# 3.5: Add and Subtract Fractions with Common Denominators - Mathematics

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Skills to Develop

• Add fractions with a common denominator
• Model fraction subtraction
• Subtract fractions with a common denominator

be prepared!

Before you get started, take this readiness quiz.

1. Simplify: (2x + 9 + 3x − 4). If you missed this problem, review Example 2.2.10.
2. Draw a model of the fraction (dfrac{3}{4}). If you missed this problem, review Example 4.1.2.
3. Simplify: (dfrac{3 + 2}{6}). If you missed this problem, review Example 4.3.12.

How many quarters are pictured? One quarter plus (2) quarters equals (3) quarters.

Figure (PageIndex{1})

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, (dfrac{1}{4} + dfrac{2}{4}).

So again, we see that

[dfrac{1}{4} + dfrac{2}{4} = dfrac{3}{4} onumber ]

Use a model to find the sum (dfrac{3}{8} + dfrac{2}{8}).

Solution

There are five (dfrac{1}{8}) pieces, or five-eighths. The model shows that (dfrac{3}{8} + dfrac{2}{8} = dfrac{5}{8}).

Exercise (PageIndex{1})

Use a model to find each sum. Show a diagram to illustrate your model. [dfrac{1}{8} + dfrac{4}{8} onumber ]

(dfrac{5}{8})

Exercise (PageIndex{2})

Use a model to find each sum. [dfrac{1}{6} + dfrac{4}{6} onumber ]

(dfrac{5}{6})

## Add Fractions with a Common Denominator

Example (PageIndex{1}) shows that to add the same-size pieces—meaning that the fractions have the same denominator—we just add the number of pieces.

If (a), (b), and (c) are numbers where (c ≠ 0), then

[dfrac{a}{c} + dfrac{b}{c} = dfrac{a + b}{c}]

To add fractions with a common denominator, add the numerators and place the sum over the common denominator.

Find the sum: (dfrac{3}{5} + dfrac{1}{5}).

Solution

 Add the numerators and place the sum over the common denominator. (dfrac{3 + 1}{5}) Simplify. (dfrac{4}{5})

Exercise (PageIndex{3})

Find each sum: (dfrac{3}{6} + dfrac{2}{6}).

(dfrac{5}{6})

Exercise (PageIndex{4})

Find each sum: (dfrac{3}{10} + dfrac{7}{10}).

(1)

Find the sum: (dfrac{x}{3} + dfrac{2}{3}).

Solution

 Add the numerators and place the sum over the common denominator. (dfrac{x + 2}{3})

Note that we cannot simplify this fraction any more. Since (x) and (2) are not like terms, we cannot combine them.

Exercise (PageIndex{5})

Find the sum: (dfrac{x}{4} + dfrac{3}{4}).

(dfrac{x+3}{4})

Exercise (PageIndex{6})

Find the sum: (dfrac{y}{8} + dfrac{5}{8}).

(dfrac{y+5}{8})

Find the sum: (− dfrac{9}{d} + dfrac{3}{d}).

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

[− dfrac{a}{b} = dfrac{−a}{b} onumber ]

 Rewrite the first fraction with the negative in the numerator. (dfrac{-9}{d} + dfrac{3}{d}) Add the numerators and place the sum over the common denominator. (dfrac{-9 + 3}{d}) Simplify the numerator. (dfrac{-6}{d}) Rewrite with negative sign in front of the fraction. (- dfrac{6}{d})

Exercise (PageIndex{7})

Find the sum: (− dfrac{7}{d} + dfrac{8}{d}).

(dfrac{1}{d})

Exercise (PageIndex{8})

Find the sum: (− dfrac{6}{m} + dfrac{9}{m}).

(dfrac{3}{m})

Find the sum: (dfrac{2n}{11} + dfrac{5n}{11}).

Solution

 Add the numerators and place the sum over the common denominator. (dfrac{2n + 5n}{11}) Combine like terms. (dfrac{7n}{11})

Exercise (PageIndex{9})

Find the sum: (dfrac{3p}{8} + dfrac{6p}{8}).

(dfrac{9p}{8})

Exercise (PageIndex{10})

Find the sum: (dfrac{2q}{5} + dfrac{7q}{5}).

(dfrac{9q}{5})

Find the sum: (− dfrac{3}{12} + left(− dfrac{5}{12} ight)).

Solution

 Add the numerators and place the sum over the common denominator. (dfrac{-3 + (-5)}{12}) Add. (dfrac{-8}{12}) Simplify the fraction. (-dfrac{2}{3})

Exercise (PageIndex{11})

Find each sum: (− dfrac{4}{15} + left(− dfrac{6}{15} ight)).

(-dfrac{2}{3})

Exercise (PageIndex{12})

Find each sum: (− dfrac{5}{21} + left(− dfrac{9}{21} ight)).

(-dfrac{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 (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} onumber ]

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?

Figure (PageIndex{2})

Again, we have five twelfths, (dfrac{5}{12}).

Example (PageIndex{7}): difference

Use fraction circles to find the difference: (dfrac{4}{5} − dfrac{1}{5}).

Solution

Start with four (dfrac{1}{5}) pieces. Take away one (dfrac{1}{5}) piece. Count how many fifths are left. There are three (dfrac{1}{5}) pieces left.

