Unit 1
Getting
Ready
Practice A
Operations with Fractions
People often use fractions and mixed numbers when making measurements.
Chefs add and multiply fractions and mixed numbers as they mix together
ingredients. Carpenters subtract and divide fractions and mixed numbers as
they cut apart pieces of lumber.
To add fractions and mixed numbers with unlike denominators, first write
equivalent fractions with like denominators. Add the numerators while keeping
the denominator the same. Regroup to simplify if possible.
Example 1
​ 3 ​  mile after school. How far did she
Laura walked __
​ 7 ​  mile before school and __
4
8
walk altogether?
Use the least common denominator (LCD) of 8 to write an
equivalent fraction for __
​ 3 ​ .
4
Write the addition problem using fractions with like
denominators.
__
​ 3 . 2 ​ = __
​ 6  ​
​ 3 ​  = ____
Step 3:
Add the numerators and write the sum over the common
denominator.
6 
13
7 ​  + __
​ 6 ​  = _____
​ 7 + ​
​ __
 = ​ ___ ​ 
8 8
8
8
Step 4:
Write the improper fraction as a mixed number.
5  ​
___
​ 13 ​ = 1​ __
Step 1:
Step 2:
© 2011 College Board. All rights reserved.
5 ​  miles.
Solution: Laura walked 1​ __
8
Example 2
3 ​  pounds of
1 ​  pounds of American cheese and 2​ __
Mr. Thompson bought 3​ __
2
8
Cheddar cheese. How many pounds of cheese did Mr. Thompson buy in all?
Step 1:
Step 4:
Use the LCD to write equivalent fractions with like
denominators.
1 ​  using the least
Write an equivalent mixed number for 3​ __
2
common denominator of 8.
3 ​ .
4 ​  and 2​ __
Add the whole-number parts of 3​ __
8
8
Add the numerators of the fractions.
Step 5:
Add the sums of the whole-number parts and the fractions.
Step 2:
Step 3:
7 ​  pounds of cheese.
Solution: Mr. Thompson bought 5​ __
8
4
4 . 2
8
__
​ 6  ​
​ 7 ​  + __
8
8
8
8
​ 3 ​  is 8.
The LCD of __
​ 1 ​  and __
2
8
1 ​  = 3​ __
4  ​
3​ __
2
8
3+2=5
4 ​  + __
​ __
​ 3 ​  = __
​ 7  ​
8 8 8
7  ​
5 + __
​ 7 ​  = 5​ __
8
8
Level 3, Unit 1 • Patterns and Numerical Relationships A-1
Unit 1
Practice A
Getting
Ready
Operations with Fractions continued
To subtract fractions that have unlike denominators, first write equivalent
fractions with like denominators. Subtract the numerators while keeping the
denominators the same. Regroup to simplify if possible. When subtracting
mixed numbers, it may be necessary to rename more than once in order
to subtract.
Step 3:
Step 4:
There are not enough twentieths to subtract, so regroup one
5  ​ as ___
whole of 5​ ___
​ 20  ​.
20 20
Subtract the numerators of the fractions.
Step 5:
Subtract the whole numbers.
Step 6:
Write the differences.
7  ​ miles left to run.
Solution: Michael has 1​ ___
20
9  ​ is 20.
1 ​  and 3​ ___
The LCD of 5​ __
4
10
9  ​ = 3​ ___
5  ​ 18  ​
1 ​  = 5​ ___
5​ __
3​ ___
 
4
20
10
20
5  ​ = 4 + ___
25  ​
5 + ​ ___
​ 20  ​+ ___
​  5  ​ = 4​ ___
20
20 20
20
−  ​
18 
7
___
______
​ 25  ​− ___
​ 18  ​= ​ 25
 = ​ ___  ​ 
20
20
20
4−3=1
7  ​ 
1 + ___
​  7  ​ = 1​ ___
20
20
20
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 4
7 ​  lb. How much does __
​ 2 ​  of the stack weigh?
A stack of newspapers weighs 3​ __
8
3
7 ​  to an improper fraction.
Step 1: Convert 3​ __
8
Step 2:
Simplify. Then multiply the numerators and the denominators.
Step 3:
Convert the improper fraction to a mixed number.
7  ​ pounds.
Solution: Two-thirds of the stack weighs 2​ ___
12
A-2 Getting Ready Practice A Operations with Fractions
7 ​  = ___
​ 24 ​ + __
​ 7 ​  = ___
​ 31 ​ 
3​ __
8
8
8
8
1
___
​ 2  ​  = _____
​ 31 · 1 ​ = ___
​ 31  ​
​ 31 ​ . __
8
3
4 · 3
12
4
7
31
​ ___  ​= 2​ ___  ​ 
12
12
© 2011 College Board. All rights reserved.
Example 3
9  ​ miles, he stops to tie his
1 ​  miles. After running 3​ ___
Michael plans on running 5​ __
4
10
shoes. How much farther does Michael have to run?
9  ​ .
1 ​  and 3​ __
Step 1: Find the LCD of 5​ __
4
10
Step 2: Write equivalent mixed numbers using fractions with
denominators of 20.
Unit 1
Practice A
Getting
Ready
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 5
3 ​  inches long. How
1 ​  inches of wood into pieces that are 3​ __
Carlos is cutting 47​ __
4
8
many pieces will there be after Carlos finishes cutting the wood?
Step 1:
Convert both numbers to improper fractions.
Step 2:
Multiply the first fraction by a reciprocal of the second fraction.
8  ​ .
The reciprocal of ___
​ 27 ​ is ​ ___
8 27
Simplify the terms and multiply.
Step 3:
3 ​  = ___
1 ​  = ____
​ 189 ​ and 3​ __
​ 27 ​ 
47​ __
4
4
8
8
189
189
27
8
____
​   ​ = ____
​    ​ 
​   ​ . ___
​   ​ ÷ ___
4
4 27
8
7
189 ​  . ​ ___
82  ​ = ____
​ 7 . 2 ​ = 14
​ ____
4
27
1 . 1
1
1
Solution: There will be 14 pieces after Carlos finishes cutting.
Try These
Solve. Write your answers in simplest form.
​ 2 ​ 
1. ___
​  7  ​ + __
10 5
3 ​
2 ​− 1​ __
4. 2​ __
4
3
3 ​  1 ​  . 1​ __
7. 4​ __
3 8
11  ​− __
2. ​ ___
​ 3 ​ 
12 4
​  3   ​
5. __
​ 5 ​. ___
6 10
1 ​ 
8. __
​ 1 ​  ÷ 3​ __
4
2
1  ​+ __
3. 4​ __
​ 7 ​ 
2 8
​ 7 ​
6. __
​ 1 ​÷ __
2 8
1  ​ − 4​ __
1 ​ 
9. 8​ ___
10
8
© 2011 College Board. All rights reserved.
3 ​  cups of tomato sauce. If Jerry makes 2​ __
1 ​ 
10. A recipe that serves 4 calls for 1​ __
4
2
times that amount, how many cups of tomato sauce will he need?
2  ​miles to work each day. Yesterday, she drove ___
11. Ms. Wilkins drives 14​ __
​  3  ​ 
5
10
of the way to work before she got a flat tire. How far away was she from
work when the tire went flat? Explain your steps.
Level 3, Unit 1 • Patterns and Numerical Relationships A-3