Answers - Head-Royce

Answer Sheet to Mixture Problems Day 1
1)
x = amount Max invested at 5%
y = amount Max invested at 8%
x + y = 9000
0.5x + 0.8y = 525
0.5x + 0.8(9000 ! x) = 525[substitution]
x = $6500
2)
x = kg of chocolate
y = kg of candy
(the $ you pay for x kg of chocolate is 8x, and the $ you pay for y kg of candy
is 10y, and the total you pay for the mix is $8.50 per kg times 12kg)
8x + 10y = 8.50 (12)
Since you have two variables, you need two equations. We also know that
x + y = 12
Substitution or comb. is probably easiest here
x = 9 kg
3)
d = # of dimes
q = # of quarters
(do ALL equations in cents!)
10d + 25q = 550
d + 8 = q
q = 18 quarters
4) At an amusement park, you get 5 point for each bull’s eye you hit, but you lose 10
points for every miss. After 30 tries, Julia lost 90 points. How many bull’s eyes
did she hit
x = # of bullseyes hit
y = # of misses

x + y = 30
5x – 10y = -90
[use comb or substitution – I used combinations here]
10 (x + y) = 30(10)
10x + 10y = 300
5x – 10y = -90
(add eqns together)
15x = 210
x = 14
so she made 14 hits and 16 misses
5) Michael bought a car dealership with all of his pro soccer earnings. He recently
received a shipment of 18 cars that weighs 30 tons. Some cars weigh 3000 lb
apiece and others 5000 lb each. Help Michael find the number of each kind of
car.
X = # of cars weighing 3000 lbs each
Y = # of cars weighing 5000 lbs each
X + Y = 18
3000X + 5000Y = 30(2000) there are 2000 lbs in a ton
(using substitution here)
X = 18 – Y
3000 (18 – Y) + 5000Y = 30(2000)
54000 – 3000Y + 5000Y = 60000
54000 + 2000Y = 60000
2000Y = 6000
Y = 3
So X = 15
6) Bonnie has 800 g of a dye solution that is 20% of its original strength. How much dye
must be added to increase the strength to 50%
Dye strength x Amount (g) = amt (g) of pure dye
Solution 1 0.02 800 800(0.20)
Pure dye 1.00 x x(1.00)
Total 0.50 y 0.50(y)
800 + x = y (the grams you start with plus how much you add equals y)

800(0.20) + x(1.00) = 0.50(y) (the amount of pure dye you start with plus the amount
of pure dye you add equals the amount of pure dye you end up with)
800 + x = y
160 + 1x = 0.5y
(substitution is easy here)
160 + 1x = 0.5(800 + x)
160 + 1x = 400 + 0.5x
0.5x = 240
x = 480 grams of pure dye
Harder problems – give them a shot!
(HINT: for problems 1 and 2, write three systems with three variables)
1) I have 30 coins (nickels, dimes, and quarters), worth $4.60. There are two more
dimes than quarters. How many of each kind of coin do I have
N = # of nickels
D = # of dimes
Q = # of quarters
5N + 10D + 25Q = 460
Q + 2 = D
N + Q + D = 30 (N = 30 – Q – D)
(substitution’s easier here)
5N + 10(Q + 2) + 25Q = 460 (substitute for D)
5(30 – Q – D) + 10(Q + 2) + 25Q = 460 (substitute for N, and in next step for D)
5[30 – Q – (Q + 2)] + 10(Q + 2) + 25Q = 460
5[30 – 2Q – 2] + 10Q + 20 + 25Q = 460
150 – 10Q – 10 + 10Q + 20 + 25Q = 460
160 + 25Q = 460
25Q = 300

Q = 12, D = 14, N = 4
2) Kyra, Davida, and Cai have $46 together. Kyra has half as much as Cai, and
Davida has $2 less than Cai. How much does each have
x = Kyra’s $
y = Davida’s $
z = Cai’s $
x + y + z = 46
x = z /2 or x = ½ z
y = z – 2
½ z + (z – 2) + z = 46
2 ½ z – 2 = 46
2 ½ z = 48
z = $19.20
x = $9.60
y = $17.20
3) If Harper gives Elisa 30 cents, they will have equal amounts of money. But if
Elisa then gives Harper 50 cents, Harper will have twice as much money as Elisa
does. How much money does each have now
x = cents Harper has
y = cents Elisa has
x – 30 = y + 30
2(y + 30 – 50) = x – 30 + 50
solving the first eqn for x: x = y + 60
2(y + 30 – 50) = y + 60 – 30 + 50
2(y – 20) = y + 80
2 y – 40 = y + 80
y = 120 cents or $1.20
x = 180 cents or $1.80

