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Problem 1 – Mathematics (Algebra)
Find three numbers such that the second number is 3 more than twice the
first number, and the third number is four times the first number. The sum
of the three numbers is 164.
Solution:
x + (2x + 3) + 4x = 164
x = 23
2x + 3 = 49
4x = 92
The three numbers are 23, 49, 92.
Problem 2 – Mathematics (Algebra)
Suppose that a computer store just announced an 8% decrease in the
price of a particular computer model. If this computer sells for $2162 after
the decrease, find the original price of this computer.
Solution:
x – 0.08x = 2162
x = 2350
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Problem 3 – Mathematics (Algebra)
A pennant in the shape of an isosceles triangle is to be constructed for the Slidell
High School Athletic Club and sold at a fund-raiser. The company manufacturing
the pennant charges according to perimeter, and the athletic club has determined
that a perimeter of 149 centimeters should make a nice profit. If each equal side of
the triangle is twice the length of the third side, increased by 12 centimeters, find
the lengths of the sides of the triangle pennant.
Solution:
(2x + 12) + (2x + 12) + x = 149
x = 25
2x + 12 = 62
Therefore, the length of the sides of the triangle are 25, 62, 62.
Problem 4 – Mathematics (Algebra)
Kelsey Ohleger was helping her friend Benji Burnstine studey for an algebra exam.
Kelsey told Benji that her three latest art history quiz scores are three consecutive
even integers whose sum is 264. Help Benji find the scores.
Solution:
x + (x + 2) + (x + 4) = 264
x = 86
x + 2 = 88
x + 4 = 90
The three consecutive even integers are 86, 88 and 90
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Problem 5 – Mathematics (Algebra)
In 2010, 8476 earthquakes occurred in the United States. Of these, 91.4% were
minor tremors with magnitudes of 3.9 or less on the Richter scale. How many
minor earthquakes occur in the United States in 2010? Round to the nearest
whole number.
Solution:
8476(0.914) = 7747
Problem 6 – Mathematics (Algebra)
Karen Estes just received an inheritance of $10,000 and plans to place all the
money in a savings account that pays 5% compounded quarterly to help her son
go to college in 3 years. How much money will be in the account in 3 years?
Solution:
⎛
0.05 ⎞
A = 10000 ⎜ 1 +
4 ⎟⎠
⎝
12
= 11,607.55
Problem 7 – Mathematics (Algebra)
The fastest average speed by a cyclist across the continental United States is 15.4
mph, by Pete Penseyres. If he traveled a total distance of about 3107.5 miles at
this speed find his time cycling in days, hours and minutes.
Solution:
3107.5 = 15.4t
201.79 = t
t = 8 days, 9 hours, 50 minutes
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Problem 8 – Mathematics (Algebra)
A serving of cashew contains 14 grams of fat, 7 grams of carbohydrate,
and 6 grams of protein. How many calories are in this serving of
cashews?
Solution:
C = 4h + 9f + 4p
h = number of grams of carbohydrate
f = number of grams of fat
p = number of grams of protein
C = 4(7) + 9(14) + 4(6)
C = 178 calories
Problem 9 – Mathematics (Algebra)
In the United States, the annual consumption of cigarettes is declining.
The consumption c in billions of cigarettes per year since the year 2000
can be approximated by the formula c = - 9.4t + 431 where t is the number
of years after 2000. Use this formula to predict the years that the
consumption of cigarettes will be less than 200 billion per year.
Solution:
c = - 9.4(20) + 431 = 243
- 9.4t + 431 < 200
9.4t = 231
t = 24.6 say 25
The annual consumption of cigarettes will be less than 200 billion in
2025.
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Problem 10 – Mathematics (Algebra)
The cost C of producing x number of scientific calculators is given by
C = 4.50x + 3000, and the revenue R from selling them is given by R = 16.50x.
Find the number of calculators that must be sold to break even.
Solution:
C = 4.50x + 3000
R = 16.50x
4.50x + 3000 = 16.50x
x = 250 calculators
Problem 11 – Mathematics (Algebra)
Chine, the United States, and France are predicted to be the top tourist
destinations by 2020. In this year, the United States is predicted to have 9 million
more tourists than France, and China is predicted to have 44 million more tourists
than France. If the total number of tourist predicted for these three countries is 332
million, find the number predicted for each country in 2020.
Solution:
C = no. of tourist in China
F = no. of tourist in France
U = no. of tourist in United States
U=9+F
C = F + 44
C + F + U = 332
F + F + 44 + 9 + F = 332
3F = 279
F = 93
C = 93 + 44 = 137
U = 9 + 93 = 102
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Problem 12 – Mathematics (Algebra)
Th number of cars manufactured on an assembly line at a General Motors
plant varies jointly as the number of workers and the time they work. If 200
workers can produce 60 cars in 2 hours, find how many cars 240 workers
should be able to make in 3 hours?
