4.4 Add and Subtract Fractions with Common Denominators

Learning objectives

Learning objectives

• Model fraction addition

• Add fractions with a common denominator

• Model fraction subtraction

• Subtract fractions with a common denominator

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

Let’s use fraction circles to model the same example, $\frac{1}{4}+\frac{2}{4}.$

So again, we see that

Example 4.52

Solution

1. Identify the unit pieces. The fractions $\frac{3}{8}$ and $\frac{2}{8}$ both count eighths, so they can be combined in one circle divided into eight equal segments.
2. Combine the counts. Three eighths and two eighths make $3+2=5$ eighth-size pieces, so the sum is $\frac{5}{8}$.
3. Interpret the model. The figures show five of the eight equal sectors shaded, together with the labels $\frac{3}{8}$, $\frac{2}{8}$, and $\frac{5}{8}$, confirming that the shaded amount is five eighths.
4. State the result. Therefore, $\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$.

Takeaway: When denominators match, add the numerators and keep the common denominator, then simplify if the numerator and denominator share a factor.

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.

Example 4.53

Solution

1. Match the denominators. Both $\frac{3}{5}$ and $\frac{1}{5}$ use fifths, so they represent portions of the same size and can be combined directly.
2. Add the numerators. Keeping the common denominator $5$, add $3+1$ to obtain a numerator of $4$ and write the result as $\frac{4}{5}$.
3. Check for simplification. The numerator $4$ and denominator $5$ have no common factor greater than $1$, so the fraction is already in simplest form.
4. State the sum. Therefore, $\frac{3}{5}+\frac{1}{5}=\frac{4}{5}$.

Takeaway: When denominators are identical, add only the numerators and retain the shared denominator, simplifying the result if possible.

Example 4.54

Solution

1. Identify the common denominator. The fractions $\frac{x}{3}$ and $\frac{2}{3}$ both have denominator $3$, so the denominator does not change in the sum.
2. Add the numerators. Combine the terms above the shared denominator to form $x+2$, giving the intermediate result $\frac{x+2}{3}$.
3. Check the numerator. The terms $x$ and $2$ are unlike terms, so they cannot be combined further, and no common factor can be removed from the whole fraction.
4. State the sum. Therefore, $\frac{x}{3}+\frac{2}{3}=\frac{x+2}{3}$.

Takeaway: With equal denominators, add the numerators and combine only like terms, leaving unlike terms unchanged in the result.

Example 4.55

Solution

1. Rewrite the negative fraction. Move the negative sign from the first fraction into its numerator, so $-\frac{9}{d}+\frac{3}{d}$ becomes $\frac{-9}{d}+\frac{3}{d}$.
2. Keep the denominator fixed. Both fractions already have denominator $d$, so add their signed numerators and retain $d$ in the denominator.
3. Evaluate the numerator. Adding $-9$ and $3$ gives $-6$, so the combined fraction is $\frac{-6}{d}$.
4. Record the sign. Rewrite the negative numerator in the conventional form $-\frac{6}{d}$, which is the requested sum.

Takeaway: Place a negative sign in the numerator before adding equal-denominator fractions, then simplify the signed numerator.

Example 4.56

Solution

1. Confirm the common denominator. Both $\frac{2n}{11}$ and $\frac{5n}{11}$ have denominator $11$, so their numerators can be added without changing the denominator.
2. Add the numerator terms. Form the numerator $2n+5n$ over $11$, giving the intermediate fraction $\frac{2n+5n}{11}$.
3. Combine like terms. The coefficients of the matching variable term $n$ add to $7$, so the numerator simplifies to $7n$.
4. State the result. The final fraction is $\frac{7n}{11}$, and no further common factor can be removed from it.

Takeaway: For equal denominators, add the numerators and combine matching variable terms before checking whether the fraction reduces.

Example 4.57

Solution

1. Place both signs in the numerators. Rewrite $-\frac{3}{12}+(-\frac{5}{12})$ as $\frac{-3}{12}+\frac{-5}{12}$ so each term has a signed numerator.
2. Keep the common denominator. Since both denominators are $12$, add the numerators while retaining $12$ as the denominator.
3. Calculate the signed sum. Adding $-3$ and $-5$ gives $-8$, so the combined fraction is $\frac{-8}{12}$.
4. Reduce the fraction. Divide numerator and denominator by their greatest common factor $4$ to obtain $-\frac{2}{3}$, the simplest form.

