Download Operations with Fractions

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class
date
Operations with Fractions
Fractions and mixed numbers are often used to make measurements. Chefs add and multiply
fractions and mixed numbers to determine how much of each ingredient to use. Carpenters
subtract and divide fractions and mixed numbers as they cut lumber.
To add fractions and mixed numbers, the denominators must be the same. If denominators are
not the same, first write equivalent fractions with like denominators. Add the numerators while
keeping the denominator the same. Regroup to simplify if possible.
Example a
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cup of strawberries and cup of blueberries for a smoothie
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recipe. How many cups of berries does she need in all?
Melanie needs
Step 1: Use the least common denominator (LCD) of 12 to write an
3
equivalent fraction for and 1 .
4
4
Step 2: Rewrite the addition problem using the fractions with like
denominators..
Step 3: Add the numerators and convert the improper fraction to a
mixed number.
1
Solution: Melanie needs 1 cups of berries for her recipe.
12
1 1⋅ 4
4
3 3⋅3 9
5
5
5
5
4 4 ⋅ 3 12 3 3 ⋅ 4 12
9
4
1
12 12
9
4 914 13
1
1 5
5 51
12 12
12
12
12
Example B
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Mrs. Perkins bought 5 pounds of apples and 4 pounds of bananas.
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How many pounds of fruit did Mrs. Perkins buy altogether?
Step 1: Use the LCD to write equivalent fractions with like denominators.
1
Step 2: Write an equivalent mixed number for 4 using the LCD of 8.
2
7
4
Step 3: Add the whole-number parts of 5 and 4 .
8
8
Step 4: Add the numerators of the fractions.
Step 5: Add the sum of the whole-number parts and the sum of the fractions.
Solution: Mrs. Perkins bought 10 3 pounds of fruit.
8
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1
The LCD of and is 8.
8
2
11
44
44 5
5 44
22
88
55 1
1 44 5
5 99
77 44 11
11
1
1 5
5
88 88 88
11
8 33
3
1 11 5
5 99 1
181
1 5
5 10
10 3
99 1
88
88 88
88
SpringBoard Algebra 1, Unit 1
Operations with Fractions (continued)
To subtract fractions and mixed numbers, the denominators must be the same. If denominators
are not the same, write equivalent fractions using the LCD. Subtract the numerators and keep the
denominators the same. When subtracting mixed numbers, it may be necessary to rename more
than once in order to subtract.
Example C
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Allen has 4 yards of wire. He uses 2 yards of wire for a project. How many yards of wire
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does he have left?
2
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and 2 .
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Step 2: Write equivalent mixed numbers using fractions with
denominators of 12.
2
3
The LCD of 4 and 2 is 12.
3
4
3
9
2
8
4 5 4 2 5 2
4
12
3
12
Step 3: There are not enough twelfths to subtract, so regroup one
8
12
whole of 4 as .
12
12
Step 4: Subtract the numerators of the fractions.
41
Step 1: Find the LCD of 4
Step 5: Subtract the whole numbers.
Step 6: Add the differences.
Solution: Allen has 1
8
12 8
20
531 1 53
12
12 12
12
20 9 20 2 9 11
2 5
5
12 12
12
12
3 2 2 51
11
11
1 1 51
12
12
11
yards of wire left.
12
To multiply with fractions and mixed numbers, first convert any mixed numbers to improper
fractions. Simplify the terms, if possible. Next, multiply the numerators and then multiply the
denominators. If necessary, convert the improper fraction back to a mixed number.
Example D
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One batch of chocolate chip cookies calls for teaspoon of salt. How many teaspoons
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of salt would Monique need for 4 batches of cookies?
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Step 1: Convert 4 to an improper fraction.
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Step 2: Before multiplying, cancel out common factors.
Step 3: Multiply the numerators and then multiply the denominators.
Solution: Monique would need 3 teaspoons of salt for 4
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batches of cookies.
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2
1 8 1 9
41 5 1 5
2 2 2 2
1
2 93
⋅
13
21
1
2 93 1 3
⋅ 5 ⋅ 53
13
21 1 1
SpringBoard Algebra 1, Unit 1
Operations with Fractions (continued)
To divide with fractions and mixed numbers, first convert any mixed numbers or whole numbers
to improper fractions. Then multiply the first fraction by the reciprocal of the second fraction.
Simplify if possible.
Example E
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Raul is cutting a board that measures 12 inches into pieces that are 2 inches
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long. How many pieces will Raul have when he finishes?
Step 1: Convert both mixed numbers to improper fractions.
Step 2: Multiply the first fraction by the reciprocal of the second fraction.
17 8
The reciprocal of is .
8 17
Step 3: Simplify by cancelling out the common factors. Then multiply.
3 48 3 51
12 5 1 5
4
4
4 4
1 16 1 17
2 5 1 5
8 8 8 8
51 17 51 8
4 5 ⋅
4
8
4 17
51 82 3 ⋅ 2 6
3
5
5 56
14
171 1 ⋅ 1 1
3
Solution: There will be 6 pieces after Raul is done cutting.
PRACTICE
Find the sum, difference, product, or quotient. Write your answers in
simplest form.
1. 4 1 1
5 6 2. 5 2 3
8 10 3. 5 2 1 1 3
3
4 4. 4 1 2 3 3
4
5
5. 7 ⋅ 2
8 9 6. 1 4 3
3 8 7. 2 1 ⋅ 3 7
2 12
8. 5 3 4 5
8 6 9. 7 4 2 3 1
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8 10. Jerry has a length of rope that he wants to cut into 5 equal pieces.
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If the rope is 21 feet long, how long should he cut each piece? Explain your steps. 8
1 3
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11. Paul said that 3 ⋅ 2 5 6 . Is Paul correct? Explain your answer. 5 8
40
5
12. Hitomi, Ben, and Gayle bought 3 pumpkins that weighed 15 pounds altogether. Ben and
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Gayle’s pumpkins each weighed the same amount.
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Hitomi’s pumpkin weighed 6 pounds. How much did Gayle’s pumpkin weigh? 3
Explain your steps.
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SpringBoard Algebra 1, Unit 1