible by both of those numbers.

Thus, the least common denominator of and is simply the least common multiple
of 8 and 10.
We can find the LCM of 8 and 10 by listing multiples of the larger number, 10,
until we find one that is divisible by the smaller number, 8. (This method is explained
in Example 2 of Section 1.8.)

Multiples of 10: 10, 20, 30, 40 , 50, 60,...

This is the first multiple of 10 that

is divisible by 8 (no remainder).

Since the LCM of 8 and 10 is 40, it follows that the LCD of and is 40.
We can also find the LCM of 8 and 10 using prime factorization. We begin by
prime factoring 8 and 10. (This method is explained in Example 4 of Section 1.8.)

10  2 ~ 5

8  2  2  2

1

10

3

8



1

10

3

8

1

10.

3

8

3

3.4 Adding and Subtracting Fractions 247

Self Check 7

Add:

Now TryProblem 45

 6 

3

8

EXAMPLE 7

Add:

Strategy We will write as the fraction Then we will follow the steps for
adding fractions that have different denominators.

WHY The fractions and have different denominators.

Solution
Since the smallest number the denominators 1 and 4 divide exactly is 4, the LCD is 4.

Write 5 as

To build so that its denominator is 4, multiply it by a

form of 1.

Write the result with the sign in front:

This fraction is in simplest form.

 17

4 ^

17

  17 4.

4

Use the rule for adding two integers with different signs:

 (^)  20  3 17.

 17

4

Add the numerators and write the sum over the

 (^) common denominator 4.

 20  3

4

Multiply the numerators. Multiply the denominators.

 (^) The denominators are now the same.

 20

4



3

4

 5

 1

 5

1



4

4



3

4

 5

 5  1.

3

4



 5

1



3

4

3

4

 5

1

 5

 5 1.

 5 

3

4