Algebra Word Problems
1. If 8 men take 12 days to assemble 16 machines, how many days will it take 15 men to assemble 50
machines?
2. The force of wind on a sail varies jointly as the area of the sail and the square of the wind velocity.
On a square foot of sail the force is 1 lb when the wind velocity is 15 mi/hr. Find the force of a 45
mi/hr wind on a sail of area 20 square yards.
3. A man has 40 ft of wire fencing with which to form a rectangular garden. The fencing is to be used
only on three sides of the garden, his house providing the fourth side. Determine the maximum area
which can be enclosed.
4. A rectangular piece of tin has dimensions 12 in. by 18 in. It is desired to make an open box from this
material by cutting out equal squares from the corners and then bending up the sides. What are the
dimensions of the squares cut out if the volume of the box is to be as large as possible?
5. A cylindrical can is to have a volume of 200 cubic inches. Find the dimensions of the can which is
made of the least amount of material.
6. There is available 120 ft of wire fencing with which to enclose two equal rectangular gardens A and
B. If no wire fencing is used along the sides formed by the house, determine the maximum
combined area of the gardens.
7. Find the area of the largest rectangle which can be inscribed in a right triangle whose legs are 6 and
8 in. respectively
8. If one pump can fill a pool in 16 hours and if two pumps can fill the pool in 6 hours, how fast can the
second pump fill the pool?
9. How many liters of pure alcohol must be added to 15 liters of a 60% alcohol solution to obtain an
80% alcohol solution?
10. The sum of two numbers is 21, and one number is twice the other. Find the numbers.
11. Ten less than four times a certain number is 14. Determine the number.
12. The sum of three consecutive integers is 24. Find the integers.
13. The sum of two numbers is 37. If the larger is divided by the smaller, the quotient is 3 and the
remainder is 5. Find the numbers.
14. A man is 41 years old and his son is 9. In how many years will the father be three times as old as
the son?
15. Ten years ago Jane was four times as old as Bianca. Now she is only twice as old as Bianca. Find
their present ages.
16. Robert has 50 coins, all in nickels and dimes, amounting to $3.50. How many nickels does he
have?
17. In a purse are nickels, dimes, and quarters amounting to $1.85. There are twice as many dimes as
quarters, and the number of nickels is two less than twice the number of dimes. Determine the
number of coins of each kind.
18. The tens digit of a certain two-digit number exceeds the units digit by 4 and is 1 less than twice the
units digit. Find the two-digit number.
19. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 4/7
times the original number. Determine the original number.
20. The sum of the digits of a two-digit number is 12. If the digits are reversed, the new number is 4/7
times the original number. Determine the original number.
21. What amount should an employee receive as bonus so that she would net $500 after deducting
30% for taxes?
22. At what price should a merchant mark a sofa that costs $120 in order that it may be offered at a
discount of 20% on the marked price and still make a profit of 25% on the selling price?
23. When each side of a given square is increased by 4 feet the area is increased by 64 square feet.
Determine the dimensions of the original square.
24. One leg of a right triangle is 20 inches and the hypotenuse is 10 inches longer than the other leg.
Find the lengths of the unknown sides.
25. Temperature Fahrenheit = 9/5(temperature Celsius) + 32. At what temperature have the Fahrenheit
and Celsius readings the same value?
26. A mixture of 40 lb of candy worth 60¢ a pound is to be made up by taking some worth 45¢/lb and
some worth 85¢/lb. How many pounds of each should be taken?
27. A tank contains 20 gallons of a mixture of alcohol and water which is 40% alcohol by volume. How
much of the mixture should be removed and replaced by an equal volume of water so that the
resulting solution will be 25% alcohol by volume?
28. What weight of water must be evaporated from 40 lb of a 20% salt solution to produce a 50%
solution? All percentages are by weight.
29. How many quarts of a 60% alcohol solution must be added to 40 quarts of a 20% alcohol solution to
obtain a mixture which is 30% alcohol? All percentages are by volume.
30. Two unblended manganese (Mn) ores contain 40% and 25% of manganese respectively. How many
tons of each must be mixed to give 100 tons of blended ore containing 35% of manganese? All
percentages are by weight.
31. Two cars A and B having average speeds of 30 and 40 mi/hr respectively are 280 miles apart. They
start moving toward each other at 3:00 P.M. At what time and where will they meet?