Exercise (PageIndex{13})

Use a model to find each difference. (dfrac{7}{8} − dfrac{4}{8})

(dfrac{3}{8}), models may differ.

Exercise (PageIndex{14})

Use a model to find each difference. (dfrac{5}{6} − dfrac{4}{6})

(dfrac{1}{6}), models may differ.

## 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.

Definition: Fraction Subtraction

If (a), (b), and (c) are numbers where (c ≠ 0), then

[dfrac{a}{c} - dfrac{b}{c} = dfrac{a-b}{c}]

To subtract fractions with a common denominator, we subtract the numerators and place the difference over the common denominator.

Example (PageIndex{8}): difference

Find the difference: (dfrac{23}{24} − dfrac{14}{24}).

Solution

 Subtract the numerators and place the difference over the common denominator. (dfrac{23 - 14}{24}) Simplify the numerator. (dfrac{9}{24}) Simplify the fraction by removing common factors. (dfrac{3}{8})

Exercise (PageIndex{15})

Find the difference: (dfrac{19}{28} − dfrac{7}{28}).

(dfrac{3}{7})

Exercise (PageIndex{16})

Find the difference: (dfrac{27}{32} − dfrac{11}{32}).

(dfrac{1}{2})

Example (PageIndex{9}): difference

Find the difference: (dfrac{y}{6} − dfrac{1}{6}).

Solution

 Subtract the numerators and place the difference over the common denominator. (dfrac{y - 1}{6})

The fraction is simplified because we cannot combine the terms in the numerator.

Exercise (PageIndex{17})

Find the difference: (dfrac{x}{7} − dfrac{2}{7}).

(dfrac{x-2}{7})

Exercise (PageIndex{18})

Find the difference: (dfrac{y}{14} − dfrac{13}{14}).

(dfrac{y-13}{14})

Example (PageIndex{10}): difference

Find the difference: (− dfrac{10}{x} − dfrac{4}{x}).

Solution

Remember, the fraction (− dfrac{10}{x}) can be written as (dfrac{−10}{x}).

 Subtract the numerators. (dfrac{-10 - 4}{x}) Simplify. (dfrac{-14}{x}) Rewrite with the negative sign in front of the fraction. (- dfrac{14}{x})

Exercise (PageIndex{19})

Find the difference: (− dfrac{9}{x} − dfrac{7}{x}).

(-dfrac{16}{x})

Exercise (PageIndex{20})

Find the difference: (− dfrac{17}{a} − dfrac{5}{a}).

(-dfrac{22}{a})

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

Example (PageIndex{11}): simplify

Simplify: (dfrac{3}{8} + left(- dfrac{5}{8} ight) − dfrac{1}{8}).

Solution

 Combine the numerators over the common denominator. (dfrac{3 + (-5) - 1}{8}) Simplify the numerator, working left to right. (dfrac{-2 - 1}{8}) Subtract the terms in the numerator. (dfrac{-3}{8}) Rewrite with the negative sign in front of the fraction. (- dfrac{3}{8})

Exercise (PageIndex{21})

Simplify: (dfrac{2}{5} + left(− dfrac{4}{5} ight) − dfrac{3}{5}).

(-1)

Exercise (PageIndex{22})

Simplify: (dfrac{5}{9} + left(− dfrac{4}{9} ight) − dfrac{7}{9}).

(-dfrac{2}{3})

## Key Concepts

• If (a,b,)$,$ and (c) are numbers where (c eq 0), then (dfrac{a}{c} + dfrac{b}{c} = dfrac{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 eq 0), then (dfrac{a}{c} - dfrac{b}{c} = dfrac{a-b}{c})
• To subtract fractions, subtract the numerators and place the difference over the common denominator.

## Practice Makes Perfect

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

1. (dfrac{2}{5} + dfrac{1}{5})
2. (dfrac{3}{10} + dfrac{4}{10})
3. (dfrac{1}{6} + dfrac{3}{6})
4. (dfrac{3}{8} + dfrac{3}{8})

### Add Fractions with a Common Denominator

In the following exercises, find each sum.

1. (dfrac{4}{9} + dfrac{1}{9})
2. (dfrac{2}{9} + dfrac{5}{9})
3. (dfrac{6}{13} + dfrac{7}{13})
4. (dfrac{9}{15} + dfrac{7}{15})
5. (dfrac{x}{4} + dfrac{3}{4})
6. (dfrac{y}{3} + dfrac{2}{3})
7. (dfrac{7}{p} + dfrac{9}{p})
8. (dfrac{8}{q} + dfrac{6}{q})
9. (dfrac{8b}{9} + dfrac{3b}{9})
10. (dfrac{5a}{7} + dfrac{4a}{7})
11. (dfrac{-12y}{8} + dfrac{3y}{8})
12. (dfrac{-11x}{5} + dfrac{7x}{5})
13. (− dfrac{1}{8} + left(− dfrac{3}{8} ight))
14. (− dfrac{1}{8} + left(− dfrac{5}{8} ight))
15. (− dfrac{3}{16} + left(− dfrac{7}{16} ight))
16. (− dfrac{5}{16} + left(− dfrac{9}{16} ight))
17. (− dfrac{8}{17} + dfrac{15}{17})
18. (− dfrac{9}{19} + dfrac{17}{19})
19. (− dfrac{6}{13} + left(− dfrac{10}{13} ight) + left(- dfrac{12}{13} ight))
20. (− dfrac{5}{12} + left(− dfrac{7}{12} ight) + left(- dfrac{11}{12} ight))

### Model Fraction Subtraction

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

1. (dfrac{5}{8} − dfrac{2}{8})
2. (dfrac{5}{6} − dfrac{2}{6})

### Subtract Fractions with a Common Denominator

In the following exercises, find the difference.

