More Practice Problems for Algebra 1 Final Exam
KEY
Solve these problems using Algebra.
1. Find two consecutive positive integers whose product is 132.
I am finding two consecutive positive integers whose product is 132.
Let n = 1st positive integer and n + 1 = 2nd positive integer
n(n + 1) = 132
(n + 12)(n - 11) = 0
n2 + n = 132
n = -12 or n = 11.
n2 + n – 132 = 0
Check
11(12) = 132
Since the answer is a positive integer, throw out -12 answer.
The two consecutive positive integers whose product is 132 are 11 and 12.
2. How many liters of water must be added to 30 L of 65% acid solution to dilute it to a
20% solution? I am finding out how many liters of water must be added to dilute the
solution.
Original solution
Water
Final solution
Quantity
30
x
30 + x
%
.65
0
.20
Amount of acid in solution
(.65)30 = 19.5
0
.20(30 + x)
19.5 = .20(30 + x)
Check
19.5 = 6 + .2x
.65(30) = 19.5
13.5 = .2x
.20(30 + 67.5) = .2 (97.5) = 19.5
x = 67.5
I must add 67.5 L of water to a 65% acid solution to dilute it to a 20% solution.
3. The sum of the squares of two consecutive negative odd integers is 130. Find the integers.
I am finding 2 consecutive negative odd integers.
Check
(-9)2 = 81
Let n = the 1st negative odd integer and n+2 = the 2nd negative odd integer
(-7)2 = 49
sum= 130
n2 + (n + 2)2 = 130
2n2 + 4n – 126 = 0
n = -9 or 7
n2+ (n + 2)(n+2) = 130
2(n2 + 2n – 63) = 0
n2+ n2 + 4n + 4 = 130
Since answer is - throw
out 7.
2(n + 9)(n-7) = 0
The two negative odd integers whose sum of their squares is 130 are -9 and -7.
4. Two trains leave the station at the same time going in opposite directions. One was going
45 mph and the other was going 40 mph. How long will it take the trains to be 340 miles
apart?
Rate ×
Train 1
Train 2
Time =
45t
40t
340
I am finding how long it will take the trains to be 340 miles apart.
Let t = time
45t + 40t = 340
45
40
Distance
t
t
t=4
85t = 340
Check
4(40) = 160
4(45) = 180
total = 340
It took 4 hours for the trains to be 340 miles apart.
5. Steve has twice as many dimes as quarters. He has 1 more nickel than quarters. All
together he has $3.05. Find the number of each coin that he has.
Type of coins
Number of coins Value
Dimes
2x
10
Nickels
x+1
5
quarters
x
25
I am finding the number of each coin.
20x + 5x + 5 + 25x = 305
50x = 300
50x + 5 = 305
x=6
Todd has 6 quarters, 12 dimes and 7 nickels.
Total worth or total
value
10(2x) = 20x
5(x + 1) = 5x + 5
25x
6 quarters = 1.50
12 dimes = 1.20
7 nickels = .35
total =
$3.05
Check
6. Todd wants to sell a popcorn and peanut mix at basketball games. Popcorn sells for 30
cents a pound and peanuts sell for $1.95 a pound. He wants to make 50 pounds of the
mixture. How many pounds of each should he mix to sell the mixture at 63 cents a pound?
I am finding how many pounds of each he should use to make a mixture worth 63 cents/lb.
Type of
Amount of
Value
Total worth
1500 – 30x + 195x = 3150
ingredients
ingredient
or total value 165x + 1500 = 3150
Check
popcorn
50 - x
30
30(50 – x)
165x= 1650
.30(40) = 12.00
peanuts
x
195
195x
x = 10
10(1.95) = 19.50
mixture
50
63
50(63)=3150 50 -10 - 40
31.50
Todd should mix 10 pounds of peanuts with 40 pounds of popcorn.
50(.63) = 31.50
7. A salmon swims 100 m in 8 minutes downstream. Going upstream, it would take the fish
20 minutes to swim the same distance. What is the rate of the salmon in still water? What is
the rate of the current? Give final answer in m/minute.
