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Add and Subtract Fractions with Common Denominators
Model fraction addition
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
\[\begin{array}{lllll}\frac{1}{4} & & \frac{2}{4} & & \frac{3}{4} \\ \text{one quarter} & + & \text{two quarters} & = & \text{three quarters}\end{array}\]Let’s use fraction circles to model the same example, \(\frac{1}{4}+\frac{2}{4}.\)
| Start with one \(\frac{1}{4}\) piece. | ||
| Add two more \(\frac{1}{4}\)pieces. | ||
| The result is \(\frac{3}{4}\). |
So again, we see that
\[\frac{1}{4}+\frac{2}{4}=\frac{3}{4}\]Example
Try it.
Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)
Solution
| Start with three \(\frac{1}{8}\) pieces. | ||
| Add two \(\frac{1}{8}\)pieces. | ||
| How many \(\frac{1}{8}\)pieces are there? |
There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)
Add Fractions with a Common Denominator
shows that to add the same-size pieces, meaning that the fractions have the same denominator, we just add the number of pieces.
Example
Try it.
Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)
Solution
| \(\frac{3}{5}+\frac{1}{5}\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{3+1}{5}\) |
| Simplify. | \(\frac{4}{5}\) |
Example
Try it.
Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)
Solution
| \(\frac{x}{3}+\frac{2}{3}\) | |
| 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
Try it.
Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)
Solution
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
Try it.
Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)
Solution
| \(\frac{2n}{11}+\frac{5n}{11}\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{2n+5n}{11}\) |
| Combine like terms. | \(\frac{7n}{11}\) |
Example
Try it.
Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)
Solution
| \(-\frac{3}{12}+(-\frac{5}{12})\) | |
| Add the numerators and place the sum over the common denominator. | \(\frac{-3+(-5)}{12}\) |
| Add. | \(\frac{-8}{12}\) |
| Simplify the fraction. | \(-\frac{2}{3}\) |
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 \(\frac{7}{12}\) of the pizza) left in the box. If Leonardo eats \(2\) of these remaining pieces (or \(\frac{2}{12}\) of the pizza), how much is left? There would be \(5\) pieces left (or \(\frac{5}{12}\) of the pizza).
\[\frac{7}{12}-\frac{2}{12}=\frac{5}{12}\]Let’s use fraction circles to model the same example, \(\frac{7}{12}-\frac{2}{12}.\)
Start with seven \(\frac{1}{12}\) pieces. Take away two \(\frac{1}{12}\) pieces. How many twelfths are left?
Again, we have five twelfths, \(\frac{5}{12}.\)
Example
Try it.
Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)
Solution
Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.
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.
Example
Try it.
Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)
Solution
| \(\frac{23}{24}-\frac{14}{24}\) | |
| 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
Try it.
Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)
Solution
| \(\frac{y}{6}-\frac{1}{6}\) | |
| 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
Try it.
Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)
Solution
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. | \(-\frac{14}{x}\) |
Now lets do an example that involves both addition and subtraction.
Example
Try it.
Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)
Solution
| \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\) | |
| 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. | \(-\frac{3}{8}\) |
Key Concepts
- Fraction Addition
- If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}+\frac{b}{c}=\frac{a+b}{c}\).
- To add fractions, add the numerators and place the sum over the common denominator.
- Fraction Subtraction
- If \(a,b,\) and \(c\) are numbers where \(c\ne 0\), then \(\frac{a}{c}-\frac{b}{c}=\frac{a-b}{c}\).
- To subtract fractions, subtract the numerators and place the difference over the common denominator.
Add and Subtract Fractions with Common Denominators
Model Fraction Addition
In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.
Try it.
\(\frac{2}{5}+\frac{1}{5}\)
Try it.
\(\frac{3}{10}+\frac{4}{10}\)
Solution
\(\frac{7}{10}\)
Try it.
\(\frac{1}{6}+\frac{3}{6}\)
Try it.
\(\frac{3}{8}+\frac{3}{8}\)
Solution
\(\frac{3}{4}\)
Add Fractions with a Common Denominator
In the following exercises, find each sum.
Try it.
\(\frac{4}{9}+\frac{1}{9}\)
Try it.
\(\frac{2}{9}+\frac{5}{9}\)
Solution
\(\frac{7}{9}\)
Try it.
\(\frac{6}{13}+\frac{7}{13}\)
Try it.
\(\frac{9}{15}+\frac{7}{15}\)
Solution
\(\frac{16}{15}\)
Try it.
\(\frac{x}{4}+\frac{3}{4}\)
Try it.