Takeaway: Add signed numerators over a common denominator, then divide both parts by their greatest common factor to simplify.

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).

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 4.58

Solution

1. Model the minuend. Draw a circle divided into five equal sectors and shade four of them to represent $\frac{4}{5}$.
2. Remove the subtrahend. Take away one of the five equal sectors to represent the $\frac{1}{5}$ being subtracted.
3. Count what remains. Four shaded sectors minus one shaded sector leaves three equal sectors, so the difference is $\frac{3}{5}$.
4. State the result. Therefore, $\frac{4}{5}-\frac{1}{5}=\frac{3}{5}$.

Takeaway: Subtracting fractions with a common denominator subtracts the numerators while keeping the shared denominator unchanged.

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 4.59

Solution

1. Recognize the common denominator. Both fractions have denominator $24$, so subtract their numerators and keep $24$ unchanged in the denominator.
2. Subtract the numerators. Computing $23-14$ gives $9$, so the difference first appears as $\frac{9}{24}$.
3. Find the greatest common factor. The numbers $9$ and $24$ share the factor $3$, so divide both numerator and denominator by $3$.
4. State the reduced result. The numerator becomes $3$ and the denominator becomes $8$, giving $\frac{3}{8}$.

Takeaway: Subtract numerators when denominators match, then reduce the result by dividing both parts by their greatest common factor.

Example 4.60

Solution

1. Use the shared denominator. The fractions $\frac{y}{6}$ and $\frac{1}{6}$ have the same denominator $6$, so the denominator stays $6$ in their difference.
2. Subtract the numerators. Form the numerator $y-1$, which gives the fraction $\frac{y-1}{6}$.
3. Check for further simplification. The variable term $y$ and the constant term $1$ cannot be combined, so there is no common factor to remove.
4. State the difference. Therefore, $\frac{y}{6}-\frac{1}{6}=\frac{y-1}{6}$.

Takeaway: Subtract the numerators of equal-denominator fractions, and simplify only when the resulting numerator has a removable factor.

Example 4.61

Solution

1. Rewrite the first sign. Express $-\frac{10}{x}$ as $\frac{-10}{x}$, so both fractions in the difference have denominator $x$ and signed numerators.
2. Subtract the numerators. Keeping the denominator $x$, compute the numerator as $-10-4$ to obtain $-14$.
3. Form the difference. Place that signed numerator over the common denominator to write the result as $\frac{-14}{x}$.
4. Record the final sign. Rewrite the fraction as $-\frac{14}{x}$, with no additional cancellation possible because $14$ and $x$ need not share a factor.

Takeaway: Move the negative sign into the numerator before subtracting equal-denominator fractions, then simplify if a common factor exists.

Now lets do an example that involves both addition and subtraction.

Example 4.62

Solution

1. Collect the signed numerators. Because every denominator is $8$, combine the terms as $\frac{3+(-5)-1}{8}$ without changing the shared denominator.
2. Work from left to right. First compute $3+(-5)=-2$, so the numerator temporarily becomes $\frac{-2-1}{8}$.
3. Finish the numerator. Subtract $1$ from $-2$ to obtain $-3$, giving the fraction $\frac{-3}{8}$.
4. Check and state the result. The numerator and denominator have no common factor, so the simplified expression is $-\frac{3}{8}$.

Takeaway: Combine signed numerators over an identical denominator from left to right, then place the final negative sign outside.

Section 4.4 Exercises

Section 4.4 Exercises

Model Fraction Addition

In the following exercises, use a model to add the fractions. Show a diagram to illustrate your model.

Add Fractions with a Common Denominator

In the following exercises, find each sum.

Model Fraction Subtraction

In the following exercises, use a model to subtract the fractions. Show a diagram to illustrate your model.

Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

Mixed Practice

In the following exercises, perform the indicated operation and write your answers in simplified form.

