2.4 Arithmetic Operations with Fractions
Least or Lowest Common Denominator (LCD)
The Least Common Denominator (LCD) of a set of two or more fractions is the smallest whole number divisible by each denominator. It is the least common multiple (LCM) of the denominators of the fractions. Two methods of finding the LCM are explained in Chapter 1, Section 1.3.
In performing addition and subtraction of fractions, it is necessary to determine equivalent fractions with common denominators. The LCD is the best choice for a common denominator because it makes further simplification easier.
Example 2.4-a: Determining the Least Common Denominator
Determine the LCD of
Solution
The LCD of the fractions
Using one of the methods from Chapter 1, Section 1.3:
- The largest number, 15, is not divisible by 9.
- Multiples of 15 are: 15, 30, 45, …
- 45 is divisible by 9.
- Thus, the LCM of 9 and 15 is 45.
Therefore, the LCD of
Comparing Fractions
Fractions can easily be compared when they have the same denominator. If they do not have the same denominator, first determine the LCD of the fractions and convert them into equivalent fractions with the LCD as the denominator.
Example 2.4-b: Comparing Fractions
Which of the fractions,
Solution
Step 1: Since the fractions do not have the same denominator, we need to first determine the LCD of the fractions, which is the same as the LCM of the denominators. The LCM of 12 and 8 is 24.
Step 2: Convert each fraction to its equivalent fraction with 24 as the denominator.
To convert
5 portions of 12 equal parts of a whole are equal to 10 portions of 24 equal parts of that whole.
Similarly, convert
3 portions of 8 equal parts of a whole is equal to 9 portions of 24 equal parts of that whole.
Step 3: Since the denominators are the same, we can compare the fractions’ numerators to identify the greater fraction.
Since 10 > 9, it implies:
Therefore,
Addition of Fractions
The denominator of a fraction indicates the number of parts into which an item is divided. Therefore, adding fractions requires that every fraction’s denominators be the same. If the denominators are different, they must first be made the same by determining the LCD and changing each fraction to its equivalent fraction with the LCD as the denominator.
When the fractions have the same denominator, the numerators of each of the fractions may be added. The numerator of the resulting fraction is equal to this sum, and the denominator is equal to the common denominator of the fractions being added.
Express the final answer reduced to the lowest terms and as a mixed number, where applicable.
Example 2.4-c: Adding Fractions that Have the Same Denominator
Add
Solution
The denominators of the fractions are the same. Adding the numerators and keeping
the common denominator,
Example 2.4-d: Adding Fractions that have Different Denominators
Add
Solution
LCM of 4 and 3 is 12 (i.e., LCD = 12). Determining the equivalent fractions with a denominator of 12,
Adding the numerators and keeping the common denominator,
Converting the improper fraction to a mixed number,
Therefore, the result from adding
Example 2.4-e: Adding a Mixed Number and a Proper Fraction
Add
Solution
Method 1:
Converting the mixed number to an improper fraction,
LCM of 6 and 9 is 18 (i.e., LCD = 18). Determining the equivalent fractions with a denominator of 18,
Adding the numerators and keeping the common denominator,
Converting the improper fraction to a mixed number,
Method 2:
Separating the whole number and the fractions,
LCM of 6 and 9 is 18 (i.e., LCD = 18). Determining the equivalent fractions with a denominator of 18,
Adding the numerators and keeping the common denominator,
Converting the improper fraction to a mixed number,
Adding the whole numbers and then the fraction,
Therefore, the result from adding
Example 2.4-f: Adding Mixed Numbers
Add:
and and
Solution
-
LCM of 6 and 4 is 12 (i.e., LCD = 12). Determining the equivalent mixed numbers with a denominator of 12,
Separating the whole numbers and the fractions,
Adding the whole numbers and the fractions,
Therefore, the result from adding
and is . -
LCM of 3 and 5 is 15 (i.e., LCD = 15). Determining the equivalent mixed numbers with a denominator of 15,
Separating the whole numbers and the fractions,
Adding the whole numbers and the fractions,
Converting the improper fraction to a mixed number,
Adding the whole numbers and then the fraction,
Therefore, the result from adding
and is .
Subtraction of Fractions
The process for the subtraction of fractions is the same as that of the addition of fractions. First, determine a common denominator and change each fraction to its equivalent fraction with the common denominator. When the fractions have the same denominator, the numerators of the fractions may be subtracted. The numerator of the resulting fraction is equal to this difference, and the denominator is equal to the common denominator of the fractions being subtracted.
Express the final answer reduced to the lowest terms and as a mixed number, where applicable.
Example 2.4-g: Subtracting Fractions that have the Same Denonimator
Subtract
Solution
The denominators of the fractions are the same. Subtracting the numerators and keeping the common denominator,
Reducing to the lowest terms,
Therefore, the result from subtracting
Example 2.4-h: Subtracting Fractions that Have Different Denominators
Subtract
Solution
LCM of 8 and 10 is 40 (i.e., LCD = 40). Determining the equivalent fractions with a denominator of 40,
Subtracting the numerators and keeping the common denominator,
Reducing to the lowest terms,
Therefore, the result from subtracting
Example 2.4-i: Subtracting Mixed Numbers
Subtract
Solution
Method 1:
Converting the mixed numbers to improper fractions,
LCM of 2 and 3 is 6 (i.e., LCD = 6). Determining the equivalent fractions with a denominator of 6,
Subtracting the numerators and keeping the common denominator,
Converting the improper fraction to a mixed number,
Method 2:
LCM of 2 and 3 is 6 (i.e., LCD = 6). Determining the equivalent mixed numbers with a denominator of 6,
The fraction
Subtracting the whole numbers and then the fractions,
Therefore, the result from subtracting
Multiplication of Fractions
First, convert any mixed number to its improper fraction to multiply two or more fractions. Then, multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.