32. A and B start from a given point and travel on a straight road at average speeds of 30 and 50 mi/hr
respectively. If B starts 3 hr after A, find (a) the time and (b) the distance they travel before meeting.
33. A and B can run around a circular mile track in 6 and 10 minutes respectively. If they start at the
same instant from the same place, in how many minutes will they pass each other if they run around
the track (a) in the same direction, (b) in opposite directions?
34. A boat, propelled to move at 25 mi/hr in still water, travels 4.2 mi against the river current in the
same time that it can travel 5.8 mi with the current. Find the speed of the current.
35. A can do a job in 3 days, and B can do the same job in 6 days. How long will it take them if they
work together?
36. A tank can be filled by three pipes separately in 20, 30, and 60 minutes respectively. In how many
minutes can it be filled by the three pipes acting together?
37. A and B working together can complete a job in 6 days. A works twice as fast as B. How many days
would it take each of them, working alone, to complete the job?
38. A’s rate of doing work is three times that of B. On a given day A and B work together for 4 hours;
then B is called away and A finishes the rest of the job in 2 hours. How long would it take B to do the
complete job alone?
39. A man is paid $18 for each day he works and forfeits $3 for each day he is idle. If at the end of 40
days he nets $531, how many days was he idle?
40. A father is 24 years older than his son. In 8 years, he will be twice as old as his son. Determine their
present ages.
41. Mary is fifteen years older than her sister Jane. Six years ago, Mary was six times as old as Jane.
Find their present ages.
42. Larry is now twice as old as Bill. Five years ago, Larry was three times as old as Bill. Find their
present ages.
43. The tens digit of a certain two-digit number exceeds the units digit by 3. The sum of the digits is 1/7
of the number. Find the number.
44. The sum of the digits of a certain two-digit number is 10. If the digits are reversed, a new number is
formed which is one less than twice the original number. Find the original number.
45. The tens digit of a certain two-digit number is 1/3 of the units digit. When the digits are reversed, the
new number exceeds twice the original number by 2 more than the sum of the digits. Find the
original number.
46. Two years ago a man was six times as old as his daughter. In 18 years he will be twice as old as his
daughter. Determine their present ages.
47. If the numerator of a certain fraction is increased by 2 and the denominator is increased by 1, the
resulting fraction equals 1/2. If, however, the numerator is increased by 1 and the denominator
decreased by 2, the resulting fraction equals 3/5. Find the fraction.
48. Find the two-digit number satisfying the following two conditions. (1) Four times the units digit is six
less than twice the tens digit. (2) The number is nine less than three times the number obtained by
reversing the digits.
49. Tank A contains a mixture of 10 gallons water and 5 gallons pure alcohol. Tank B has 12 gallons
water and 3 gallons alcohol. How many gallons should be taken from each tank and combined in
order to obtain an 8 gallon solution containing 25% alcohol by volume?
50. A given alloy contains 20% copper and 5% tin. How many pounds of copper and of tin must be
melted with 100 lb of the given alloy to produce another alloy analyzing 30% copper and 10% tin?
All percentages are by weight.
51. Determine the rate of a woman’s rowing in still water and the rate of the river current, if it takes her 2
hours to row 9 miles with the current and 6 hours to return against the current.
52. Two particles move at different but constant speeds along a circle of circumference 276 ft. Starting
at the same instant and from the same place, when they move in opposite directions they pass each
other every 6 sec and when they move in the same direction they pass each other every 23 sec.
Determine their rates.
53. A picture frame of uniform width has outer dimensions 12 in. by 15 in. Find the width of the frame (a)
if 88 square inches of picture show, (b) if 100 square inches of picture show.
54. A pilot flies a distance of 600 miles. He could fly the same distance in 30 minutes less time by
increasing his average speed by 40 mi/hr. Find his actual average speed.
55. A retailer bought a number of shirts for $180 and sold all but 6 at a profit of $2 per shirt. With the
total amount received she could buy 30 more shirts than before. Find the cost per shirt.
56. A and B working together can do a job in 10 days. It takes A 5 days longer than B to do the job when
each works alone. How many days would it take each of them, working alone, to do the job?
57. A ball projected vertically upward with initial speed v0 ft/sec is at time t sec at a distance s ft from
the point of projection as given by the formula s ¼ v0t 216t2. If the ball is given an initial upward
speed of 128 ft/sec, at what times would it be 100 ft above the point of projection?
58. A slide of uniform grade is to be built on a level surface and is to have 10 supports equidistant from
each other. The heights of the longest and shortest supports will be 421 2 feet and 2 feet
respectively. Determine the required height of each support.