\(\frac{y}{3}+\frac{2}{3}\)
Solution
\(\frac{y+2}{3}\)
Try it.
\(\frac{7}{p}+\frac{9}{p}\)
Try it.
\(\frac{8}{q}+\frac{6}{q}\)
Solution
\(\frac{14}{q}\)
Try it.
\(\frac{8b}{9}+\frac{3b}{9}\)
Try it.
\(\frac{5a}{7}+\frac{4a}{7}\)
Solution
\(\frac{9a}{7}\)
Try it.
\(\frac{-12y}{8}+\frac{3y}{8}\)
Try it.
\(\frac{-11x}{5}+\frac{7x}{5}\)
Solution
\(\frac{-4x}{5}\)
Try it.
\(-\frac{1}{8}+(-\frac{3}{8})\)
Try it.
\(-\frac{1}{8}+(-\frac{5}{8})\)
Solution
\(-\frac{3}{4}\)
Try it.
\(-\frac{3}{16}+(-\frac{7}{16})\)
Try it.
\(-\frac{5}{16}+(-\frac{9}{16})\)
Solution
\(-\frac{7}{8}\)
Try it.
\(-\frac{8}{17}+\frac{15}{17}\)
Try it.
\(-\frac{9}{19}+\frac{17}{19}\)
Solution
\(\frac{8}{19}\)
Try it.
\(\frac{6}{13}+(-\frac{10}{13})+(-\frac{12}{13})\)
Try it.
\(\frac{5}{12}+(-\frac{7}{12})+(-\frac{11}{12})\)
Solution
\(-\frac{13}{12}\)
Model Fraction Subtraction
In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.
Try it.
\(\frac{5}{8}-\frac{2}{8}\)
Try it.
\(\frac{5}{6}-\frac{2}{6}\)
Solution
\(\frac{1}{2}\)
Subtract Fractions with a Common Denominator
In the following exercises, find the difference.
Try it.
\(\frac{4}{5}-\frac{1}{5}\)
Try it.
\(\frac{4}{5}-\frac{3}{5}\)
Solution
\(\frac{1}{5}\)
Try it.
\(\frac{11}{15}-\frac{7}{15}\)
Try it.
\(\frac{9}{13}-\frac{4}{13}\)
Solution
\(\frac{5}{13}\)
Try it.
\(\frac{11}{12}-\frac{5}{12}\)
Try it.
\(\frac{7}{12}-\frac{5}{12}\)
Solution
\(\frac{1}{6}\)
Try it.
\(\frac{4}{21}-\frac{19}{21}\)
Try it.
\(-\frac{8}{9}-\frac{16}{9}\)
Solution
\(-\frac{8}{3}\)
Try it.
\(\frac{y}{17}-\frac{9}{17}\)
Try it.
\(\frac{x}{19}-\frac{8}{19}\)
Solution
\(\frac{x-8}{19}\)
Try it.
\(\frac{5y}{8}-\frac{7}{8}\)
Try it.
\(\frac{11z}{13}-\frac{8}{13}\)
Solution
\(\frac{11z-8}{13}\)
Try it.
\(-\frac{8}{d}-\frac{3}{d}\)
Try it.
\(-\frac{7}{c}-\frac{7}{c}\)
Solution
\(-\frac{14}{c}\)
Try it.
\(-\frac{23}{u}-\frac{15}{u}\)
Try it.
\(-\frac{29}{v}-\frac{26}{v}\)
Solution
\(-\frac{55}{v}\)
Try it.
\(\frac{6c}{7}-\frac{5c}{7}\)
Try it.
\(\frac{12d}{11}-\frac{9d}{11}\)
Solution
\(\frac{3d}{11}\)
Try it.
\(\frac{-4r}{13}-\frac{5r}{13}\)
Try it.
\(\frac{-7s}{3}-\frac{7s}{3}\)
Solution
\(-\frac{14s}{3}\)
Try it.
\(-\frac{3}{5}-(-\frac{4}{5})\)
Try it.
\(-\frac{3}{7}-(-\frac{5}{7})\)
Solution
\(\frac{2}{7}\)
Try it.
\(-\frac{7}{9}-(-\frac{5}{9})\)
Try it.
\(-\frac{8}{11}-(-\frac{5}{11})\)
Solution
\(-\frac{3}{11}\)
Mixed Practice
In the following exercises, perform the indicated operation and write your answers in simplified form.
Try it.
\(-\frac{5}{18}\cdot \frac{9}{10}\)
Try it.