When multiplying fractions, you can reduce any numerator term with any denominator term. Reduce as much as possible before multiplying the numerators and the denominators together to keep the numbers as simple as possible.
Express the final answer reduced to the lowest terms and as a mixed number, where applicable.
Note: When multiplying mixed numbers, it is incorrect to multiply the whole number parts separately from the fractional parts to arrive at the answer.
Example 2.4-j: Multiplying Fractions
Multiply:
Solution
-
Reducing the fractions,
Multiplying the numerators together and denominators together,
Therefore, the result of
is . -
Reducing the fractions,
Multiplying the numerators together and denominators together,
Therefore, the result of
is 6. -
Converting the mixed numbers to improper fractions,
Reducing the fractions,
Multiplying the numerators together and denominators together,
Converting the improper fraction to a mixed number,
Therefore, the result of
is
Division of Fractions
When dividing fractions, as in multiplication, first convert any mixed number to its improper fraction. The division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction. Then, follow the procedure used in multiplication to arrive at the final result.
Express the final answer reduced to the lowest terms and as a mixed number, where applicable.
Note:
Dividing by 2 is the same as multiplying by the reciprocal of 2, which is
When multiplying or dividing mixed numbers, it is incorrect to multiply or divide the whole number parts separately from the fractional parts to arrive at the answer.
Example 2.4-k: Dividing Fractions
Divide
Solution
Multiplying
Reducing the fractions,
Multiplying the numerators together and denominators together,
Converting the improper fraction to a mixed number,
Therefore, the result of
Example 2.4-l: Dividing Mixed Numbers
Divide
Solution
Converting the mixed numbers to improper fractions,
Multiplying
Reducing the fractions,
Converting the improper fraction to a mixed number,
Therefore, the result of
Complex Fractions
A complex fraction is a fraction in which one or more fractions are found in the numerator or denominator.
For example,
is a complex fraction because it has a fraction in the denominator. is a complex fraction because it has a fraction in the numerator. is a complex fraction because it has two fractions in the numerator. is a complex fraction because it has a fraction in both the numerator and the denominator.
A complex fraction can be simplified by dividing the numerator by the denominator and then, following the rule(s) for dividing fractions.
Example 2.4-m: Simplifying Complex Fractions
Add:
Solution
Powers and Square Roots of Fractions
Powers of fractions are expressed the same way as whole numbers. When the base of a power is a fraction, it is written within brackets.
For example,
- This means that
is used as a factor two times. - i.e.,
A mixed number that is raised to a power is evaluated by first converting it into an improper fraction and then following the same procedure explained above.
For example,
Example 2.4-n: Evaluating Powers of Fractions
Evaluate the following powers:
Solution
-
Expanding by using
as a factor four times, -
Converting the mixed number into an improper fraction,
Expanding by using
as a factor five times,Converting back to a mixed number,
Square roots of fractions are calculated the same way as square roots of whole numbers, but the numerators and denominators are evaluated separately.
For example,
Example 2.4-o: Evaluating Square Roots of Fractions
Evaluate the square root:
Solution
Determining the square root of the numerator and denominator separately,
2.4 Exercises
Answers to the odd-numbered problems are available at the end of the textbook.
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- Peter spent
of his money on rent and on food. What fraction of his money did he spend on rent and food? - Alan walked
km to his friend’s house and from there, he walked another km to his school. How far did Alan walk? - Last night, Amy spent
hours on her math project and hours on her design project. How much time did she spend on both projects altogether? - A bag contains
kg of red beans and kg of green beans. What is the total weight of the bag? - Thomas baked a
pound cake. He gave pounds of it to his friend Yan. How much was left? - Alexander bought
litres of milk and drank litres of it. How much milk was left? - Sarah had
kg of cheese. She used kg of the cheese while baking. How many kilograms of cheese was left? - Cassidy bought
litres of olive oil and used litre of the oil while cooking. What quantity of olive oil was left? - David spent
of his money on toys and of the remainder on food. What fraction of his money was spent on food? - Mary spent
of her money on a school bag. She then spent of the remainder on shoes. What fraction of her money was spent on shoes? - After selling
of its textbooks, a bookstore had 810 books left. How many textbooks were in the bookstore initially? - Rose travelled
of her journey by car and the remaining 20 km by bus. How far did she travel by car? - Cheng can walk
km in hours. How many kilometres can he walk in 1 hour? litres of juice weighs kg. Determine the weight (in kilograms) of 1 litre of juice.- A chain of length
metres is cut into pieces measuring metres each. How many pieces are there? - A cake that weighs
kg is cut into slices weighing kg each. How many slices are there? - A bottle of medicine contains 80 mg of medicine. Each dose of the medicine is
mg. How many doses are there in the bottle? - A box of cereal contains 917 grams of cereal. How many bowls of cereal will there be if each serving is
grams? - Out of 320 bulbs,
of the bulbs are defective. How many of them are not defective? - If
of the 1,800 students in a school enrolled for a mathematics course, how many students did not enroll for the course? - The product of two numbers is 9. If one number is
, what is the other number? - If a wire that is
cm long is cut into several cm equal pieces, how many pieces would exist?
Unless otherwise indicated, this chapter is an adaptation of the eTextbook Foundations of Mathematics (3rd ed.) by Thambyrajah Kugathasan, published by Vretta-Lyryx Inc., with permission. Adaptations include supplementing existing material and reordering chapters.