Solution:
60 = K(200)(2)
60
K=
400
60
N=
(240)(3) = 108
400
Problem 13 – Mathematics (Algebra)
Boyle’s law says that if the temperature stays the same, the pressure P of
a gas is inversely proportional to the volume V. If a cylinder in a steam
engine has a pressure of 960 kilopascals when the volume is 1.4 cubic
meters, find the pressure when the volume increases to 2.5 cubic meters.
Solution:
k
P=
V
k
960 =
1.4
1344 = k
P=
1344
= 537.6 kPa
2.5
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Problem 14 – Mathematics (Algebra)
Hooke’s law states that the distance a spring stretches is directly
proportional to the weight attached to the spring. If a 40-pound weight
attached to the spring stretches the spring 5 inches, find the distance that
a 65-pound weight attached to the spring stretches the spring.
Solution:
d = kw
5 = k(40)
1
=k
8
1
d = (65) = 8.125
8
Problem 15 – Mathematics (Algebra)
If a certain number is subtracted from the numerator and added to the
denominator of 9/19, the new fraction is equivalent to 1/3. Find the
number.
Solution:
9-n
1
=
19 + n 3
3(9 - n) = 19 + n
27 - 3n = 19 + n
n=2
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Problem 16 – Mathematics (Algebra)
If the cost, C(x), for manufacturing x units of a certain product is given by
C(x) = x2 – 15x + 50, find the number of units manufactured at a cost of
$9500.
Solution:
C(x) = x2 – 15x + 50
9500 = x2 – 15x + 50
x2 – 15x – 9450 = 0
x = 105
Problem 17 – Mathematics (Algebra)
The world’s highest bridge, the Millau Viaduct in France, is 1125 feet
above the River Tarn. An object is dropped from the top of this bridge.
Neglecting air resistance, the height of the object at time t seconds is
given by the polynomial function P(t) = - 16t2 + 1125. Find the height of
the object when t = 8 seconds.
Solution:
P(t) = - 16t2 + 1125
P(8) = - 16(8)2 + 1125
P(8) = 101
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Problem 18 – Mathematics (Algebra)
A company that manufactures boxes recently purchased $2000 worth of
new equipment to offer gift boxes to its customers. The cost of producing
a package of gift boxes is $1.50 and it is sold for $4.00. Find the number
of packages that must be sold for the company to break even.
Solution:
2000 + 1.50x = 4x
2.50x = 2000
x = 800
Problem 19 – Mathematics (Algebra)
Mr.s Laser agrees to give her son Mark an allowance of $0.1 on the first
day of his 14-day vacation, $0.20 on the second day, $0.40 on the third
day, and so on. Write an equation of a sequence whose terms correspond
to Mark’s allowance. Find the allowance Mark will receive on the last day
of his vacation.
Solution:
0.20
r=
=2
0.1
a n = a1 r n−1
a n = 0.1(2)n−1
a14 = 819.20
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Problem 20 – Mathematics (Algebra)
If the third term of an arithmetic sequence is 23 and the eighth term is 63.
Find the sixth term.
Solution:
an = a1 + (n – 1)d
23 = a1 + (3 – 1)d
23 = a1 + 2d
Solving for a1 = 7, d = 8
Therefore the sixth term is
a6 = 7 + (6 – 1)(8)
a6 = 47
63 = a1 + (8 – 1)d
63 = a1 + 7d
Problem 21 – Mathematics (Algebra)
Find the first four terms of a geometric sequence whose first term is 8 and
whose common ratio is – 3.
Solution:
an = a1 rn-1
a1 = 8
a2 = 8(-3)2-1
a2 = - 24
a3 = 8(-3)3-1
a3 = 72
The four terms are 8, - 24, 72, - 216.
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Problem 22 – Mathematics (Algebra)
The research department of a company that manufactures children’s fruit
drinks is experimenting with a new flavor. A 17.5% fructose solution is
needed but only 10% and 20% solutions are available. How many gallons
of the 10% fructose solution should be mixed with the 20% solution to
obtain 20 gallons of a 17.5% fructose solution?
Solution:
x
y
20
20%
17.5%
10%
+
=
80%
82.5%
90%
10x + 20y = 20(17.5)
90x + 80y = 20(82.5)
Solving for:
x = 5 gallons of 10% fructose solution
y = 15 gallons of 20% fructose solution
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Problem 23 – Mathematics (Algebra)
A motel in New Orleans charges $90 per day for double occupancy and
$80 per day for single occupancy during off-season. If 80 rooms are
occupied for a total of $6930, how many rooms of each kind are
occupied?
