10-8 Applying Rational Equations
Preview
Warm Up
California Standards
Lesson Presentation
10-8 Applying Rational Equations
Warm Up
Multiply. Simplify your answer.
1.
3.
Solve
4.
5.
2.
1
10-8 Applying Rational Equations
California
Standards
15.0 Students apply algebraic
techniques to solve rate problems, work
problems, and percent mixture problems.
10-8 Applying Rational Equations
When two people team up to complete a job,
each person completes a fraction of the whole
job. You can use this idea to write and solve a
rational equation to find out how long it will
take to complete a job.
10-8 Applying Rational Equations
Additional Example 1: Application
Jessica can clean her apartment in 5 hours.
Her roommate can clean the apartment in 4
hours. How long will it take to clean the
apartment if they work together?
Let h be the number of hours Jessica and her
roommate need to clean the apartment. Jessica
cleans the apartment in 5 hours, so she cleans
of the apartment per hour. The roommate cleans
the apartment in 4 hours, so she cleans of the
apartment per hour.
10-8 Applying Rational Equations
Additional Example 1 Continued
The table shows the part of the apartment that
each person cleans in h hours.
Jessica’s
part
+
+
+
=1
Roommate’s
part
=
=
Whole
Apartment
1
Solve this equation for h.
10-8 Applying Rational Equations
Additional Example 1 Continued
Multiply both sides by the
LCD, 20.
5h + 4h = 20
9h = 20
Distribute 20 on the left side.
Combine like terms.
Divide by 9 on both sides.
Working together Jessica and her roommate can
clean the apartment in
hours or about 2
hours 13 minutes.
10-8 Applying Rational Equations
Additional Example 1 Continued
Check
Jessica cleans
of her apartment per hour, so in
hours, she cleans
roommate cleans
so in
of the apartment. Her
of the apartment per hour,
hours, she cleans
apartment. Together, they clean
apartment.
of the
10-8 Applying Rational Equations
Check It Out! Example 1
Cindy mows a lawn in 50 minutes. It takes
Sara 40 minutes to mow the same lawn. How
long will it take them to mow the lawn if they
work together?
Let m be the number of minutes Cindy and Sara
need to mow the same lawn. Cindy mows the
lawn in 50 minutes, so she mows
of the lawn
per minute. Sara mows the lawn in 40 minutes, so
she mows
of the lawn per minute.
10-8 Applying Rational Equations
Check It Out! Example 1 Continued
The table shows the part of the lawn that each
person mows in m hours.
Cindy’s part
+
+
Sara’s part
=
Whole lawn
=
1
Solve this equation for m.
10-8 Applying Rational Equations
Check It Out! Example 1 Continued
Multiply both sides by the
LCD, 200.
4m + 5m = 200
9m = 200
Distribute 200 on the
left side.
Combine like terms.
Divide by 9 on both
sides.
Working together Cindy and Sara can mow the
lawn in
minutes or about 22 minutes and
13 seconds.
10-8 Applying Rational Equations
Check It Out! Example 1 Continued
Check
Cindy mows
of the lawn per minute, so in
minutes, she mows
mows
of the lawn. Sara
of the lawn per minute, so in
minutes, she mows
Together, they mow
of the lawn.
lawn.
10-8 Applying Rational Equations
Additional Example 2: Application
Omar has 20 oz of a snack mix that is half
peanuts and half raisins. How many ounces
of peanuts should he add to make a mix that
is 70% peanuts?
Let p be the number of ounces of peanuts that
Omar should add.
The table shows the amount of peanuts and the
total amount of the mixture.
Peanuts (oz)
Original Solution
New Solution
10
10 + p
Total (oz)
20
20 + p
10-8 Applying Rational Equations
Additional Example 2 Continued
The new mixture is 70% peanuts, so
Solve the equation for p.
10 + p = 0.7(20 + p)
Multiply both sides by 20 + p.
10 + p = 14 + 0.7p
Distribute 0.7 on the right
side.
Subtract 10 from both sides
and 0.7p from both sides.
Divide both sides by 0.3.
0.3p = 4
p = 13.33…
Omar should add
oz of peanuts to the mixture.
10-8 Applying Rational Equations
Check It Out! Example 2
Suppose the chemist wants a solution that is
80% alcohol. How many milliliters of alcohol
should he add in this case?
The new solution is 80% alcohol, so
Solve the equation for a.
250 + a = 0.8(500 + a)
Multiply both sides by 500 + a.
250 + a = 400 + 0.8a
Distribute 0.8 on the right side.
0.2a = 150
a = 750
Subtract 250 from both sides and
0.8a from both sides.
Divide both sides by 0.2.
10-8 Applying Rational Equations
Additional Example 3: Application
Amber runs along a 16-mile trail while Dave
walks. Amber runs 4 mi/h faster than Dave
walks. It takes Dave 2 hours longer than
Amber to cover the 16 miles. How long does
it take Amber to complete the trip?
Let t be the time it takes Amber to run the 16 miles.
Distance (mi)
Amber
16
Dave
16
Time (h)
t
t+2
Rate (mi/h)
10-8 Applying Rational Equations
Additional Example 3 Continued
Amber is 4 mi/h faster than Dave so,
Multiply both sides by
the LCD.
Distribute t(t + 2) on
the right side.
16t = (t + 2)16 + t(t + 2)(–4) Simplify.
16t = 16t + 32 –
4t2
0 = –4t2 – 8t + 32
– 8t
Distribute and
multiply.
Subtract 16t from both sides
and rearrange terms.
10-8 Applying Rational Equations
Additional Example 3 Continued
0 = 4t2 + 8t – 32
Multiply by –1.
0 = t2 + 2t – 8
Divide both sides by 4.
0 = (t + 4)(t – 2)
Factor.
t = –4 or 2
Time must be
nonnegative, so –4 is
an extraneous solution.
Amber makes the trip in 2 hours.
10-8 Applying Rational Equations
Check It Out! Example 3
Ryan drives 10 mi/h slower than Maya, and it
takes Ryan 1 hour longer to travel 300 miles.
How long does it take Maya to make the trip?
Let t be the time it takes Maya to drive the 300 miles.
Distance (mi)
Time (h)
Ryan
300
t+1
Maya
300
t
Rate (mi/h)
10-8 Applying Rational Equations
Check It Out! Example 3 Continued
Maya is 10 mi/h faster than Ryan so,
Multiply both sides
by the LCD.
Distribute t(t + 1) on
the left side.
300t + t(t + 1)10 = (t + 1)(300) Simplify.
300t
+10t2
+ 10t = 300t + 300
Distribute and
multiply.
10-8 Applying Rational Equations
Check It Out! Example 3 Continued
10t2 + 10t – 300 = 0
t2 + t – 30 = 0
(t + 6)(t – 5) = 0
t = –6, 5
Subtract 300t from both sides
and 300 from both sides.
Divide both sides by 10.
Factor.
Time must be nonegative, so
–6 is an extraneous
solution.
Maya makes the trip in 5 hours.
10-8 Applying Rational Equations
Lesson Quiz
1. Pipe A can fill a storage tank in 40 minutes.
Pipe B can fill the tank in 80 minutes. How long
does it take to fill the tank using both pipes at
the same time?
min, or 26 min 40 s
2. An 8 oz smoothie is made up of 50%
strawberries and 50% yogurt. How many
2 oz
ounces of strawberries should be added to
make a smoothie that is 60% strawberries?
3. Carlos bikes 3 mi/h faster than Tim. It takes
Tim 1 hour longer than Carlos to bike 36
miles. How long does it take Carlos to bike 36
miles? 3 h