1. (dfrac{4}{5} − dfrac{1}{5})
2. (dfrac{4}{5} − dfrac{3}{5})
3. (dfrac{11}{15} − dfrac{7}{15})
4. (dfrac{9}{13} − dfrac{4}{13})
5. (dfrac{11}{12} − dfrac{5}{12})
6. (dfrac{7}{12} − dfrac{5}{12})
7. (dfrac{4}{21} − dfrac{19}{21})
8. (- dfrac{8}{9} − dfrac{16}{9})
9. (dfrac{y}{17} − dfrac{9}{17})
10. (dfrac{x}{19} − dfrac{8}{19})
11. (dfrac{5y}{8} − dfrac{7}{8})
12. (dfrac{11z}{13} − dfrac{8}{13})
13. (- dfrac{8}{d} − dfrac{3}{d})
14. (- dfrac{7}{c} − dfrac{7}{c})
15. (- dfrac{23}{u} − dfrac{15}{u})
16. (- dfrac{29}{v} − dfrac{26}{v})
17. (- dfrac{6c}{7} − dfrac{5c}{7})
18. (- dfrac{12d}{11} − dfrac{9d}{11})
19. (dfrac{-4r}{13} − dfrac{5r}{13})
20. (dfrac{-7s}{3} − dfrac{7s}{3})
21. (- dfrac{3}{5} − left(- dfrac{4}{5} ight))
22. (- dfrac{3}{7} − left(- dfrac{5}{7} ight))
23. (- dfrac{7}{9} − left(- dfrac{5}{9} ight))
24. (- dfrac{8}{11} − left(- dfrac{5}{11} ight))

## Mixed Practice

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

1. (− dfrac{5}{18} cdot dfrac{9}{10})
2. (− dfrac{3}{14} cdot dfrac{7}{12})
3. (dfrac{n}{5} − dfrac{4}{5})
4. (dfrac{6}{11} − dfrac{s}{11})
5. (- dfrac{7}{24} − dfrac{2}{24})
6. (- dfrac{5}{18} − dfrac{1}{18})
7. (dfrac{8}{15} div dfrac{12}{5})
8. (dfrac{7}{12} div dfrac{9}{28})

## Everyday Math

1. Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has (dfrac{6}{10}) of a pound of nuts and (dfrac{3}{10}) of a pound of raisins. How much trail mix can he make?
2. Baking Janet needs (dfrac{5}{8}) of a cup of flour for a recipe she is making. She only has (dfrac{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?

## Writing Exercises

1. Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know.

Bits in case: (dfrac{1}{16}, dfrac{1}{8}), ___, ___, (dfrac{5}{16}, dfrac{3}{8}), ___, (dfrac{1}{2}, dfrac{9}{16}, dfrac{5}{8}).

Bits that fell out: (dfrac{7}{16}, dfrac{3}{16}, dfrac{1}{4}).

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

## Self Check

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

(b) 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?

## 23 Add and Subtract Fractions with Common Denominators

1. Simplify:
If you missed this problem, review (Figure).
2. Draw a model of the fraction
If you missed this problem, review (Figure).
3. Simplify:
If you missed this problem, review (Figure).

How many quarters are pictured? One quarter plus quarters equals 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,

Use a model to find the sum

There are five pieces, or five-eighths. The model shows that

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

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

### Add Fractions with a Common Denominator

(Figure) shows that to add the same-size pieces—meaning that the fractions have the same denominator —we just add the number of pieces.

If are numbers where then

To add fractions with a common denominators, add the numerators and place the sum over the common denominator.

Find the sum:

 Add the numerators and place the sum over the common denominator. Simplify.

Find each sum:

Find each sum:

Find the sum:

 Add the numerators and place the sum over the common denominator.

Note that we cannot simplify this fraction any more. Since are not like terms, we cannot combine them.

Find the sum:

Find the sum:

Find the sum:

We will begin by rewriting the first fraction with the negative sign in the numerator.

 Rewrite the first fraction with the negative in the numerator. Add the numerators and place the sum over the common denominator. Simplify the numerator. Rewrite with negative sign in front of the fraction.

Find the sum:

Find the sum:

Find the sum:

 Add the numerators and place the sum over the common denominator. Combine like terms.

Find the sum:

Find the sum:

Find the sum:

 Add the numerators and place the sum over the common denominator. Add. Simplify the fraction.

Find each sum:

Find each sum:

### Model Fraction Subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or of the pizza) left in the box. If Leonardo eats of these remaining pieces (or of the pizza), how much is left? There would be pieces left (or of the pizza).

Let’s use fraction circles to model the same example,

Start with seven />pieces. Take away two />pieces. How many twelfths are left?

Again, we have five twelfths,

Use fraction circles to find the difference:

Start with four pieces. Take away one piece. Count how many fifths are left. There are three pieces left.