Solution:
Let x = no. of double room
80 – x = no. of single room
90x + 80(80 – x) = 6930
x = 53 double rooms
80 – x = 27 single rooms
Problem 24 – Mathematics (Algebra)
An endangered species of sparrow had an estimated population of 800 in
2000 and scientists predicted that its population would decrease by half
each year. Estimate the population in 2004.
Solution:
an = a1 rn-1
r = 0.5
n=5
a1 = 800
an = 800(0.5)5-1
an = 50 sparrows
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Problem 25 – Mathematics (Algebra)
The number of cases of new infectious disease is doubling every year such that
the number of cases is modeled by a sequence whose general term is
an = 75(2)n-1, where n is the number of the year just beginning. Find how many
cases there will be at the beginning of the sixth year. Find how many cases there
were at the beginning of the first year.
Solution:
No. of cases at the beginning of the sixth year:
n=6
an = 75(2)n-1,
an = 75(2)6-1
an = 2,400 cases
No. of cases at the beginning of the first year.
n=1
an = 75(2)n-1,
an = 75(2)1-1
an = 75 cases
Problem 25 – Mathematics (Algebra)
Keith Robinson bought two Siamese fighting fish. But when he got home, he found
he only had one rectangular tank that was 12 in. long, 7 in. wide, and 5 in. deep.
Since the fish must be kept separated, he
7 in
12 in
needed to insert a plastic divider in the diagonal
of the tank. He already has a piece that is 5 in. in
5 in
one dimension but how long must it be to fit
corner to corner in the tank?
Solution:
x2 = (7)2 + (12)2
x = 13.89 in.
5 in
x
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Problem 26 – Mathematics (Algebra)
The value of an automobile bought in 2012 continues to decrease as time
passes. Two years after the car was bought, it was worth $20600; four
years after, it was bought, it was worth $14600.
➀ Assuming that this relationship between the number of years past
2012 and the value of the car is linear, write an equation describing
this relationship. Hint: Use ordered pairs of the form (year past 2012,
value of the automobile).
Solution:
an = a1 + (n – 1)d
After 2 yrs, n = 2
20600 = a1 + (2 – 1)d
20600 = a1 + d
After 4 yrs, n = 4
14600 = a1 + (4 – 1)d
14600 = a1 + 3d
Solving for:
a1 = 23600
d = - 3000
General equation:
an = 23600 + (n – 1)(- 3000)
an = 26600 – 3000n
➁ Use this equation to estimate the value of the automobile in 2018.
Solution:
At 2018, n = 6
a6 = 26600 – 3000(6)
a6 = $8600
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Problem 27 – Mathematics (Algebra)
A starting salary for a consulting company is $57000 per year with
guaranteed annual increase of $2200 for the next 4 years. Write the
general term for the arithmetic sequence that models the potential annual
salaries and find the salary for the third year.
Solution:
an = a1 + (n – 1)d
an = 57000 + (n – 1)(2200)
an = 57000 + n(2200) – 2200
an = 54,800 + 2200n
an = 54,800 + 2200n
when n = 3
an = 54,800 + 2200(3)
an = $61,400
Problem 28 – Mathematics (Algebra)
Find the earthquake’s magnitude on the Richter scale if a recording
station measures an amplitude of 300 micrometers and 2.5 seconds
between waves. Assume that B is 4.2. Approximate the solution to the
nearest tenth.
Solution:
⎛ a⎞
R = log ⎜ ⎟ + B
⎝ T⎠
⎛ 300 ⎞
R = log ⎜
+ 4.2
⎟
⎝ 2.5 ⎠
R = log(120) + 4.2
R = 6.3
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Problem 29 – Mathematics (Algebra)
The Rogun Dam in Tajikistan (part of the former USSR) is the tallest dam
in the world at 1100 feet. How long would it take an object to fall from the
top to the base of the dam?
Solution:
S = Vot + ½ at2
S = - 1100 ft.
a = - 32 ft/sec2
Vo = 0
- 1100 = 0 + ½ (- 32)t2
t = 8.29 seconds
Problem 30 – Mathematics (Algebra)
After applying a test antibiotic, the population of a bacterial culture is
reduced by one-half every day. Predict how large the culture will be at the
start of day 7 if it measures 4800 units at the beginning of day 1.
Solution:
an = a1 rn-1
r = 0.5
n=7
a1 = 4800
an = 4800(0.5)7-1
an = 75 units
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Problem 31 – Mathematics (Algebra)
If a radioactive element has a half-life of 3 hours, then x grams of element
dwindles to x/2 grams after 3 hours. If a nuclear reactor has 400 grams of that
radioactive element, find the amount of radioactive material after 12 hours.
