RATES & RATIOS WITH COMPLEX FRACTIONS
LESSON 2-F
A
complex fraction is a fraction that contains a fractional expression in its numerator, denominator or
both. The following are examples of complex fractions.
Complex Fractions
Fraction in the numerator
Fraction in the denominator
_
​  2 ​
3
6
10
1 ​
​ _
2
Fraction in the numerator
AND fraction in the
denominator
3 ​
​ _
4
3 ​
​ _
8
Sometimes a rate or ratio is a complex fraction when it is first written. For example, if Jean walked 1​ _12 ​​ miles
in _​  14 ​hour, her rate would be:
3
miles
2
1
hour
4
What does this rate mean? Although accurate, this rate is hard to understand when it is written as a complex
fraction. The complex fraction needs to be simplified so the rate makes more sense. There are two ways to
simplify a complex fraction.
Method 1 - Division
Simplify
3
2
1
4
Method 2 - Least Common Denominator
Simplify
3
2
1
4
1.
Rewrite the fraction using division:
1.
Find the least common denominator (LCD) for
each fraction in the numerator and denominator:
LCD = 4
2.
Simplify:
2.
Multiply the numerator and denominator of the
complex fraction by the LCD and simplify:
3 1
÷
2 4
3 1 3 4 12
÷ = ⋅ = =6
2 4 2 1 2
This means
3
2
1
4
is equal to 6.
3
2
1
4
This means
3
2
1
4
⋅
3 4
⋅
4 2 1 6
=
= =6
4 1 4 1
⋅
4 1
is equal to 6.
Each method shows Jean walked at a rate of 6 miles per hour.
Lesson 2-F ~ Rates & Ratios With Complex Fractions 23
EXAMPLE 1
Simplify each complex fraction.
a.
Solutions
9
10
3
b.
Method 1 - Division
Answer:
Method 2 - Least Common Denominator
9
10
3
1
9
÷3
10
a. Find LCD of __  ​  and _
  : 
39 1
Multiply the numerator
and denominator by the
LCD. Simplify.
a. Rewrite using division:
Simplify:
4
4
7
⋅
10 3 1
3
​
10
b. Rewrite using division: 4 ÷
9
10
3
Simplify:
1
4 7
⋅
1 4
Answer:
7
1
⋅
10 9
3
=
=
10 30 10
3
10
Answer:
4
7
LCD = 10
4
1
4
7
b. Find the LCD of _  and _:  LCD = 7
Multiply the numerator
and denominator by the
LCD. Simplify.
Answer:
4
4
7
7 28
⋅ =
= 7
7 4
7
Anytime a rate or ratio problem involves a complex fraction, simplify the complex fraction to best answer the
question.
EXAMPLE 2
Solution
2  of one of his aquariums. Find
Ryan has many aquariums. He spent _
 1  hour filling _
3
3
the unit rate of hours per aquarium to find how long it takes Ryan to fill each one.
Write the rate.
Rewrite the complex fraction using division. 1
hour
3
2
aquarium
3
_
​  13 ÷ _​  32 
1 _
3 1
Simplify._
 ∙ ​   = _ 
3 2 2
1 hour
This can be written as
which means it takes Ryan 1 hour to fill 2 aquariums
2 aquariums
at this rate. But, as a unit rate, this is
24 1 hour ÷ 2
=
2 aquarium ÷ 2
1
hour
1
_
2
or  2 hour per
1 aquarium
aquarium. The simplified complex fraction of _
 1  can be written as the unit rate.
2 Ryan fills the aquariums at a rate of _
 1  hour per aquarium.
2
Lesson 2-F ~ Rates & Ratios With Complex Fractions
EXAMPLE 3
Find the scale factor of the similar squares.
1​ _23 ​ yards
_
​  49 ​ yard
Solution
4
9
5
3
Write the ratio of the sides of the squares as a complex fraction.
4
5
4 31 4
÷ = ⋅ = Simplify the complex fraction.
9 3 3 9 5 15
The scale factor is
4
or 4 : 15.
15
EXPLORE!
A CHANGE OF PACE
Kevin walked 13,200 feet in 30 minutes. Follow the directions below to find Kevin’s rate in miles per hour
three different ways.
Step 1: a. Fill in the conversion needed to change Kevin’s speed to miles per hour.
13200 feet
⋅
30 min
mile
⋅
feet
b. Calculate Kevin’s speed in miles per hour.
min
miles
=
hours
hours
Step 2: a. Convert 13,200 feet to miles. Write your answer as a decimal.
13,200 feet = _________ miles
b. Convert 30 minutes to hours. Write your answer as a decimal.
30 minutes = _________ hour
c. Find Kevin’s speed in miles per hour.
Step 3: a. Convert 13,200 feet to miles. Write your answer as a fraction.
13,200 feet = _________ miles
b. Convert 30 minutes to hours. Write your answer as a fraction.
30 minutes = _________ hour
c. Find Kevin’s speed in miles per hour.
Step 4: In Step 1 you converted feet per minute to miles per hour in one conversion
equation. In Steps 2 and 3, you converted feet to miles and minutes to hours
first and then found Kevin’s speed. In Step 2 you used decimals and in Step 3
you used fractions. Which of the three methods did you like best to find Kevin’s
speed? Why?
Lesson 2-F ~ Rates & Ratios With Complex Fractions 25
EXERCISES
Simplify each complex fraction.
_
 8 
4
5
3
1.
2.
3.
1
3  
4
_
__
  