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

, models may differ.

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

, models may differ

### 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.

If are numbers where then

To subtract fractions with a common denominators, we subtract the numerators and place the difference over the common denominator.

Find the difference:

 Subtract the numerators and place the difference over the common denominator. Simplify the numerator. Simplify the fraction by removing common factors.

Find the difference:

Find the difference:

Find the difference:

 Subtract the numerators and place the difference over the common denominator.

The fraction is simplified because we cannot combine the terms in the numerator.

Find the difference:

Find the difference:

Find the difference:

Remember, the fraction can be written as

 Subtract the numerators. Simplify. Rewrite with the negative sign in front of the fraction.

Find the difference:

Find the difference:

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

Simplify:

 Combine the numerators over the common denominator. Simplify the numerator, working left to right. Subtract the terms in the numerator. Rewrite with the negative sign in front of the fraction.

Simplify:

Simplify:

### Key Concepts

• If and are numbers where , then .
• To add fractions, add the numerators and place the sum over the common denominator.
• If and are numbers where , then .
• To subtract fractions, subtract the numerators and place the difference over the common denominator.

#### Practice Makes Perfect

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

Trail Mix Jacob is mixing together nuts and raisins to make trail mix. He has of a pound of nuts and of a pound of raisins. How much trail mix can he make?

Baking Janet needs of a cup of flour for a recipe she is making. She only has 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?

#### Writing Exercises

Greg dropped his case of drill bits and three of the bits fell out. The case has slots for the drill bits, and the slots are arranged in order from smallest to largest. Greg needs to put the bits that fell out back in the case in the empty slots. Where do the three bits go? Explain how you know.
Bits in case: , , ___, ___, , , ___, , , .
Bits that fell out: , , .

After a party, Lupe has of a cheese pizza, of a pepperoni pizza, and of a veggie pizza left. Will all the slices fit into pizza box? Explain your reasoning.

#### 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?

## More Florida Topics

MAFS.4.NBT.1.3 Use place value understanding to round multi-digit whole numbers to any place.

(a) Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. (b) Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 3/8 = 1/8 + 2/8 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8.

MAFS.4.MD.1.1 Know relative sizes of measurement units within one system of units including km, m, cm kg, g lb, oz. l, ml hr, min, sec. Within a single system of measurement, express measurements in a larger unit in terms of a smaller unit. Record measurement equivalents in a two-column table. For example, know that 1 ft is 12 times as long as 1 in. Express the length of a 4 ft snake as 48 in. Generate a conversion table for feet and inches listing the number pairs (1, 12), (2, 24), (3, 36),

MAFS.4.MD.1.2 Use the four operations to solve word problems* involving distances, intervals of time, and money, including problems involving simple fractions or decimals*. Represent fractional quantities of distance and intervals of time using linear models. (*See glossary Table 1 and Table 2) (*Computational fluency with fractions and decimals is not the goal for students at this grade level.)

MAFS.4.MD.1.3 Apply the area and perimeter formulas for rectangles in real world and mathematical problems. For example, find the width of a rectangular room given the area of the flooring and the length, by viewing the area formula as a multiplication equation with an unknown factor.

## 24 Add and Subtract Fractions with Different Denominators

1. Find two fractions equivalent to
If you missed this problem, review (Figure).
2. Simplify:
If you missed this problem, review (Figure).

### Find the Least Common Denominator

In the previous section, we explained how to add and subtract fractions with a common denominator. But how can we add and subtract fractions with unlike denominators?

Let’s think about coins again. Can you add one quarter and one dime? You could say there are two coins, but that’s not very useful. To find the total value of one quarter plus one dime, you change them to the same kind of unit—cents. One quarter equals cents and one dime equals cents, so the sum is cents. See (Figure).

Together, a quarter and a dime are worth cents, or of a dollar.

Similarly, when we add fractions with different denominators we have to convert them to equivalent fractions with a common denominator. With the coins, when we convert to cents, the denominator is Since there are cents in one dollar, cents is and cents is So we add to get which is cents.

You have practiced adding and subtracting fractions with common denominators. Now let’s see what you need to do with fractions that have different denominators.

First, we will use fraction tiles to model finding the common denominator of and

We’ll start with one tile and tile. We want to find a common fraction tile that we can use to match both and exactly.

If we try the pieces, of them exactly match the piece, but they do not exactly match the piece.

If we try the pieces, they do not exactly cover the piece or the piece.

If we try the pieces, we see that exactly of them cover the piece, and exactly of them cover the piece.

If we were to try the pieces, they would also work.

Even smaller tiles, such as and would also exactly cover the piece and the piece.

The denominator of the largest piece that covers both fractions is the least common denominator (LCD) of the two fractions. So, the least common denominator of and is

Notice that all of the tiles that cover and have something in common: Their denominators are common multiples of and the denominators of and The least common multiple (LCM) of the denominators is and so we say that is the least common denominator (LCD) of the fractions and

The least common denominator (LCD) of two fractions is the least common multiple (LCM) of their denominators.

To find the LCD of two fractions, we will find the LCM of their denominators. We follow the procedure we used earlier to find the LCM of two numbers. We only use the denominators of the fractions, not the numerators, when finding the LCD.