I am finding the speed (m/min) of the salmon in still water and the speed of the current.
Let s = speed of the salmon in still water and c = speed of the water
rate time distance
upstream
s - c 20
20 (s – c) = 100
downstream
s+c 8
8(s + c) = 100
Check
20 (s – c) = 100
s–c=5
8.75 – c = 5
20(8.75-3.75) = 20(5)
20
20
s + c = 12.5
c = 3.75
=100
s–c=5
2s = 17.5
2
2
8(8.75
+
3.75) = 8(12.5)
8(s + c) = 100
s = 8.75
=100
8
8
s + c = 12.5
The speed of the salmon in still water is 8.75 m/min and the current’s speed is 3.75
m/min.
8. A plane left an airport and flew with the wind for 4 hours, covering 2000 miles. It then
returned over the same route to the airport flying against the wind in 5 hours. Find the rate of
the plane in still air and the speed of the wind.
I am finding the speed of the plane in still air and the speed of the wind.
Let p = speed of the plane in still air and w = speed of the wind
rate
time distance
Check
Going with the wind
p+w 4
4(p + w) = 2000
4(450+50) = 4(500)
=2000
Returning against the wind P - w 5
5(p - w) = 2000
4(p + w) = 2000
4
4
p + w = 500
p – w = 400
2p = 900
2
2
p = 450
5(450 - 50) = 5(400)
=2000
450 + w = 500
w = 50
5(p - w) = 2000
5
5
The speed of the plane in still air is 450 mph and the wind’s speed is 50 mph.
9. Six boxes of oranges and 5 boxes of grapefruit cost $61. At the same time and place 3
boxes of oranges and 2 boxes of grapefruit cost $28. Find the cost of one box of each.
I am finding the cost of one box of grapefruit and one box of oranges.
Let o = cost of 1 box of oranges and g = the cost of one box of grapefruit
6o + 5g = 61
3o + 2g = 28
6o + 5g = 61
-2(3o + 2g) = 28(-2)
6o + 5g = 61
-6o – 4g = -56
g=5
A box of grapefruit cost $5 and a box of oranges cost $6.
3o + 2(5) = 28
3o + 10 = 28
3o = 18
o=6
Check
6(6) + 5(5)=
36 + 35 = 61
3(6) + 2(5) =
18 + 10 = 28
10. A farmer raises chickens and cows. There are 34 animals in all. He counts 110 legs on
theses animals. How many of each animal does he have?
I am finding out how many chickens and how many cows the farmer has.
Let 34 -x = the number of chickens and
Let x = the number of cows
2(34 – x) + 4x= 110
34 – 21 = 13
Check
68 – 2x + 4x = 110
68 + 2x = 110
2(13) = 26
68 - 68 + 2x = 110 - 68
4(21) = 84
2x = 42
2
2
26 + 84 = 110
x = 21
The farmer has 21 cows and 13 chickens.
11. The perimeter of a rectangle is 114 feet. Its length is three more than twice its width.
Find the dimensions of the rectangle.
w + 2w + 3 + w + 2w + 3 = 114
2w + 3
6w + 6 = 114
Check
w
w
6w + 6 – 6 = 114 – 6
39
6w = 108
2w + 3
39
6
6
w = 18
18
18
114
2(18) + 3 = 39
The width of the rectangle is 18 feet and the length is 39 feet.
12. Tina can clean the rooms on one office floor in 2 hours. Her brother Mark can clean the
same rooms in 3 hours. How long will it take them working together to clean the rooms?
Tina
Mark
t
t
+ =1
2 3
t
2
6( +
t
) = 1(6)
3
Hourly Work rate
time
Work done
1
2
1
3
t
t
2
t
3
t
5t = 6
t=
Check
1 6 6 3
i =
=
2 5 10 5
1 6 6 2
i =
=
3 5 15 5
3 2 5
+ = =1
5 5 5
6
1
=1
5
5
3t + 2t = 6
1
5
It will take Tina and Mark working together 1 hours to clean the rooms.