4
10
4.
_
 2 
5
7  
__
10
7. Trevon insists
reasoning.
16
_
 2 
8
_
 2 
3
_
 5 
6
5.
​
is equivalent to
6.
__
​ 16  ​
2
4
_
 4 
9
_
 8 
3
. Pedro disagrees. Who is correct? Explain your
Find the unit rate.
_
​  43 ​ inches
_______
8. ​  _4
 
 ​
 
​  9 ​ minute
2​ _1 ​ pages
_
1​ _12 ​ miles
​  57 ​ foot
________
______
9. ​  __
   ​
 
10. ​  _1
  
​
​  4 ​ hour
​  15
 ​ seconds
14
7
1​ __
  ​ cookies
4​ _2 ​ innings
​  5 ​ hour
3​  3 ​ games
2
11. _______
​  4 minutes
 
 ​
 
12. _________
​  10_4
   
 
​
13. ________
​  3_1
  
​
Solve each problem. Show all work.
14. Luke wrote 12 entries in his journal. It took him 1​ _13 ​hours to write them all.
Assume each entry took the same amount of time. How many entries did
he write per hour?
15. During a snowstorm, 3​ _34 ​feet of snow fell in 5 hours. Assume the snow fell
at the same rate throughout the storm. How much snow fell per hour?
16. Sasha walked 6​ _13 ​miles at a constant rate in 1​ _12 ​hours. How fast did she walk
in miles per hour?
17. Victor read 2​ _13 ​books over 14 days last summer. Assume it took him the
same amount of time to read each book. How many books did he read each day?
18. Rodrigo and his family drove to Disneyland for their vacation. In the
first _​ 12 ​hour of the trip, they drove 30 miles. If they drive at the same
rate for 5​ _12 ​hours total, how far will they travel?
19. Lucy spent ​ _14 ​hour shooting baskets. She made 15 baskets. At that rate,
how many hours will it take Lucy to make 90 baskets?
26 Lesson 2-F ~ Rates & Ratios With Complex Fractions
20. A car traveled 15 miles in 20 minutes. Corin and Alejandro found the speed of the car in miles per hour.
One of them made a mistake. Identify who made the mistake and fix his solution.
Corin
Alejandro
15 miles 3
= mile per minute
20 min 4
15 1
15 miles
= ÷
1
1 3
hour
3
= 15 ⋅ 3
1 1
3 miles 1 hour
1
mile per hour
⋅
=
4 min 60 min 80
= 45 miles per hour
21. Find the scale factor of the similar rectangles.
_
​  78 ​ inch
2​ _14 ​ inches
22. Find the scale factor of the similar triangles.
_
​  16 ​ foot
3​ _13 ​ feet
23. Find the ratio of the areas of the squares.
_
​  23 ​ cm
1​ _23 ​ cm
Lesson 2-F ~ Rates & Ratios With Complex Fractions 27