Find the LCD for the fractions and

 Factor each denominator into its primes. List the primes of 12 and the primes of 18 lining them up in columns when possible. Bring down the columns. Multiply the factors. The product is the LCM. The LCM of 12 and 18 is 36, so the LCD of and is 36. LCD of and is 36.

Find the least common denominator for the fractions: and

Find the least common denominator for the fractions: and

To find the LCD of two fractions, find the LCM of their denominators. Notice how the steps shown below are similar to the steps we took to find the LCM.

1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.

Find the least common denominator for the fractions and

To find the LCD, we find the LCM of the denominators.

Find the LCM of and

The LCM of and is So, the LCD of and is

Find the least common denominator for the fractions: and

Find the least common denominator for the fractions: and

### Convert Fractions to Equivalent Fractions with the LCD

Earlier, we used fraction tiles to see that the LCD of is We saw that three pieces exactly covered and two pieces exactly covered so

We say that are equivalent fractions and also that are equivalent fractions.

We can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The Equivalent Fractions Property is repeated below for reference.

If are whole numbers where

To add or subtract fractions with different denominators, we will first have to convert each fraction to an equivalent fraction with the LCD. Let’s see how to change to equivalent fractions with denominator without using models.

Convert to equivalent fractions with denominator their LCD.

 Find the LCD. The LCD of and is 12. Find the number to multiply 4 to get 12. Find the number to multiply 6 to get 12. Use the Equivalent Fractions Property to convert each fraction to an equivalent fraction with the LCD, multiplying both the numerator and denominator of each fraction by the same number. Simplify the numerators and denominators.

We do not reduce the resulting fractions. If we did, we would get back to our original fractions and lose the common denominator.

Change to equivalent fractions with the LCD:

and

Change to equivalent fractions with the LCD:

and

1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply both the numerator and denominator by the number you found in Step 2.
4. Simplify the numerator and denominator.

Convert and to equivalent fractions with denominator their LCD.

 The LCD is 120. We will start at Step 2. Find the number that must multiply 15 to get 120. Find the number that must multiply 24 to get 120. Use the Equivalent Fractions Property. Simplify the numerators and denominators.

Change to equivalent fractions with the LCD:

and LCD

Change to equivalent fractions with the LCD:

and LCD

### Add and Subtract Fractions with Different Denominators

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.

 Find the LCD of 2, 3. Change into equivalent fractions with the LCD 6. Simplify the numerators and denominators. Add.

Remember, always check to see if the answer can be simplified. Since and have no common factors, the fraction cannot be reduced.

Subtract:

 Find the LCD of 2 and 4. Rewrite as equivalent fractions using the LCD 4. Simplify the first fraction. Subtract. Simplify.

One of the fractions already had the least common denominator, so we only had to convert the other fraction.

Simplify:

Simplify:

 Find the LCD of 12 and 18. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. Add.

Because is a prime number, it has no factors in common with The answer is simplified.

When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.

The LCD, has factors of and factors of

Twelve has two factors of but only one of —so it is ‘missing‘ one We multiplied the numerator and denominator of by to get an equivalent fraction with denominator

Eighteen is missing one factor of —so you multiply the numerator and denominator by to get an equivalent fraction with denominator We will apply this method as we subtract the fractions in the next example.

Subtract:

 Find the LCD. 15 is ‘missing’ three factors of 2 24 is ‘missing’ a factor of 5 Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. Subtract. Rewrite showing the common factor of 3. Remove the common factor to simplify.

Subtract:

Subtract:

 Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify each numerator and denominator. Add. Rewrite showing the common factor of 2. Remove the common factor to simplify.

In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

The fractions have different denominators.

 Find the LCD. Rewrite as equivalent fractions with the LCD. Simplify the numerators and denominators. Add.

We cannot add and since they are not like terms, so we cannot simplify the expression any further.

### Identify and Use Fraction Operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

Fraction multiplication: Multiply the numerators and multiply the denominators.

Fraction division: Multiply the first fraction by the reciprocal of the second.

Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

First we ask ourselves, “What is the operation?”

Do the fractions have a common denominator? No.

 Find the LCD. Rewrite each fraction as an equivalent fraction with the LCD. Simplify the numerators and denominators. Add the numerators and place the sum over the common denominator. Check to see if the answer can be simplified. It cannot.

ⓑ The operation is division. We do not need a common denominator.

 To divide fractions, multiply the first fraction by the reciprocal of the second. Multiply. Simplify.

ⓐ The operation is subtraction. The fractions do not have a common denominator.

 Rewrite each fraction as an equivalent fraction with the LCD, 30. Subtract the numerators and place the difference over the common denominator.

ⓑ The operation is multiplication no need for a common denominator.

 To multiply fractions, multiply the numerators and multiply the denominators. Rewrite, showing common factors. Remove common factors to simplify.

1. ⓐ />
2. ⓑ />

### Use the Order of Operations to Simplify Complex Fractions

In Multiply and Divide Mixed Numbers and Complex Fractions, we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

Now we will look at complex fractions in which the numerator or denominator can be simplified. To follow the order of operations, we simplify the numerator and denominator separately first. Then we divide the numerator by the denominator.

1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator.
4. Simplify if possible.

Simplify:

 Simplify the numerator. Simplify the term with the exponent in the denominator. Add the terms in the denominator. Divide the numerator by the denominator. Rewrite as multiplication by the reciprocal. Multiply.