Everyday Math

Writing Exercises

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

1. Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{8}+\frac{4}{8}$

2. Use a model to find each sum. Show a diagram to illustrate your model.

$\frac{1}{6}+\frac{4}{6}$

3. Find each sum: $\frac{3}{6}+\frac{2}{6}.$

4. Find the sum: $\frac{x}{4}+\frac{3}{4}.$

1. Find the sum: $\frac{y}{8}+\frac{5}{8}.$

2. Find the sum: $-\frac{7}{d}+\frac{8}{d}.$

3. Find the sum: $-\frac{6}{m}+\frac{9}{m}.$

4. Find the sum: $\frac{3p}{8}+\frac{6p}{8}.$

1. Find the sum: $\frac{2q}{5}+\frac{7q}{5}.$

2. Find each sum: $-\frac{4}{15}+(-\frac{6}{15}).$

3. Find each sum: $-\frac{5}{21}+(-\frac{9}{21}).$

4. Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{7}{8}-\frac{4}{8}$

1. Use a model to find each difference. Show a diagram to illustrate your model.

$\frac{5}{6}-\frac{4}{6}$

2. Find the difference: $\frac{19}{28}-\frac{7}{28}.$

3. Find the difference: $\frac{27}{32}-\frac{11}{32}.$

4. Find the difference: $\frac{x}{7}-\frac{2}{7}.$

1. Find the difference: $\frac{y}{14}-\frac{13}{14}.$

2. Find the difference: $-\frac{9}{x}-\frac{7}{x}.$

3. Find the difference: $-\frac{17}{a}-\frac{5}{a}.$

4. Simplify: $\frac{2}{5}+(-\frac{4}{5})-\frac{3}{5}.$

1. Simplify: $\frac{5}{9}+(-\frac{4}{9})-\frac{7}{9}.$

2. $\frac{3}{10}+\frac{4}{10}$

3. $\frac{3}{8}+\frac{3}{8}$

4. $\frac{2}{9}+\frac{5}{9}$

1. $\frac{9}{15}+\frac{7}{15}$

2. $\frac{y}{3}+\frac{2}{3}$

3. $\frac{8}{q}+\frac{6}{q}$

4. $\frac{5a}{7}+\frac{4a}{7}$

1. $\frac{-11x}{5}+\frac{7x}{5}$

2. $-\frac{1}{8}+(-\frac{5}{8})$

3. $-\frac{5}{16}+(-\frac{9}{16})$

4. $-\frac{9}{19}+\frac{17}{19}$

1. $\frac{5}{12}+(-\frac{7}{12})+(-\frac{11}{12})$

2. $\frac{5}{6}-\frac{2}{6}$

3. $\frac{4}{5}-\frac{3}{5}$

4. $\frac{9}{13}-\frac{4}{13}$

1. $\frac{7}{12}-\frac{5}{12}$

2. $-\frac{8}{9}-\frac{16}{9}$

3. $\frac{x}{19}-\frac{8}{19}$

4. $\frac{11z}{13}-\frac{8}{13}$

1. $-\frac{8}{d}-\frac{3}{d}$

2. $-\frac{7}{c}-\frac{7}{c}$

3. $-\frac{29}{v}-\frac{26}{v}$

4. $\frac{12d}{11}-\frac{9d}{11}$

1. $\frac{-7s}{3}-\frac{7s}{3}$

2. $-\frac{3}{7}-(-\frac{5}{7})$

3. $-\frac{8}{11}-(-\frac{5}{11})$

4. $-\frac{3}{14}\cdot\frac{7}{12}$

1. $\frac{6}{11}-\frac{s}{11}$

2. $-\frac{5}{18}+\frac{1}{18}$

3. $\frac{7}{12}\mathrm{÷}\frac{9}{28}$

4. Baking Janet needs $\frac{5}{8}$ of a cup of flour for a recipe she is making. She only has $\frac{3}{8}$ of a cup of flour and will ask to borrow the rest from her next-door neighbor. How much flour does she have to borrow?

1. After a party, Lupe has $\frac{5}{12}$ of a cheese pizza, $\frac{4}{12}$ of a pepperoni pizza, and $\frac{4}{12}$ of a veggie pizza left. Will all the slices fit into $1$ pizza box? Explain your reasoning.