3.6 Adding and Subtracting Mixed Numbers

Add and simplify, if possible: 75

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EXAMPLE 1

	3.6 Adding and Subtracting Mixed Numbers			271
S E C T I O N 3.6		Objectives
S E C T I O N 3.6		Objectives
Adding and Subtracting Mixed Numbers			Add mixed numbers.
Adding and Subtracting Mixed Numbers		1	Add mixed numbers.
In this section, we discuss several methods for adding and subtracting mixed numbers.			Add mixed numbers in vertical

		2
			form.
			form.
		3	Subtract mixed numbers.

1 Add mixed numbers.

We can add mixed numbers by writing them as improper fractions. To do so, we follow these steps.

4Solve application problems by adding and subtracting mixed numbers.

Adding Mixed Numbers: Method 1

1.Write each mixed number as an improper fraction.

2.Write each improper fraction as an equivalent fraction with a denominator that is the LCD.

3.Add the fractions.

4.Write the result as a mixed number, if desired.

Method 1 works well when the whole-number parts of the mixed numbers are small.

1 3 Add: 4 6 2 4

Strategy We will write each mixed number as an improper fraction, and then use the rule for adding two fractions that have different denominators.

WHY We cannot add the mixed numbers as they are; their fractional parts are not similar objects.

4	1	2	3
	6		4
Four and one-sixth	T		T	Two and three-fourths

Solution

4	1	2	3		25		11	Write 4	1	and 234 as improper fractions.
	6		4		6		4		6

By inspection, we see that the lowest common denominator is 12.

256 ##22 114 ##33

5012 3312

8312

6 1112

To build 256 and 114 so that their denominators are 12, multiply each by a form of 1.

Multiply the numerators.

Multiply the denominators.

Add the numerators and write the
sum over the common denominator 12.	6
The result is an improper fraction.	6
	12		83
Write the improper fraction 8312	12		83
Write the improper fraction 8312		72
as a mixed number.	11
as a mixed number.	11

Self Check 1

Add: 3 23 1 15

Now Try Problem 13

EXAMPLE 2

272	Chapter 3 Fractions and Mixed Numbers

Self Check 2

1 1

Add: 4 12 2 4

Now Try Problem 17

Success Tip We can use rounding to check the results when adding (or subtracting) mixed numbers. To check the answer 61112 from Example 1, we proceed as follows:

1		3			Since	1	is less than	1	, round 4			1	down to 4.
				4 3 7		6		2				6
4		2				3 is greater than				1			23 up to 3.
	6		4		Since						, round
										2
						4							4

Since 61112 is close to 7, it is a reasonable answer.

Add: 3 18 112

Strategy We will write each mixed number as an improper fraction, and then use the rule for adding two fractions that have different denominators.

WHY We cannot add the mixed numbers as they are; their fractional parts are not similar objects.

	1	1
3		1
	8		2
Negative three and one-eighth	T		T	One and one-half

Solution

3	1	1	1	25	3	Write 3	1	and 1	1	as improper fractions.

	8	2		8	2		8		2

Since the smallest number the denominators 8 and 2 divide exactly is 8, the LCD is 8.
We will only need to build an equivalent fraction for 3 .
						2
		25		3	4	To build 32 so that its denominator is 8,
		8		2	4	multiply it by a form of 1.
		25		12		Multiply the numerators.
		8		8		Multiply the denominators.
	25 12					Add the numerators and write the sum
		8				over the common denominator 8.
		8				over the common denominator 8.
	13					Use the rule for adding integers that have
	8					different signs: 25 12 13.
	8					different signs: 25 12 13.
1			5			Write 13 as a negative mixed number by dividing 13 by 8.
1						Write 13 as a negative mixed number by dividing 13 by 8.
		8				8

We can also add mixed numbers by adding their whole-number parts and their fractional parts. To do so, we follow these steps.

Adding Mixed Numbers: Method 2

1.Write each mixed number as the sum of a whole number and a fraction.

2.Use the commutative property of addition to write the whole numbers together and the fractions together.

3.Add the whole numbers and the fractions separately.

4.Write the result as a mixed number, if necessary.

Method 2 works well when the whole number parts of the mixed numbers are large.