Simplify: .

Simplify: .

Simplify:

 Rewrite numerator with the LCD of 6 and denominator with LCD of 12. Add in the numerator. Subtract in the denominator. Divide the numerator by the denominator. Rewrite as multiplication by the reciprocal. Rewrite, showing common factors. Simplify. 2

Simplify: .

Simplify: .

### Evaluate Variable Expressions with Fractions

We have evaluated expressions before, but now we can also evaluate expressions with fractions. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify.

Evaluate when

ⓐ To evaluate when substitute for in the expression.

 Simplify.

ⓑ To evaluate when we substitute for in the expression.

 Rewrite as equivalent fractions with the LCD, 12. Simplify the numerators and denominators. Add.

Evaluate: when

Evaluate: when

Evaluate when

We substitute for in the expression.

 Rewrite as equivalent fractions with the LCD, 6. Subtract. Simplify.

Evaluate: when

Evaluate: when

Evaluate when and

Substitute the values into the expression. In the exponent applies only to

 Simplify exponents first. Multiply. The product will be negative. Simplify. Remove the common factors. Simplify.

Evaluate. when and

Evaluate. when and

Evaluate when and

We substitute the values into the expression and simplify.

 Add in the numerator first. Simplify.

Evaluate: when

Evaluate: when

### Key Concepts

• Find the least common denominator (LCD) of two fractions.
1. Factor each denominator into its primes.
2. List the primes, matching primes in columns when possible.
3. Bring down the columns.
4. Multiply the factors. The product is the LCM of the denominators.
5. The LCM of the denominators is the LCD of the fractions.
• Equivalent Fractions Property
• If , and are whole numbers where , then
and
1. Find the LCD.
2. For each fraction, determine the number needed to multiply the denominator to get the LCD.
3. Use the Equivalent Fractions Property to multiply the numerator and denominator by the number from Step 2.
4. Simplify the numerator and denominator.
1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.

1. Simplify the numerator.
2. Simplify the denominator.
3. Divide the numerator by the denominator.
4. Simplify if possible.

#### Practice Makes Perfect

Find the Least Common Denominator (LCD)

In the following exercises, find the least common denominator (LCD) for each set of fractions.

Convert Fractions to Equivalent Fractions with the LCD

In the following exercises, convert to equivalent fractions using the LCD.

Add and Subtract Fractions with Different Denominators

In the following exercises, add or subtract. Write the result in simplified form.

Identify and Use Fraction Operations

In the following exercises, perform the indicated operations. Write your answers in simplified form.

Use the Order of Operations to Simplify Complex Fractions

In the following exercises, simplify.

Mixed Practice

In the following exercises, simplify.

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary.

when

when

when

when

when

when

when

when

when

#### Everyday Math

Decorating Laronda is making covers for the throw pillows on her sofa. For each pillow cover, she needs yard of print fabric and yard of solid fabric. What is the total amount of fabric Laronda needs for each pillow cover?

Baking Vanessa is baking chocolate chip cookies and oatmeal cookies. She needs cups of sugar for the chocolate chip cookies, and cups for the oatmeal cookies How much sugar does she need altogether?

#### Writing Exercises

Explain why it is necessary to have a common denominator to add or subtract fractions.

Explain how to find the LCD of two fractions.

#### Self Check

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

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?

## ASVAB Math Study Guide Part 2

This is the second part of our free ASVAB Math Study Guide. This part covers fractions, percentages and decimals. Read through the material carefully and be sure to study the examples that are given. At the bottom of the page there is a review quiz to test your knowledge of these topics.

### Multiplying Fractions

Recall that the top number in a fraction is called the numerator and the bottom number is called the denominator:

Suppose we want to multiply the following fractions:

Multiply the numerators to find the numerator of the answer. Multiply the denominators to find the denominator of the answer:

Always reduce the fraction to lowest terms if possible. In this case, because the numerator and denominator share no factors in common (e.g. we cannot divide both by 2) the fraction cannot be reduced.

### Dividing Fractions

Suppose we want to divide the following fractions:

Begin by converting the division problem into a multiplication problem.

We do this by changing the divide (÷) into a multiply (×) and flipping the second fraction over (taking its reciprocal). When you see fractions being divided think “Flip It and Multiply”:

Using the Multiplication technique from above:

This fraction can be reduced no further.

### Adding and Subtracting Fractions with a Common Denominator

When adding or subtracting fractions, the fractions must first be changed so that they share a common denominator—the same number in the denominator—if they do not already.

In this first example the denominator is already the same in both fractions in other words they share a common denominator.

When adding fractions with a common denominator, we add the numerators, but leave the denominators the same:

Subtracting fractions with a common denominator follows a similar approach. We subtract the numerators, but leave the denominators the same:

### Adding and Subtracting Fractions with Different Denominators

Let’s now consider an example where the denominators of the fractions being added are not the same:

The first step is to find a common denominator, a number that is a multiple of both of the denominators.

15 is a common multiple of both 5 and 3, so it would be a good choice for our common denominator. We could also have used 30, but the lower the common multiple is, the less reducing we will need to perform at the end.

We have found our common denominator, but now we need to convert each of the original fractions to use the common denominator.

We have two fractions. Let’s start with $dfrac<2><5>$.

