Complicated Systems Word Problems
The following problems are slightly more difficult to set up before solving, but can still be solved using the
graphing, substitution, or elimination methods.
Mixture Systems
Beverly mixes a solution of 20% acid is mixed with a solution of 40% acid to
make 200 mL of a 25% acid. How many mL of each of the original acid solutions
did she use?
The most important thing to do is still defining our variables. We’ll let be the mL of 20% acid used and be the mL of 40% acid used. Since we know there is a total of 200 mL, we can write the following equation:
+ = 200
We know that 20% of is actually acid. Since “of” means to multiply, we know that is . 2 after we
convert the percent to a decimal. Similarly we get . 4 for the amount of acid in the 40% solution. Our total
amount of acid is 25% of the 200 mL which is . 25(200) or just 50. That gives us the following equation:
. 2 + .4 = 50
Solving by any method, we get that = 150 and = 50.
Distance, Rate, and Time Systems
Miles canoed upstream (against the current) 25 miles in 10 hours. The next day
he canoed downstream (with the current) the same distance in 5 hours. How
fast does Miles canoe in still water and what is the speed of the current?
Defining our variables first, let be the speed of the vehicle (canoe in this case) in miles per hour and let
be the speed of the water current in miles per hour. We know that distance equals rate times time. We also
know that when traveling against the current, the speed of the current is subtracted from the rowing speed.
When going with the current, the speed of the current is added to the rowing speed. That gives us the following:
25 = 10( − )
25 = 5( + )
In this case, it might be helpful to do some manipulation to both equations before trying to solve.
Dividing the first equation by 10 and the second equation by 5 gives us the following two equations:
2.5 = − 5=+
These are easier equations to work with using the elimination method, and we will Miles’ canoe speed to
be 3.75 mph and the speed of the current as 1.25 mph.
Practice on Your Own
Write and solve a system of equations using any method for each of the following situations.
(Problems 1-9 were taken from Valencia Community College.)
1. Lou wants to make a coffee mixture to sell. He is going to mix Sumatra coffee which costs $2.50 per pound
with Columbian coffee which costs $3.75 per pound. He wants to make 50 pounds of mix and he wants the cost
of the mix to be $3.35 per pound. How many pound of each will he need?
2. Jackie wants to make a mixture of nuts to sell in her store consisting of hazelnuts and cashews. Hazelnuts cost
$6.50 per pound and cashews cost $4.50 per pound. If Jackie wants 60 pounds total of mixture and the cost to be
$5.10 per pound, how many pounds of each will she need?
3. A chemist wants to make 40 liters of 22.5% acid solution. She is going to make it by mixing a 10% acid and a
30% acid solution. How many liters of each will she need?
4. A chemist wants to make 75 liters of 16% acid solution. He is going to make it by mixing a 10% acid and a 25%
acid solution. How many liters of each will he need?
5. A plane can travel 1,015 miles in 7 hours traveling against the wind. Traveling with the same wind, the plane
can travel 820 miles in 4 hours. How fast can the plane travel in still air and how fast is the wind current?
6. Lance Armstrong can ride 162 miles on flat ground in 6 hours with a good breeze at his back. It takes him 10
hours to go 90 miles with the same breeze working against him. How fast is Lance going on a bike and how fast is
the wind speed?
7. It takes Bob the Boy Scout 10 hours to paddle upstream (against the current) a distance of 15 miles. When he
turns around he finds it only takes him 5 hours to paddle 22.5 miles with the current. How fast is Bob’s boat going
in still water and what is the speed of the current of the river?
8. A boat going upstream (against the current) travels 105 miles in 15 hours. It takes the same boat 7.5 hours to
make the same trip when it is traveling back downstream (with the current). What is the speed of the boat in still
water and what is the speed of the current of the river?
9. Flying from Tokyo to London is approximately 6175 miles. On the way to London from Tokyo (against the
wind) the flight took 13 hours. The return flight (with the wind) took 9.88 hours. Find the speed of the plane in
still air and the speed of the wind current.