Name
Date
Mixture Practice Worksheet
For the following problems,
a) define a variable or variables if you need more than one
b) write an equation or system (use a rate chart if it’s appropriate)
c) solve the equation or system to find the answer to the problem.
Remember to include units! These are real life problems – if your answer comes out
negative, for example, you must have made a mistake!
1. Miles has 30 coins in nickels and dimes. In all, he has $2.10. How many nickels
and dimes does he have
Variable: n = # of nickels d = # of dimes
N + d = 30
5n + 10d = 210
D = 30-n
5n + 10(30 – n) = 210
5n + 300 – 10n = 210
N = 18 and d = 12
2. The Head-Royce Jazz Band had a concert where adult tickets costs $4 and student
tickets cost $2.50. How many of each kind of ticket was sold if 125 tickets were
bought for $413
Variable: a = cost of adult tickets s = cost of student tickets
4a + 2.5s = 413 and a + s = 125
A = 125 - s
4(125 – s) + 2.5s = 413
87 = 1.5s
S = 58 and A = 67
3. Alexis is doing a chemistry experiment. She needs to make a 40% solution of
copper sulfate. She has 60 ml of a 25% solution. How many ml of a 70%
solution should she add to end up with the 40% solution she needs
Variable: x = ml of 70% solution
60(.25) + (.7)x = (.4)(x+60)
15 + 0.7x = 0.4x + 24
0.3x = 9
X = 27 ml

4. Sweet Dreams, the candy store, has hired Tauqeer to figure out how to end up
with a mix of jelly beans that cost $3.40 a pound. He is going to mix 10 pounds
of one kind of jelly beans that cost $3.50 a pound with some jelly beans that cost
$3 a pound. How many pounds should he mix (use a rate table!)
Variable: x = # of lbs of jelly beans used that cost $3 a pound
10(3.50) + x(3) = (3.40)(10 + x)
35 + 3x = 34 + 3.4x
1 = 0.4 x
X = 2.5 pounds
5. Because he made some money from winning American Idol, Aaron has $15,000
to invest. He invested part of it at 10% annual interest and the rest of it at 6%
annual interest. If he is going to earn $1260 in interest for the year from the two
accounts, how much did he invest at 10%
Variable: x = $ invested at 10% y = $ invested at 6%
X + y = 15000
0.1x + 0.06y = 1260
0.1x + (0.06)(15000 – x) = 1260
0.1x + 900 – 0.06x = 1260
0.04x = 360
X = $9000 and y = $6000
6. A Jamba Juice drink claims it contains 15% pineapple juice. How much pure
pineapple juice would have to be added to 8 quarts of the drink to obtain a
mixture containing 50% pineapple juice
Variable: x = quarts of pure juice added
8 (0.15) + (1.00)(x) = (0.50)(x + 8)
1.2 + x = 0.5x + 4
0.5 x = 2.8
X = 5.6 quarts
7. Several families went to the movies for a matinee, when adult tickets only cost
$5.50 and kid tickets only cost $3.50. How many adults and children went if 21
tickets were bought for $83.50
Variable: a = # of adult tickets c = # of kid tickets
A + c = 21 and 5.5A + 3.5C = 83.50
5.5 (21 – c) + 3.5C = 83.50
115.5 – 5.5C + 3.5C = 83.50
32 = 2C c = 16 tickets and a = 5 tickets

8. A liter of cream has 9.2% butterfat. How much skim milk containing 2%
butterfat should be added to the cream to obtain a mixture with 6.4% butterfat
Variable: x = liters of skim milk to add
0.092 (1) + (0.02)(x) = (0.064)(x + 1)
0.092 + 0.02x = 0.064x + 0.064
0.028 = 0.044x
X = 0.64 liters
9. Ms. Brown’s favorite coffee is a mix of two different types of coffee beans – one
that costs $6.40 per pound and one that costs $7.28 per pound. The mix sells for
$6.95 a pound. How many pounds of $7.28 coffee are mixed with 9 pounds of
the $6.40 coffee to make the mixture that sells for $6.95 a pound
Variable: x = lbs of 7.28 coffee
7.28(x) + 9(6.40) = (x + 9)(6.95)
10. Bryce and his family are going to Marine World for the day. The total cost of
tickets for their group of two adults and three children is $79.50. If an adult ticket
costs $6 more than a child ticket, find the cost of each type of ticket.
Variable: c = child ticket a = c + 6
3c + 2(c + 6) = 79.50
5c = 57.50
C = $11.50 and A = $17.50
11. Extra Credit: A car radiator has a capacity of 16 quarts and is filled with a 25%
antifreeze solution. How much antifreeze must be drained off and replaced with
pure antifreeze to obtain a 40% antifreeze solution
Variable:
X = quarts to be drained off
16(0.40) = x + (16-x)(0.25)
6.4 = x + 4 – 0.25x
2.4 = 0.75x
X = 3.2 quarts