What do we need to multiply the denominator by to change it to our common denominator of 15?

We need to multiply it by 3. But whatever we do to the denominator, we must also do to the numerator:

We need to also convert our second fraction using the same approach:

We can now rewrite the original problem into one with fractions sharing a common denominator:

Subtracting follows a similar process:

### Mixed Fractions (aka “Mixed Numbers”)

Consider the following mixed number:

To work with mixed numbers, it is typically best to convert them to an improper fraction (a special type of fraction where the numerator is greater than the denominator).

To convert $3frac<2><5>$ to an improper fraction, we must first determine what the new numerator will be. We do so by multiplying the whole number by the denominator and add the numerator to it:

The denominator of the improper fraction will be the same as the denominator from the original mixed number $= 5$.

### Converting Percentages to Decimals (and vice versa)

The percentage symbol, %, literally means “per hundred (100)”.

So, 12% means “12 per 100”. We can write “12 per 100” as the fraction $frac<12><100>$.

Taking this a step farther, we can convert 12% into a decimal by simply moving the decimal point two positions to the left:

$12\% = 0.12$

Let’s try one more. Convert 7.5% to a decimal:

Notice that we must fill any “gaps” with the 0 digit.

Now let’s look at converting from a decimal to a percentage.

Convert 0.371 to a percentage:

.371 = 37.1\%$Convert 0.3 to a percentage: Again, we moved the decimal two positions to the right, but this time we needed to fill the “gaps” with the 0 digit. ### Percentage Calculations Let’s say we have been asked to find 30% of 120. When you see “of”, translate it as “times”. Next, we must convert the percentage to a decimal. Suppose we want to calculate the sales tax on an item that costs$49. Assume our sales tax is 7.5%.

$7.5\% ext < of >$49 = 7.5\% × $49 = 0.075 ×$497.5\% ext < of >$49 =$3.675$(Which would be rounded to$$3.68$)

### Converting Improper Fractions to Mixed Numbers

Though Improper Fractions are generally easier to work with, there are times when you will be interested in converting them to a Mixed Number.

Consider the following Improper Fraction:

To convert this to a Mixed Number, we divide the numerator by the denominator:

$23 ÷ 5 = 4$ with a remainder of $3 = 4 In other words,$5$goes into$23$four times with$3$left over. We take the$4$and put it as the whole number, the remainder of$3$becomes the numerator of the fraction, and the denominator of the fraction remains unchanged as$5\$:

Now that you’ve read more of our lessons and tips for the Math section of the ASVAB, put your skills to practice with the review quiz below. Try not to reference the above information and treat the questions like a real test.

## 3.5: Add and Subtract Fractions with Common Denominators - Mathematics

Adding fractions with COMMON denominators is simple. Just add the top numbers (the numerators) together, and place the resulting answer in the top of a fraction using the existing denominator for the bottom number. Then reduce the fraction, if possible

 + =

No reduction is possible, so we have found the answer!

Example 2: Reducing the fraction answer

 + =

Then reduce:
 =

Example 3: Converting the answer to a mixed number

 + =

Then convert the improper fraction to a mixed number:
 =

Example 1: If we have the fraction 2/3, we can multiply the top and bottom by 2, and not change its value: (2/2) x (2/3) = 4/6 Then if we reduce 4/6, we still get the original number, 2/3

Example 2: If we have the fraction 2/3, we could multiply top and bottom by 5, and not change its value: (5/5) x (2/3) = 10/15. Then if we reduce 10/15, we still get the original number, 2/3.

You can only add together fractions which have the same denominator, so you must first change one or both of the fractions so that you end up with two fractions having a common denominator. The easiest way to do this, is to simply select the opposite fraction's denominator to use as a top and bottom multiplier.

Example 1: Say you have the fractions 2/3 and 1/4
Select the denominator of the second fraction (4) and multiply the top and bottom of the first fraction (2/3) by that number:

 x =

Select the denominator of the first fraction (3) and multiply the top and bottom of the second fraction (1/4) by that number:

 x =

These two fractions (8/12 and 3/12) have common denominators - the number 12 on the bottom of the fraction.

Add these two new fractions together:

 + =

Example 2: Say you have the fractions 3/5 and 2/7
Select the denominator of the second fraction (7) and multiply the top and bottom of the first fraction (3/5) by that number

 x =

Select the denominator of the first fraction (5) and multiply the top and bottom of the second fraction (2/7) by that number

 x =

These two fractions (21/35 and 10/35) have common denominators -- the number 35 on the bottom of the fraction.

We can now add these two fractions together, because they have common denominators:

 + =

Got it? Great! Then go to the SuperKids Math Worksheet Creator for Basic Fractions, and give it a try!

## Anchor Chart and Free Printable

After teaching the four strategies for converting fractions to like denominators, I will bring out the anchor chart that serves as a review and also as a way to discuss a “strategy” for which strategy to use.

I have about half of the chart prepared beforehand (everything but the example work). We work on the same anchor chart for a couple of days (pacing depends on the students). We practice the strategies several times as review on markerboards before writing the example problem together on the chart and on their printables. The printables stay in their math notebooks for them to refer back to as needed.

## Adding more than two fractions

At this point in the article, we hope that you have mastered the art of adding two fractions. If you have to add more than two fractions, the procedure is the same. Let’s do an example to illustrate it.