EXAMPLE 3

3.6 Adding and Subtracting Mixed Numbers

273

Add: 168 37 85 29

Strategy We will write each mixed number as the sum of a whole number and a fraction. Then we will add the whole numbers and the fractions separately.

WHY If we change each mixed number to an improper fraction, build equivalent fractions,and add,the resulting numerators will be very large and difﬁcult to work with.

Self Check 3

Add: 275 16 81 35

Now Try Problem 21

Solution

We will write the solution in horizontal form.

168	3	85	2	168			3		85			2

	7		9				7					9

										3			2
					168	85				3			2
					168	85				7		9
										7		9
				253		3			2
				253		7			9
				253		3		9			2	7

						7		9			9			7
				253		27				14
				253		63				63
						63				63
				253		41
				253		63
						63

	253	41
	253	63
		63

Write each mixed number as the sum of a whole number and a fraction.

Use the commutative property
of addition to change the order
of the addition so that the
whole numbers are together
and the fractions are together.	1 1
	1 1
	168
Add the whole numbers.		85
	253
Prepare to add the fractions.	253
Prepare to add the fractions.
To build 37 and 92 so that their
denominators are 63, multipy
each by a form of 1.

Multiply the numerators.
Multiply the denominators.
	1
	1
Add the numerators and write	27
the sum over the common		14
denominator 63.	41

Write the sum as a mixed number.

Caution! If we use method 1 to add the mixed numbers in Example 3, the numbers we encounter are very large. As expected, the result is the same: 253 4163 .

168	3	85	2	1,179		767				Write 1683 and 85	2 as improper fractions.
168	7	85	9	7						Write 1683 and 85	2 as improper fractions.
	7		9	7			9			7	9
				1,179	9			767	7	The LCD is 63.

				7	9			9	7
				10,611			5,369			Note how large the numerators are.
				63				63		Note how large the numerators are.
				63				63
				15,980
				15,980						Add the numerators and write the sum over the
				63						common denominator 63.
				63						common denominator 63.

253	41	To write the improper fraction as a
253	63	mixed number, divide 15,980 by 63.
		mixed number, divide 15,980 by 63.

Generally speaking, the larger the whole-number parts of the mixed numbers, the more difﬁcult it becomes to add those mixed numbers using method 1.

2 Add mixed numbers in vertical form.

We can add mixed numbers quickly when they are written in vertical form by working in columns. The strategy is the same as in Example 2: Add whole numbers to whole numbers and fractions to fractions.

EXAMPLE 4

274	Chapter 3 Fractions and Mixed Numbers

Self Check 4

Add: 7158 23 13

Now Try Problem 25

Self Check 5

Add and simplify, if possible:

1 5 1

68 6 37 18 52 9

Now Try Problem 29

Add: 25 34 3115

Strategy We will perform the addition in vertical form with the fractions in a column and the whole numbers lined up in columns.Then we will add the fractional parts and the whole-number parts separately.

WHY It is often easier to add the fractional parts and the whole-number parts of mixed numbers vertically—especially if the whole-number parts contain two or more digits, such as 25 and 31.

Solution

Write the mixed numbers in vertical form.

							Build	3 and		1	so that their denominators are 20.
							Build	3 and			so that their denominators are 20.
								4	5			Add the fractions separately.
												Add the fractions separately.
															Add the whole numbers
															Add the whole numbers
															separately.
	3													15
25	3			25	3	5		25		15			25	15
25	4			25	4	5		25		20			25	20
31	1			31	1	4		31		4			31	4
31	5			31	5	4		31		20			31	20
										19			56	19
										20			56	20
The sum is 56		19	.
		20
		20

Strategy We will write the problem in vertical form. We will make sure that the fractional part of the answer is in simplest form.

WHY When adding, subtracting, multiplying, or dividing fractions or mixed numbers, the answer should always be written in simplest form.

Solution

The LCD for	1	,	1	, and	1	is 12.
	12		4		6

Write the mixed numbers in vertical form.

						Build		1	and	1		so that their denominators are 12.
						Build	4		and	6		so that their denominators are 12.
							4			6			Add the fractions separately.


