Add and Subtract Fractions with Common Denominators
| Site: | Saylor Academy |
| Course: | RWM101: Foundations of Real World Math copy 1 |
| Book: | Add and Subtract Fractions with Common Denominators |
| Printed by: | Guest user |
| Date: | Wednesday, May 14, 2025, 7:39 AM |
Description
buttons.Add and Subtract Fractions with Common Denominators
Read this text. Pay special attention to the sections on fraction addition and subtraction. They provide an overview of how to add and subtract fractions with the same denominator. Complete the practice questions and check your answers.
Model Fraction Addition
How many quarters are pictured? One quarter plus 2 quarters equals 3 quarters.
Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that
\( \begin{array}{ccc} \dfrac{1}{4} & \dfrac{2}{4} & \dfrac{3}{4} \\ \text { one quarter +} & \text { two quarters =} & \text { three quarters } \end{array} \)
Let's use fraction circles to model the same example, \(\dfrac{1}{4}+\dfrac{2}{4}\).
| Start with one \(\dfrac{1}{4}\) piece. |  |
\(\dfrac{1}{4}\) |
| Add two more \(\dfrac{1}{4}\) pieces. |  |
\( \begin{align} +\dfrac{2}{4} \\ \text{___} \end{align} \) |
| The result is \(\dfrac{3}{4}\). |  |
\(\dfrac{3}{4}\) |
So again, we see that
\(\dfrac{1}{4}+\dfrac{2}{4}=\dfrac{3}{4}\)
Example 4.52
Use a model to find the sum \(\frac{3}{8} + \frac{2}{8}\).
Solution
| Start with one three \(\dfrac{1}{8}\) pieces. |  |
\(\dfrac{3}{8}\) |
| Add two \(\dfrac{1}{8}\) pieces. |  |
\(\dfrac{2}{8}\) |
| How many \(\dfrac{1}{8}\) pieces are there? |  |
\(\dfrac{5}{8}\) |
Add Fractions with a Common Denominator
Example 4.52 shows that to add the same-size pieces - meaning that the fractions have the same denominator - we just add the number of pieces.Fraction Addition
If \(a\), \(b\), and \(c\) are numbers where \(c≠0\), then
\(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\)
To add fractions with a common denominators, add the numerators and place the sum over the common denominator.
Model Fraction Subtraction
Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 12 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or \(\dfrac{7}{12}\) of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or \(\dfrac{2}{12}\) of the pizza), how much is left? There would be 5 pieces left (or \(\dfrac{5}{12}\) of the pizza).
\(\dfrac{7}{12}-\dfrac{2}{12}=\dfrac{5}{12}\)
Let's use fraction circles to model the same example, \(\dfrac{7}{12}-\dfrac{2}{12}\).
Start with seven \(\dfrac{1}{12}\) pieces. Take away two \(\dfrac{1}{12}\) pieces. How many twelfths are left?
Again, we have five twelfths, \(\dfrac{5}{12}\).
Subtract Fractions with a Common Denominator
We subtract fractions with a common denominator in much the same way as we add fractions with a common denominator.
FRACTION SUBTRACTION
If \(a, b\), and \(c\) are numbers where \(c \neq 0\), then
\(\dfrac{a}{c}-\dfrac{b}{c}=\dfrac{a-b}{c}\)
To subtract fractions with common denominators, we subtract the numerators and place the difference over the common denominator.
Source: Rice University, https://openstax.org/books/prealgebra/pages/4-4-add-and-subtract-fractions-with-common-denominators
This work is licensed under a Creative Commons Attribution 4.0 License.
Examples
EXAMPLE 4.53
Find the sum: \(\dfrac{3}{5}+\dfrac{1}{5}\).
EXAMPLE 4.54
Find the sum: \(\dfrac{x}{3}+\dfrac{2}{3}\).
EXAMPLE 4.55
Find the sum: \(-\dfrac{9}{d}+\dfrac{3}{d}\).
EXAMPLE 4.56
Find the sum: \(\dfrac{2 n}{11}+\dfrac{5 n}{11}\).
EXAMPLE 4.57
Find the sum: \(-\dfrac{3}{12}+\left(-\dfrac{5}{12}\right)\).
EXAMPLE 4.59
Find the difference: \(\dfrac{23}{24}-\dfrac{14}{24}\).
EXAMPLE 4.60
Find the difference: \(\dfrac{y}{6}-\dfrac{1}{6}\).
EXAMPLE 4.61
Find the difference: \(-\dfrac{10}{x}-\dfrac{4}{x}\).
EXAMPLE 4.62
Simplify: \(\dfrac{3}{8}+\left(-\dfrac{5}{8}\right)-\dfrac{1}{8}\).
Answers
EXAMPLE 4.53
| \(\frac{3}{5}+\frac{1}{5}\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{3+1}{5}\) |
| Simplify. | \(\frac{4}{5}\) |
EXAMPLE 4.54
| \(\frac{x}{3}+\frac{2}{3}\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{x+2}{3}\) |
Note that we cannot simplify this fraction any more. Since \(x\) and \(2\) are not like terms, we cannot combine them.
EXAMPLE 4.55
We will begin by rewriting the first fraction with the negative sign in the numerator.
\(-\frac{a}{b}=\frac{-a}{b}\)
| \(-\frac{9}{d}+\frac{3}{d}\) | |
| Rewrite the first fraction with the negative in the numerator. | \(\frac{-9}{d}+\frac{3}{d}\) |
| Add the numerators and place the sum over the common denominator. | \(\frac{-9+3}{d}\) |
| Simplify the numerator. | \(\frac{-6}{d}\) |
| Rewrite with negative sign in front of the fraction. | \(-\frac{6}{d}\) |
EXAMPLE 4.56
| \(\frac{2 n}{11}+\frac{5 n}{11}\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{2 n+5 n}{11}\) |
| Combine like terms. | \(\frac{7 n}{11}\) |
EXAMPLE 4.57
| \(-\frac{3}{12}+\left(-\frac{5}{12}\right)\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{-3+(-5)}{12}\) |
| Add. | \(\frac{-8}{12}\) |
| Simplify the fraction. | \(-\frac{2}{3}\) |
EXAMPLE 4.59
| \(\frac{23}{24}-\frac{14}{24}\) | |
| Subtract the numerators and place the difference over the common denominator. | \(\frac{23-14}{24}\) |
| Simplify the numerator. | \(\frac{9}{24}\) |
| Simplify the fraction by removing common factors. | \(\frac{3}{8}\) |
Example 4.60
| \(\frac{y}{6}-\frac{1}{6}\) | |
| Subtract the numerators and place the difference over the common denominator. | \(\frac{y-1}{6}\) |
The fraction is simplified because we cannot combine the terms in the numerator.
Example 4.61
Remember, the fraction \(-\frac{10}{x}\) can be written as \(\frac{-10}{x}\).
| \(-\frac{10}{x}-\frac{4}{x}\) | |
| Subtract the numerators. | \(\frac{-10-4}{x}\) |
| Simplify. | \(\frac{-14}{x}\) |
| Rewrite with the negative sign in front of the fraction. | \(-\frac{14}{x}\) |
Example 4.62
| \(\frac{3}{8}+\left(-\frac{5}{8}\right)-\frac{1}{8}\) | |
| Combine the numerators over the common denominator. | \(\frac{3+(-5)-1}{8}\) |
| Simplify the numerator, working left to right. | \(\frac{-2-1}{8}\) |
| Subtract the terms in the numerator. | \(\frac{-3}{8}\) |
| Rewrite with the negative sign in front of the fraction. | \(-\frac{3}{8}\) |