\(-\frac{3}{14}\cdot \frac{7}{12}\)
Solution
\(-\frac{1}{8}\)
Try it.
\(\frac{n}{5}-\frac{4}{5}\)
Try it.
\(\frac{6}{11}-\frac{s}{11}\)
Solution
\(\frac{6-s}{11}\)
Try it.
\(-\frac{7}{24}+\frac{2}{24}\)
Try it.
\(-\frac{5}{18}+\frac{1}{18}\)
Solution
\(-\frac{2}{9}\)
Try it.
\(\frac{8}{15}\div \frac{12}{5}\)
Try it.
\(\frac{7}{12}\div \frac{9}{28}\)
Solution
\(\frac{49}{27}\)
Condensed: the full section is in OpenStax Prealgebra 2e.
الآن أنت لا توجد آلة حسابية تقوم بتسوية هذا الحساب، ولكن أجزاءه قابلة للحساب. جرب أحد الحلول أدناه، أو أكتب حلك الخاص.
الممارسة (40)
حاول كل واحد على الورق أولاً. كشفت الإجابة للتحقق؛ يمكن فتح الإجابات المثبتة في الحل لكل خطوة.
-
Simplify: \(2x+9+3x-4.\)
كشفت الإجابة
\(5x+5\)
-
Draw a model of the fraction \(\frac{3}{4}.\)
-
Simplify: \(\frac{3+2}{6}.\)
كشفت الإجابة
\(\frac{5}{6}\)
-
Use a model to find the sum \(\frac{3}{8}+\frac{2}{8}.\)
كشفت الإجابة
Start with three \(\frac{1}{8}\) pieces. Add two \(\frac{1}{8}\)pieces. How many \(\frac{1}{8}\)pieces are there? There are five \(\frac{1}{8}\) pieces, or five-eighths. The model shows that \(\frac{3}{8}+\frac{2}{8}=\frac{5}{8}.\)
-
Use a model to find each sum. Show a diagram to illustrate your model.
\(\frac{1}{8}+\frac{4}{8}\)
كشفت الإجابة
\(\frac{5}{8}\)
-
Use a model to find each sum. Show a diagram to illustrate your model.
\(\frac{1}{6}+\frac{4}{6}\)
كشفت الإجابة
\(\frac{5}{6}\)
-
Find the sum: \(\frac{3}{5}+\frac{1}{5}.\)
كشفت الإجابة
\(\frac{3}{5}+\frac{1}{5}\) Add the numerators and place the sum over the common denominator. \(\frac{3+1}{5}\) Simplify. \(\frac{4}{5}\) -
Find each sum: \(\frac{3}{6}+\frac{2}{6}.\)
كشفت الإجابة
\(\frac{5}{6}\)
-
Find each sum: \(\frac{3}{10}+\frac{7}{10}.\)
كشفت الإجابة
1
-
Find the sum: \(\frac{x}{3}+\frac{2}{3}.\)
كشفت الإجابة
\(\frac{x}{3}+\frac{2}{3}\) 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.
-
Find the sum: \(\frac{x}{4}+\frac{3}{4}.\)
كشفت الإجابة
\(\frac{x+3}{4}\)
-
Find the sum: \(\frac{y}{8}+\frac{5}{8}.\)
كشفت الإجابة
\(\frac{y+5}{8}\)
-
Find the sum: \(-\frac{9}{d}+\frac{3}{d}.\)
كشفت الإجابة
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. \(-\frac{6}{d}\) -
Find the sum: \(-\frac{7}{d}+\frac{8}{d}.\)
كشفت الإجابة
\(\frac{1}{d}\)
-
Find the sum: \(-\frac{6}{m}+\frac{9}{m}.\)
كشفت الإجابة
\(\frac{3}{m}\)
-
Find the sum: \(\frac{2n}{11}+\frac{5n}{11}.\)
كشفت الإجابة
\(\frac{2n}{11}+\frac{5n}{11}\) Add the numerators and place the sum over the common denominator. \(\frac{2n+5n}{11}\) Combine like terms. \(\frac{7n}{11}\) -
Find the sum: \(\frac{3p}{8}+\frac{6p}{8}.\)
كشفت الإجابة
\(\frac{9p}{8}\)
-
Find the sum: \(\frac{2q}{5}+\frac{7q}{5}.\)
كشفت الإجابة
\(\frac{9q}{5}\)
-
Find the sum: \(-\frac{3}{12}+(-\frac{5}{12}).\)
كشفت الإجابة
\(-\frac{3}{12}+(-\frac{5}{12})\) Add the numerators and place the sum over the common denominator. \(\frac{-3+(-5)}{12}\) Add. \(\frac{-8}{12}\) Simplify the fraction. \(-\frac{2}{3}\) -
Find each sum: \(-\frac{4}{15}+(-\frac{6}{15}).\)
كشفت الإجابة
\(-\frac{2}{3}\)
-
Find each sum: \(-\frac{5}{21}+(-\frac{9}{21}).\)
كشفت الإجابة
\(-\frac{2}{3}\)
-
Use fraction circles to find the difference: \(\frac{4}{5}-\frac{1}{5}.\)
كشفت الإجابة
Start with four \(\frac{1}{5}\) pieces. Take away one \(\frac{1}{5}\) piece. Count how many fifths are left. There are three \(\frac{1}{5}\) pieces left.