															Add the whole numbers
															Add the whole numbers
															separately.
	1														1
										1				1 1
			1							1				1 1
75		75							75					75
75	12	75	12						75	12				75	12
43	1	43	1	3					43	3				43	3

	4		4		3					12					12
54	1	54	1		2		54				2			54	2
54	6	54	6	2			54			12				54	12			Simplify:
										6					6		1		1
										6				172	6	172	1	6	6		1	.
														172		172			2 6			.
										12					12	2		12	2 6	2
															12	2			1

1 The sum is 172 2.

EXAMPLE 7

EXAMPLE 6

3.6 Adding and Subtracting Mixed Numbers

275

When we add mixed numbers, sometimes the sum of the fractions is an improper fraction.

2 4 Add: 45 3 96 5

Strategy We will write the problem in vertical form. We will make sure that the fractional part of the answer is in simplest form.

WHY When adding, subtracting, multiplying, or dividing fractions or mixed numbers, the answer should always be written in simplest form.

Solution

The LCD for 23 and 45 is 15.

Write the mixed numbers in vertical form.

					Build 32 and 54 so that their denominators are 15.
					Build 32 and 54 so that their denominators are 15.
								Add the fractions separately.
								Add the fractions separately.
											Add the whole numbers
											Add the whole numbers
											separately.
	2									10
45	2	45	2	5		45	10		45	10
45	3	45	3	5		45	15		45	15
96	4	96	4	3		96	12		96	12
96	5	96	5	3		96	15		96	15
							22		141	22
							15		141	15
							15			15
												The fractional part of
												The fractional part of
												the answer is greater
												than 1.
												than 1.

Since we don’t want an improper fraction in the answer, we write 2215 as a mixed number. Then we carry 1 from the fraction column to the whole-number column.

141 2215 141 2215

141 1 157

142 157

Write the mixed number as the sum
of a whole number and a fraction.	1
of a whole number and a fraction.	1
	15
To write the improper fraction as a	15	22
To write the improper fraction as a	15
mixed number divide 22 by 15.	15
mixed number divide 22 by 15.	7
	7
Carry the 1 and add it to 141 to get 142.

Self Check 6

11 5

Add: 76 12 49 8

Now Try Problem 33

3 Subtract mixed numbers.

Subtracting mixed numbers is similar to adding mixed numbers.

Subtract and simplify, if possible: 16 107 9 158

Strategy We will perform the subtraction in vertical form with the fractions in a column and the whole numbers lined up in columns. Then we will subtract the fractional parts and the whole-number parts separately.

Self Check 7

Subtract and simplify, if possible:

12 209 8 301

Now Try Problem 37

WHY It is often easier to subtract the fractional parts and the whole-number parts of mixed numbers vertically.

EXAMPLE 8

276	Chapter 3 Fractions and Mixed Numbers
	Solution
					7		8
	The LCD for					and				is 30.
	The LCD for				10	and		15		is 30.
				Write the mixed numbers in vertical form.
				Write the mixed numbers in vertical form.
												Build	7	and		8	so that their denominators are 30.
												Build		and			so that their denominators are 30.
													10		15			Subtract the fractions separately.
																		Subtract the fractions separately.
																					Subtract the whole numbers
																					Subtract the whole numbers
																					separately.
			7																	21
	16		7				16		7		3		16		21				16	21
	16		10				16		10		3		16		30				16	30
		9	8				9		8		2		9		16				9	16
		9	15				9		15		2		9		30				9	30			Simplify:
																							5	5	1 .
															5				7	5 7 1			5	5	1 .
																								1
															30					30		6	30	5 6		6
																				30		6		1

1 The difference is 7 6 .

Self Check 8

3 15

Subtract: 258 4 175 16

Now Try Problem 41

Subtraction of mixed numbers (like subtraction of whole numbers) sometimes involves borrowing. When the fraction we are subtracting is greater than the fraction we are subtracting it from, it is necessary to borrow.

Subtract: 34 18 11 23

WHY It is often easier to subtract the fractional parts and the whole-number parts of mixed numbers vertically.

Solution

The LCD for	1	and	2	is 24.
	8		3

Write the mixed number in vertical form.

Build 81 and 32 so that their denominators are 24.


	1
34	1		34	1	3	34	3
34	8		34	8	3	34	24
11	2	11		2	8	11	16
11	3	11		3	8	11	24
								Note that	16	is greater than	3	.
											3

									24		24
									24		24

Since 2416 is greater than 243 , borrow 1

(in the form of 2424) from 34 and add it to 243 to get 2427. Subtract the fractions separately.