10. You have $10,000 to invest over the next year. You can invest in mutual funds at an interest rate of 8% per
year and a savings account at an interest rate of 3% per year. You don’t want to put everything into the mutual
fund because it’s hard to get your money out if there is an emergency need. If you want to make exactly $600 in
interest over the next year so you can buy that new TV, how much should you put in the mutual fund and how
much should you put in the savings account?
Practice on Your Own
ANSWERS
Write and solve a system of equations using any method for each of the following situations.
(Problems 1-9 were taken from Valencia Community College.)
1. Lou wants to make a coffee mixture to sell. He is going to mix Sumatra coffee which costs $2.50 per pound with
Columbian coffee which costs $3.75 per pound. He wants to make 50 pounds of mix and he wants the cost of the mix to be
$3.35 per pound. How many pound of each will he need?
2.50 + 3.75 = 3.35(50)
16 pounds of Sumatra, 34 pounds of Columbian
+ = 50
2. Jackie wants to make a mixture of nuts to sell in her store consisting of hazelnuts and cashews. Hazelnuts cost $6.50 per
pound and cashews cost $4.50 per pound. If Jackie wants 60 pounds total of mixture and the cost to be $5.10 per pound,
how many pounds of each will she need?
6.50ℎ + 4.50 = 5.10(60)
42 pounds of cashews, 18 pounds of hazelnuts
ℎ + = 60
3. A chemist wants to make 40 liters of 22.5% acid solution. She is going to make it by mixing a 10% acid and a 30% acid
solution. How many liters of each will she need?
. 10 + .30 = .225(40)
25 liters of 30% solution, 15 liters of 10% solution
+ = 40
4. A chemist wants to make 75 liters of 16% acid solution. He is going to make it by mixing a 10% acid and a 25% acid
solution. How many liters of each will he need?
. 10 + .25 = .16(75)
30 liters of 25% solution, 45 liters of 10% solution
+ = 75
5. A plane can travel 1,015 miles in 7 hours traveling against the wind. Traveling with the same wind, the plane can travel
820 miles in 4 hours. How fast can the plane travel in still air and how fast is the wind current?
− =
Plane: 175 mph, Wind: 30 mph
!
+ =
"
6. Lance Armstrong can ride 162 miles on flat ground in 6 hours with a good breeze at his back. It takes him 10 hours to go
90 miles with the same breeze working against him. How fast is Lance going on a bike and how fast is the wind speed?
$!
#+ =
Lance: 18 mph, Wind: 9 mph
$
%
#− =
7. It takes Bob the Boy Scout 10 hours to paddle upstream (against the current) a distance of 15 miles. When he turns
around he finds it only takes him 5 hours to paddle 22.5 miles with the current. How fast is Bob’s boat going in still water and
what is the speed of the current of the river?
−& =
Boat: 3 mph, River: 1.5 mph
!!.
+& =
8. A boat going upstream (against the current) travels 105 miles in 15 hours. It takes the same boat 7.5 hours to make the
same trip when it is traveling back downstream (with the current). What is the speed of the boat in still water and what is
the speed of the current of the river?
−& =
Boat: 10.5 mph, River: 3.5 mph
+& =
.
9. Flying from Tokyo to London is approximately 6175 miles. On the way to London from Tokyo (against the wind) the flight
took 13 hours. The return flight (with the wind) took 9.88 hours. Find the speed of the plane in still air and the speed of the
wind current.
$
− =
Plane: 550 mph, Wind: 75 mph
'
$
+ =
%.
10. You have $10,000 to invest over the next year. You can invest in mutual funds at an interest rate of 8% per year and a
savings account at an interest rate of 3% per year. You don’t want to put everything into the mutual fund because it’s hard to
get your money out if there is an emergency need. If you want to make exactly $600 in interest over the next year so you can
buy that new TV, how much should you put in the mutual fund and how much should you put in the savings account?
. 08 + .03 = 600
$6000 in mutual fund, $4000 in savings account
+ = 10000