Solution:
an = a1 rn-1
r = 0.5 or ½
n=5
a1 = 400
an = 400(1/2)5-1
an = 25 grams
Problem 32 – Mathematics (Algebra)
A ball is dropped from a height of 20 feet and repeatedly rebounds to a height that
is 4/5 of its previous height. Find the total distance the ball covers before it comes
to rest.
Solution:
Consider the problem as
infinite geometric sequence.
20 ft
4
r=
a1 = 20 ft.
5
a
20
S= 1 =
= 100
1 - r 1 - 4/5
Due to rebounding, double the distance minus the initial.
T = 2(100) - 20 = 180 ft.
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Problem 33 – Mathematics (Algebra)
The sum of three numbers is 40. The first number is five more than the
second number. It is also twice the third. Find the numbers.
Solution:
x = 1st no.
y = 2nd no.
z = 3rd no.
x + y + z = 40
x=y+5
x = 2z
(1)
(2)
(3)
Convert eqn. (2) and (3) in terms of x
x=y+5
y=x–5
x = 27
x
z=
(5)
2
Substitute (4) and (5) to eqn. (1)
⎡
⎤
x
x
+
(x
5)
+
=
40
2
⎥
⎢
2
⎣
⎦
2x + 2x - 10 + x = 80
5x = 90
x = 18
y = 13
z=9
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Problem 34 – Mathematics (Algebra)
The number of baby gorillas born at the San Diego Zoo is a sequence defined by
an = n(n – 1), where n is the number of years the zoo has owned gorillas. Find the
total number of baby gorillas bonr in the first 4 years.
Solution:
4
S4 = ∑ i(i - 1)
i =1
S4 = 1(1 - 1) + 2(2 - 1) + 3(3 - 1) + 4(4 - 1)
S 4 = 20
Problem 35 – Mathematics (Algebra)
A company’s cost per tee shirt for silk screening x tee shirts is given by the rational
3.2x + 400
C(x)
=
function
. Find the cost per tee shirt.
x
➀ For printing 1000 tee shirts
Solution:
3.2x + 400
C(x) =
; x = 1000
x
3.2(1000) + 400
=
1000
C = $3.60
➀ For printing 100 tee shirts
Solution:
3.2x + 400
C(x) =
; x = 100
x
3.2(100) + 400
=
100
C = $7.20
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Problem 36 – Mathematics (Algebra)
Mark Keaton’s workout consists of jogging for 3 miles and then riding his
bike for 5 miles at a speed 4 miles per hour faster than he jogs. If his total
workout time is 1 hour, find his jogging speed and his biking speed.
Solution:
x = speed in jogging
x + 4 = speed in biking
d
t1 + t 2 = 1
v=
t
3
5
d
+
=1
t=
x x +4
v
t1 = time in jogging
t2 = time in biking
x = 6 mph jogging speed
x + 4 = 10 mph biking speed
Problem 37 – Mathematics (Algebra)
Suppose that an open box is to be made from a square sheet of
cardboard by cutting out squares from each corner as shown and then
folded along the dotted lines. If the box is to have a volume of 128 cubic
inches, find the original dimensions of the sheet of cardboard.
Solution:
Assume cut-out of 2 in. by 2 in.
V=l x w x h
128 = (x – 4)(x – 4)(2)
x = 12 in.
Therefore, 12 in. by 12 in.
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Problem 38 – Mathematics (Algebra)
A sprinkler that sprays water in a circular pattern is to be used to water a
square garden. If the area of the garden is 920 square feet, find the
smallest whole number radius that the sprinkler can be adjusted to so that
the entire garden is watered.
Solution:
920 = x 2
x = 30.33 ft.
2
⎛ x⎞
⎛ x⎞
2
r =⎜ ⎟ +⎜ ⎟
⎝ 2⎠
⎝ 2⎠
r
2
r = 21.45 ≈ 22 ft.
Sprinkler
x
Problem 39 – Mathematics (Algebra)
Gary Marcus and Tony Alva work at Lombardo’s Pipe and Concrete.
Mr. Lombardo is preparing an estimate for a customer. He knows that
Gary can lay 4 slab of concrete in 6 hours. Tony can lay the same size
slab in 4 hours. If both work on the job and the cost of labor is $45.00 per
hour, determine what the labor estimate should be.
Solution:
⎛ 1 1⎞
⎜⎝ 6 + 4 ⎟⎠ t = 1
t = 2.4 hours
Labor estimate = 2.4(45) = $108