These fractions are all in simplest form, so let’s start the process of finding the LCM and adjusting each fraction so the denominators are all the same.

### Step 1

Mini-Step 1.3: Choose highest powers of common factors for LCM and any additional factors of the three values:

### Step 2

Multiply these factors and write the value as the denominator of the fraction

### Step 3

Now it is time to figure out which fractions need to be adjusted to create like denominators. For each fraction ask, “does the denominator match the LCD?”

The first fraction is (frac<4><5>) , and we can see that 5 does not equal the LCD of 20. As before, divide the LCD by the original denominator. In this case, (20div 5=4).

Now the original fraction can be adjusted by multiplying both the numerator and denominator by 4, as shown:

The second fraction, (frac<1><2>) , must also be adjusted, as 2 does not match the LCD.

Divide the LCD by the denominator, 2: (20div 2=10)

Multiply the numerator and denominator of the second fraction by 10, as shown:

Finally, determine the adjustment that needs to be made to the third fraction, (frac<1><4>) .

Divide the LCD by the denominator, 4, for a factor of 5. Adjust the third fraction by multiplying by (frac<5><5>) .

Now that each of the three denominators match the LCD, the equivalent fractions can be added:

## ATI TEAS Math Practice Test for Fractions

1. What is the sum of 2 ½ and 3 ¼?
2. Simplify the expression: 6 2/3 – 1 5/8
3. Compute the difference of 10 – 2 3/8
4. Simplify the expression: 5/9 + 1/3 =
5. What is the sum of 3 6/10 and 1 1/3?
6. Simplify the expression: 2/3 + 1 14/15 =
7. Compute the difference of 10 – 6 2/3 =
8. Simplify the expression: 2/3 – 1/5 =
9. What is the sum of 2 and 2 5/10?
10. Simplify the expression 65 ½ + 3 25/40 =

1.) 5 ¾
2.) 5 1/24
3.) 7 5/8
4.) 8/9
5.) 4 14/15
6.) 2 3/5
7.) 3 1/3
8.) 7/15
9.) 4 ½
10.) 69 1/8

## 3.5: Add and Subtract Fractions with Common Denominators - Mathematics

Beginning Algebra
Tutorial 3: Fractions

1. Know what the numerator and denominator of a fraction are.
2. Find the prime factorization of a number.
3. Simplify a fraction.
4. Find the least common denominator of given fractions.
5. Multiply, divide, add and subtract fractions.

Do you ever feel like running and hiding when you see a fraction? If so, you are not alone. But don't fear help is here. Hey that rhymes. Anyway, in this tutorial we will be going over how to simplify, multiply, divide, add, and subtract fractions. Sounds like we have our work cut out for us. I think you are ready to tackle these fractions.

b = denominator

Examples of prime numbers are 2, 3, 5, 7, 11, and 13. The list can go on and on.

Be careful, 1 is not a prime number because it only has one distinct factor which is 1.

When you rewrite a number using prime factorization, you write that number as a product of prime numbers.

For example, the prime factorization of 12 would be

That last product is 12 and is made up of all prime numbers.

 When is a Fraction Simplified?

 Writing the Fraction in Lowest Terms (or Simplifying the Fraction)

Make sure that you do reduce your answers, as shown above. You may do this before you multiply or after.

*Div. the common factor of 5 out of both num. and den.

In other words, you flip the number upside down. The numerator becomes the denominator and vice versa.

For example, 5 (which can be written as 5/1) and 1/5 are reciprocals. 3/4 and 4/3 are also reciprocals of each other.

*Write as prod. of num. over prod. of den.

*Div. the common factor of 2 out of both num. and den.

Step 2: Put the sum or difference found in step 1 over the common denominator.

Why do we have to have a common denominator when we add or subtract fractions.
Another good question. The denominator indicates what type of fraction that you have and the numerator is counting up how many of that type you have. You can only directly combine fractions that are of the same type (have the same denominator). For example if 2 was my denominator, I would be counting up how many halves I had, if 3 was my denominator, I would be counting up how many thirds I had. But, I would not be able to add a fraction with a denominator of 2 directly with a fraction that had a denominator of 3 because they are not the same type of fraction. I would have to find a common denominator first before I could combine, which we will cover after this example.

 Least Common Denominator (LCD)

You can achieve this by multiplying the top and bottom by the same number. This is like taking it times 1. You can write 1 as any non zero number over itself. For example 5/5 or 7/7. 1 is the identity number for multiplication. In other words, when you multiply a number by 1, it keeps its identity or stays the same value.

*Multiply num. and den. by 4.

An improper fraction is a fraction in which the numerator is larger than the denominator.

*Mult. den. 4 times whole number 7 and add it to num. 3.

 Adding or Subtracting Fractions Without Common Denominators

Therefore, the LCD is 4.

*Multiply num. and den. by 4.

Therefore, the LCD is 15.

*Multiply num. and den. by 5.

*Multiply num. and den. by 3.

*Write the sum and difference over the common den.

*Div. the common factor of 3 out of both num. and den.

To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.

Practice Problem 1a: Write the number as a product of primes.

Practice Problems 2a - 2b: Write the fraction in lowest terms.

Practice Problems 3a - 3e: Perform the following operations. Write answers in the lowest terms.