Subtract the whole numbers separately.


3	3	24				27
3	3	24		27		27
34			33		33
34	24	24	33	24	33	24
	24	24		24		24
	11	16	11	16	11	16
	11	24	11	24	11	24
		24		24		24
				11	22	11
				24	22	24
				24		24

The difference is 22 1124.

3.6 Adding and Subtracting Mixed Numbers

277

Success Tip We can use rounding to check the results when subtracting mixed numbers. To check the answer 22 1124 from Example 8, we proceed as follows:

1		2			Since	1	is less than	1	, round 34			1	down to 34.
				34 12 22		8		2				8
34		11
	8		3		Since	32 is greater than				1	, round 1132 up to 12.
										2

Since 221124 is close to 22, it is a reasonable answer.

	EXAMPLE 9	11
	Subtract: 419 53	11
	Subtract: 419 53	16	1in the form
		16
Strategy We will write the numbers in vertical form and borrow 1
of 1616 2from 419.

WHY In the fraction column, we need to have a fraction from which to subtract 1116.

Solution

Self Check 9

Subtract: 2,300 129 32

Now Try Problem 45

Write the mixed number in vertical form.

Borrow 1 (in the form of 1616) from 419. Then subtract the fractions separately.

Subtract the whole numbers separately. This also requires borrowing.


							16
				16		3 11	16
419		418				418
419		418		16		418	16
				16			16
53	11	53		11		53	11
53		53		16		53	16
16				16			16
		365		5		365	5
		365		16		365	16
				16			16
		5
The difference is 365					.
			16

4Solve application problems by adding and subtracting mixed numbers.

EXAMPLE 10 Horse Racing In order

to become the Triple Crown Champion, a thoroughbred horse must win three races: the Kentucky Derby (1 14 miles long), the Preakness Stakes (1 163 miles long), and the Belmont Stakes (1 12 miles long). What is the combined length of the three races of the Triple Crown?

Analyze

•The Kentucky Derby is 1 14 miles long.

•The Preakness Stakes is 1 163 miles long.

•The Belmont Stakes is 1 12 miles long.

•What is the combined length of the three races?

Focus on Sport/Getty Images

Afﬁrmed, in 1978, was the last of only 11 horses in history to win the Triple Crown.

Self Check 10

SALADS A three-bean salad calls for one can of green beans (1412 ounces), one can of garbanzo beans (10 34 ounces), and one can of kidney beans (1578 ounces). How many ounces of beans are called for in the recipe?

Now Try Problem 89

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168	3	85	2	168			3		85			2

	7		9				7					9

										3			2
					168	85				3			2
					168	85				7		9
										7		9
				253		3			2
				253		7			9
				253		3		9			2	7

						7		9			9			7
				253		27				14
				253		63				63
						63				63
				253		41
				253		63
						63


3	3	24				27
3	3	24		27		27
34			33		33
34	24	24	33	24	33	24
	24	24		24		24
	11	16	11	16	11	16
	11	24	11	24	11	24
		24		24		24
				11	22	11
				24	22	24
				24		24

168	3	85	2	168			3		85			2

	7		9				7					9

										3			2
					168	85				3			2
					168	85				7		9
										7		9
				253		3			2
				253		7			9
				253		3		9			2	7

						7		9			9			7
				253		27				14
				253		63				63
						63				63
				253		41
				253		63
						63


3	3	24				27
3	3	24		27		27
34			33		33
34	24	24	33	24	33	24
	24	24		24		24
	11	16	11	16	11	16
	11	24	11	24	11	24
		24		24		24
				11	22	11
				24	22	24
				24		24

168	3	85	2	168			3		85			2

	7		9				7					9

										3			2
					168	85				3			2
					168	85				7		9
										7		9
				253		3			2
				253		7			9
				253		3		9			2	7

						7		9			9			7
				253		27				14
				253		63				63
						63				63
				253		41
				253		63
						63


3	3	24				27
3	3	24		27		27
34			33		33
34	24	24	33	24	33	24
	24	24		24		24
	11	16	11	16	11	16
	11	24	11	24	11	24
		24		24		24
				11	22	11
				24	22	24
				24		24