Weighted Averages
A.A.5 Write algebraic equations or inequalities that represent a situation
A.A.6 Analyze and solve verbal problems whose solution requires solving a linear
equation in one variable or linear inequality in one variable
Also addresses A.N.5 and A.A.22.
Mixture Problem
Mandisha feeds her cat gourmet cat food that costs $1.75 per pound. She
combines it with cheaper food that costs $0.50 per pound. How many pounds
of cheaper food should Mandisha buy to go with 5 pounds of gourmet food, if
she wants the average price to be $1.00 per pound?
Let w = the number of pounds of cheaper cat food.
Make a table.
8.75 + 0.5w
8.75 + 0.5w
8.75 + 0.5w – 0.5w
8.75
=
=
=
=
1.00(5 + w) Original equation
5.0 + 1w
Distributive Property
5.0 + 1w – 0.5w
Subtract 0.5w from each side.
5.0 + 0.5w Simplify.
Mandisha should buy 7.5 pounds of cheaper cat food to be mixed with the 5
pounds of gourmet cat food to equal out to $1.00 per pound of cat food.
Percent Mixture Problem
A car’s radiator should contain a solution of 50% antifreeze. Bae has
2 gallons of a 35% antifreeze. How many gallons of 100% antifreeze should
Bae add to his solution to produce a solution of 50% antifreeze?
Let g = the number of gallons of 100% antifreeze to be added.
Make a table.
0.35(2) + 1.0(g) = 0.50(2 + g)
0.70 + 1g = 1 + 0.50g
0.70 + 1g – 0.50g = 1 + 0.50g – 0.50g
0.70 + 0.50g = 1
0.70 + 0.50g – 0.70 = 1 – 0.70
0.50g = 0.30
Original equation
Distributive Property
Subtract 0.50g from each side.
Simplify.
Subtract 0.70 from each side.
Simplify.
Divide each side by 0.50.
g = 0.60
Simplify.
Bae should add 0.60 gallon of 100% antifreeze to produce a 50%
solution.
Speed of One Vehicle
Nita took a non-stop flight to visit her grandmother. The 750-mile trip took
three hours and 45 minutes. Because of bad weather, the return trip took four
hours and 45 minutes. What was her average speed for the round trip?
Understand We know that Nita did not travel the same amount of time on
each portion of her trip. So, we will need to find the weighted
average of the plane’s speed. We are asked to find the average
speed for both portions of the trip.
Plan
First find the rate of the going portion, and then the return
portion of the trip. Because the rate is in miles per hour, convert
3 hours and 45 minutes to 3.75 hours and 4 hours 45 minutes to
4.75 hours.
Going
d
r
Formula for rate
t
750 miles

or 200 miles per hour
3.75 hours
Return
d
Formula for rate
t
750 miles

or 157.9 miles per hour
4.75 hours
r
Because we are looking for a weighted average we cannot just average their
speeds.We need to find the weighted average for the round trip.
Solve
Answer:
The average speed was about 176 miles per hour.
Check The solution of 176 miles per hour is between the going portion rate
200 miles per hour, and the return rate, 157.9 miles per hour. So, we know
that the answer is reasonable.
Speeds of Two Vehicles
A railroad switching operator has discovered that two trains are heading
toward each other on the same track. Currently, the trains are 53 miles apart.
One train is traveling at 75 miles per hour and the other 40 miles per hour.
The faster train will require 5 miles to stop safely, and the slower train will
require 3 miles to stop safely. About how many minutes does the operator
have to warn the train engineers to stop their trains?
Step 1 Draw a diagram.
53 miles apart
Takes 5 miles to stop
Takes 3 miles to stop
53 – (5 + 3) = 45 miles left
Step 2 Let m = the number of minutes that the operator has to warn the train
engineers to stop their trains safely. Make a table.
Step 3 Write and solve an equation using the information in the table.
The operator has about 23 minutes to warn the engineers.
Other Word Problems
The difference between two numbers is 3. The sum of the larger and twice the
smaller is 57. Find the numbers. Only an algebraic solution will be accepted.
The sum of two angles is 180. If one angle is 30 more than twice the other, find
the number of degrees in the larger angle. Only an algebraic solution will be
accepted.
The sum of three consecutive numbers is 78. Find the three numbers. Only an
algebraic solution will be accepted.