59. A freely falling body, starting from rest, falls 16 ft during the first second, 48 ft during the second
second, 80 ft during the third second, etc. Calculate the distance it falls during the fifteenth second
and the total distance it falls in 15 seconds from rest.
60. In a potato race, 8 potatoes are placed 6 ft apart on a straight line, the first being 6 ft from the
basket. A contestant starts from the basket and puts one potato at a time into the basket. Find the
total distance she must run in order to finish the race.
61. A boy agrees to work at the rate of one cent the first day, two cents the second day, four cents the
third day, eight cents the fourth day, etc. How much would he receive at the end of 12 days?
62. It is estimated that the population of a certain town will increase 10% each year for four years. What
is the percentage increase in population after four years?
63. From a tank filled with 240 gallons of alcohol, 60 gallons are drawn off and the tank is filled up with
water. Then 60 gallons of the mixture are removed and replaced with water, etc. How many gallons
of alcohol remain in the tank after 5 drawings of 60 gallons each are made?
64. A man wishes to borrow $200. He goes to the bank where he is told that the interest rate is 5%,
interest payable in advance, and that the $200 is to be paid back at the end of one year. What
interest rate is he actually paying?
65. A merchant borrows $4000 under the condition that she pay at the end of every 3 months $200 on
the principal plus the simple interest of 6% on the principal outstanding at the time. Find the total
amount she must pay.
66. Find the compound interest and amount of $2800 in 8 years at 5% compounded quarterly.
67. A student has a choice of 5 foreign languages and 4 sciences. In how many ways can he choose 1
language and 1 science?
68. In how many ways can 2 different prizes be awarded among 10 contestants if both prizes (a) may
not be given to the same person, (b) may be given to the same person?
69. In how many ways can 5 letters be mailed if there are 3 mailboxes available?
70. There are 4 candidates for president of a club, 6 for vice-president and 2 for secretary. In how many
ways can these three positions be filled?
71. In how many different orders may 5 persons be seated in a row?
72. In how many ways can 7 books be arranged on a shelf?
73. Twelve different pictures are available, of which 4 are to be hung in a row. In how many ways can
this be done?
74. It is required to seat 5 men and 4 women in a row so that the women occupy the even places. How
many such arrangements are possible?
75. In how many ways can n women be seated in a row so that 2 particular women will not be next to
each other?
76. Six different biology books, 5 different chemistry books and 2 different physics books are to be
arranged on a shelf so that the biology books stand together, the chemistry books stand together,
and the physics books stand together. How many such arrangements are possible?
77. Determine the number of different words of 5 letters each that can be formed with the letters of the
word chromate (a) if each letter is used not more than once, (b) if each letter may be repeated in
any arrangement. (These words need not have meaning.)
78. How many numbers may be formed by using 4 out of the 5 digits 1, 2, 3, 4, 5 (a) if the digits must
not be repeated in any number, (b) if they may be repeated? If the digits must not be repeated, how
many of the 4-digit numbers (c) begin with 2, (d) end with 25?
79. How many numbers between 3000 and 5000 can be formed by using the 7 digits 0, 1, 2, 3, 4, 5, 6 if
each digit must not be repeated in any number?
80. From 11 novels and 3 dictionaries, 4 novels and 1 dictionary are to be selected and arranged on a
shelf so that the dictionary is always in the middle. How many such arrangements are possible?
81. How many signals can be made with 5 different flags by raising them any number at a time?
82. Compute the sum of the 4-digit numbers which can be formed with the four digits 2, 5, 3, 8 if each
digit is used only once in each arrangement.
83. In how many ways can 5 persons be seated at a round table?
84. In how many ways can 8 persons be seated at a round table if 2 particular persons must always sit
together?
85. In how many ways can 4 men and 4 women be seated at a round table if each woman is to be
between two men?
86. A box contains black chips and red chips. A person draws two chips without replacement. If the
87. probability of selecting a black chip and a red chip is 15/56 and the probability of drawing a black
chip on the first draw is 3/4, what is the probability of drawing a red chip on the second draw, if you
know the first chip drawn was black?
88. One ball is drawn at random from a box containing 3 red balls, 2 white balls, and 4 blue balls.
Determine the probability p that it is (a) red, (b) not red, (c) white, (d ) red or blue.