-
Use a model to find each difference. Show a diagram to illustrate your model.
\(\frac{7}{8}-\frac{4}{8}\)
كشفت الإجابة
\(\frac{3}{8}\), models may differ.
-
Use a model to find each difference. Show a diagram to illustrate your model.
\(\frac{5}{6}-\frac{4}{6}\)
كشفت الإجابة
\(\frac{1}{6}\), models may differ
-
Find the difference: \(\frac{23}{24}-\frac{14}{24}.\)
كشفت الإجابة
\(\frac{23}{24}-\frac{14}{24}\) 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. \(\frac{3}{8}\) -
Find the difference: \(\frac{19}{28}-\frac{7}{28}.\)
كشفت الإجابة
\(\frac{3}{7}\)
-
Find the difference: \(\frac{27}{32}-\frac{11}{32}.\)
كشفت الإجابة
\(\frac{1}{2}\)
-
Find the difference: \(\frac{y}{6}-\frac{1}{6}.\)
كشفت الإجابة
\(\frac{y}{6}-\frac{1}{6}\) 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.
-
Find the difference: \(\frac{x}{7}-\frac{2}{7}.\)
كشفت الإجابة
\(\frac{x-2}{7}\)
-
Find the difference: \(\frac{y}{14}-\frac{13}{14}.\)
كشفت الإجابة
\(\frac{y-13}{14}\)
-
Find the difference: \(-\frac{10}{x}-\frac{4}{x}.\)
كشفت الإجابة
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. \(-\frac{14}{x}\) -
Find the difference: \(-\frac{9}{x}-\frac{7}{x}.\)
كشفت الإجابة
\(-\frac{16}{x}\)
-
Find the difference: \(-\frac{17}{a}-\frac{5}{a}.\)
كشفت الإجابة
\(-\frac{22}{a}\)
-
Simplify: \(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}.\)
كشفت الإجابة
\(\frac{3}{8}+(-\frac{5}{8})-\frac{1}{8}\) 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. \(-\frac{3}{8}\) -
Simplify: \(\frac{2}{5}+(-\frac{4}{5})-\frac{3}{5}.\)
كشفت الإجابة
−1
-
Simplify: \(\frac{5}{9}+(-\frac{4}{9})-\frac{7}{9}.\)
كشفت الإجابة
\(-\frac{2}{3}\)
-
\(\frac{2}{5}+\frac{1}{5}\)
-
\(\frac{3}{10}+\frac{4}{10}\)
كشفت الإجابة
\(\frac{7}{10}\)
-
\(\frac{1}{6}+\frac{3}{6}\)
-
\(\frac{3}{8}+\frac{3}{8}\)
كشفت الإجابة
\(\frac{3}{4}\)
الرموز المستخدمة هنا
لمس أي رمز للحصول على التعريف الكامل، والصورة، وما يعنيه كل حرف في ذلك.
كيف: Add and Subtract Fractions with Common Denominators
- Model fraction addition
- Add fractions with a common denominator
- Model fraction subtraction
- Subtract fractions with a common denominator
- If
- To add fractions, add the numerators and place the sum over the common denominator.
- If
- To subtract fractions, subtract the numerators and place the difference over the common denominator.
أسئلة يطرحها الناس
Why does the order of operations matter?
Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.
How do I check an arithmetic answer?
Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.
Why are fractions harder than decimals?
They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.
أجزاء من هذه الصفحة مستمدة من OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). لقد تم تلخيصها وإعادة شرحها هنا؛ والأخطاء هي أخطائنا. وبما أن مصدره مرخص للاستخدام غير التجاري، فإن هذا الدرس مجاني، ولا يبيع شيئاً، ويجري تقاسمه بموجب نفس الترخيص.