89. One bag contains 4 white balls and 2 black balls; another bag contains 3 white balls and 5 black
balls. If one ball is drawn from each bag, determine the probability p that (a) both are white, (b) both
are black, (c) 1 is white and 1 is black.
90. Determine the probability of throwing a total of 8 in a single throw with two dice, each of whose
faces is numbered from 1 to 6.
91. What is the probability of getting at least 1 one in 2 throws of a die?
92. Three cards are drawn from a pack of 52 cards. Determine the probability p that (a) all are aces, (b)
all are aces and drawn in the order spade, club, diamond, (c) all are spades, (d ) all are of the same
suit, (e) no two are of the same suit.
93. The probability that A can solve a given problem is 4/5, that B can solve it is 2/3, and that C can
solve it is 3/7. If all three try, compute the probability that the problem will be solved.
94.
Supplementary
95. A boat whose rate in still water is 12 miles per hour goes up a stream whose current has a rate of 2
miles per hour, and returns. The entire trip took 24 hours. Find how many hours are required for the
trip upstream and how many hours for the return trip.
96. A chemist has the same acid in two strengths. Eight liters of one acid mixed 12 liters of the second
acid gives a mixture that is 84% pure. Three liters of the first acid mixed with 2 liters of the second
acid gives a mixture that is 86% pure. Find the purity of each acid.
97. A family borrowed $6000 to repair their home, agreeing to pay $50 monthly until the principal and
interest at 6% is paid. Find the number of full payments required.
98. What is the present value of an annual pension of $1200 at 4% if the pension is to run 18 years and
the present value, V, is found by the formula V = (A/r)[1 - 1/(1 + r)n].
99. From an alphabet containing 21 consonants and 5 vowels, six-letter words are to be formed. How
many of the words will contain 4 consonants and 2 vowels with no letters repeated?
100.
In how many ways can a committee of 5 senators be appointed from the U.S. Senate, if the
majority leader is always a member of the committee?
101.
A man has in his pocket 3 pennies, 2 nickels, and a dime. If he draws two coins from his
pocket at random, what is the probability that the amount drawn exceeds 6 cents?
102.
A club has a lottery which offers two prizes. Twenty tickets are sold. If one person bought 4
of the tickets, what is the probability that they will win at least one prize?
103.
A manufacturer has in stock a quantity of metal 8 inches by 15 inches, out of which they
want to make open-top boxes by cutting equal squares out of each corner and folding up the metal
to make the sides and ends. What must the sides of the square be so as to make a box of the
greatest possible volume?
104.
Find the sum of all the integers between 46 and 100 that are divisible by 3.
105.
A colony of bacteria doubles in number each day. If there are 1000 bacteria on day 1, how
many bacteria will there be on day 7?
106.
If a, b, c, d, ... is a geometric sequence with a common ratio of r, is a2, b2, c2, d2, ... a
geometric sequence? If it is a geometric sequence, what is the common ratio?
107.
Social Security numbers in the United States have the format of NNN - NN - NNNN where
each N is one of the digits 0 through 9. If there are no additional instruction on forming the numbers,
how many Social Security numbers can be issued?
108.
At Maggie’s Marvelous Pizzas, the available toppings are pepperoni, onions, mushrooms,
green peppers, olives, pineapple, mozzarella, and anchovies. Maggie’s sells the Fancy Pizza, which
allows the customer to choose any five toppings for their Fancy Pizza. How many different Fancy
Pizzas can be ordered?
109.
A charity has 11 letters that they use to request a donation to the charity. Each month they
select 4 letters to use that month. How many different sets of letters can they use?
110.
A politician wants her constituents to rank 7 issues in order of importance to them. Replies
are put into folders, and two replies are put into the same folder when they give the same ranking to
each of the 7 issues. What is the maximum number of folders that might be needed?
111.
In how many ways can the nine justices of the U.S. Supreme Court reach a majority
decision?
112.
Brandi has applied for funding to attend a conference. She knows that her name, along with
the names of the other nine candidates, was put into a box from which two names will be drawn.
What is the probability that Brandi’s name will be selected?
113.
If the probability that Tony will like his blind date is 0.3, what is the probability he will not like
his blind date?
114.
A lottery requires a winner to correctly select 6 numbers out of 50 numbers. What is the
probability of selecting all 6 winning numbers?
115.
What is the probability that a card selected at random from a standard deck of 52 playing
cards will be a 7 or a 9?
116.
What is the probability of getting a sum of 8 when a pair of dice is rolled?