**Find below a wide variety of hard word problems in algebra. Most tricky and tough algebra word problems are covered here. If you can solve these, you can probably solve any algebra problems.**

**1.** The
cost of petrol rises by 2 cents a liter. last week a man bought 20
liters at the old price. This week he bought 10 liters at the new
price. Altogether, the petrol costs $9.20. What was the old price for
1 liter?

**2. **Teachers
divided students into groups of 3. Each group of 3 wrote a report
that had 9 pictures in it. The students used 585 pictures altogether.
How many students were there in all?

**3. **Vera
and Vikki are sisters. Vera is 4 years old and Vikki is 13 years old.
What age will each sister be when Vikki is twice as old as Vera?

**4. **A
can do a work in 14 days and working together A and B can do the same
work in 10 days. In what time can B alone do the work?

**5.** 7
workers can make 210 pairs of cup in 6 days. How many workers are required
to make 450 pairs of cup in 10 days?

**6.** Ten
years ago the ratio between the ages of Mohan and Suman was 3:5. 11
years hence it will be 11:16. What is the present age of Mohan?

**7.** The
ratio of girls to boys in class is 9 to 7 and there are 80 students in
the class. How many girls are in the class?

**8. **One
ounce of solution X contains only ingredients a and b in a ratio of 2:3.
One ounce of solution Y contains only ingredients a and b in a ratio of
1:2. If solution Z is created by mixing solutions X and Y in a ratio of
3:11, then 2520
ounces of solution Z contains how many ounces of a?

**9. **This
week Bob puts gas in his truck
when the tank was about half empty.
Five days later, bob puts gas again when the tank was about three
fourths full. If Bob Bought 24 gallons of gas, how many gallons does
the tank hold?

**10.** A
commercial
airplane flying with a speed of 700 mi/h is detected 1000 miles
away with a radar. Half an hour later an interceptor plane flying with
a speed of 800 mi/h is dispatched. How long will it take the interceptor
plane to meet with the other plane?

**11.** There
are 40 pigs and chickens in a farmyard. Joseph counted 100 legs in
all. How many pigs and how many chickens are there?

**12. **The
top of a box is a rectangle with a perimeter of 72 inches. If the box
is 8 inches high, what dimensions will give the maximum volume?

**13.** You
are raising money for a charity. Someone made a fixed donation of
500. Then, you require each participant to make a pledge of 25
dollars. What is the minimum amount of money raised if there are 224 participants.

**14.** The
sum of two positive numbers is 4 and the sum of their squares
is 28.
What are the two numbers?

**15.** Flying
against the jet stream, a jet travels 1880 mi in 4 hours. Flying with
the jet stream, the same jet travels 5820 mi in 6 hours. What is the
rate of the jet in still air and what is the rate of the jet stream?

**16. **Jenna
and her friend, Khalil, are having a contest to see who can save the
most money. Jenna has already saved $110 and every week she saves an
additional $20. Khalil has already saved $80 and every week he saves an
additional $25. Let x represent the number of weeks and y represent the
total amount of money saved. Determine in how many weeks Jenna and Khalil
will have the same amount of money.

**17.** The
sum of three consecutive terms of a geometric sequence is 104 and their
product is 13824.find the terms.

**18.** The
sum of the first and last of four consecutive odd integers is 52. What
are the four integers?

**19.** A
health club charges a one-time initiation fee and
a monthly fee. John paid
100 dollars for 2 months of membership. However, Peter paid 200 for 6
months of membership. How much will Sylvia pay for 1 year of
membership?

**20.** The
sum of two positive numbers is 4 and the sum of their cubes is 28. What
is the product of the two numbers?

**21.** A
man selling computer parts realizes that when he sells 16 computer parts,
his earning is $1700. When he sells 56 computer parts, his earning
is $4300. What will the earning be if the man sells 30 computers parts?

**22.** A
man has 15 coins in his pockets. These coins are dimes and quarters
that add
up 2.4 dollars. How many quarters and how many dimes does the man have?

**23. **The
lengths of the sides of a triangle are in the ratio 4:3:5. Find
the lengths
of the sides if the perimeter is 18 inches.

**24.** The
ratio of base to height of a equilateral
triangle is 3:4. If the area of
the triangle is 6, what is the perimeter of the triangle?

**25.** The
percent return rate of a growth fund, income fund, and money market are
10%, 7%, and 5% respectively. Suppose you have 3200 to invest and you want
to put twice as much in the growth fund as in the money market to maximize
your return. How should you invest to get a return of 250 dollars
in 1 year?

**26.** A
shark was caught whose tail weighted 200 pounds. The head of the
shark weighted
as much as its tail plus half its body. Its body weighted as much
as its head and tail. What is the weight of the shark?

**27.** The
square root of a
number plus two is the same as the
number. What
is the
number?

**28.** Suppose
you have a coupon worth 6 dollars off any item at a mall. You go to
a store at the mall that offers a 20% discount. What do you need to
do to
save the most money?

**29.** Suppose
your grades on three math exams are 80, 93, and 91. What grade do you
need on your next exam to have at least a 90 average on the four exams?

**30.** Peter
has a photograph that is 5 inches wide and 6 inches long. She enlarged
each side by the same amount. By how much was the photograph enlarged
if the new area is 182 square inches?

**31. **The
cost to produce a book is 1200 to get started plus 9
dollars per book. The
book sells for 15 dollars each. How many books must be sold to make a profit?

**32.** Store
A sells CDs
for 2 dollars each if you pay a one-time
fee of 104
dollars. Store B offers 12 free CDs and charges 10 dollars for each additional
CD. How many CD must you buy so it will cost the same under both
plans?

**33.** When
4 is added to two numbers, the ratio is 5:6. When 4 is subtracted
from the two numbers, the ratio is 1:2. Find the two numbers.

**34.** A
store owner wants to sell 200 pounds of pistachios and walnuts mixed
together. Walnuts cost 4 dollars per pound and pistachio cost 6
dollars per
pounds. How many pounds of each type of nuts should be mixed if the
store owner will charge 5 dollars for the mixture?

**35.** A
cereal box manufacturer makes 32-ounce boxes of cereal. In a perfect
world, the box will be 32-ounce every time is made. However, since the
world is not perfect, they allow a difference of 0.06 ounce. Find the
range of acceptable size for the cereal box.

**36.** A
man weighing 600 kg has been losing 3.12% of his weight each month
with some
heavy exercises and eating the right food. What will the man weigh after
20 months?

**37.** An
object is thrown into the air at a height of 60 feet. After 1 second and
2 second, the object is 88 feet and 84 feet in the air respectively. What
is the initial speed of the object?

**38.** A
transit is 200 feet from the base of a building. There is man
standing on top of the building. The angles of elevation from the top
and bottom of the man are 45 degrees and 44 degrees. What is the
height of the man?

**39.** A
lemonade consists of 6% of lemon juice and a strawberry juice
consists of 15% pure fruit juice. How much of each kind should be
mixed together to get a 4 Liters of a 10% concentration of fruit
juice?

**40.** Ellen
can wash her car in 60 minutes. Her older sister Sarah can do the
same job in 45 minutes. How long will it take if they wash the car
together?

**41.** A
plane flies 500 mi/h. The plane can travel 1100 miles with the wind
in the same amount of time as it travels 900 miles against the wind.
What is the speed of the wind?

**42.** A
company produces boxes that are 5 feet long, 4 feet wide, and 3 feet
high. The company wants to increase each dimension by the same amount
so that the new volume is twice as big. How much is the increase in
dimension?

**43.** James
invested half of his money in land, a tenth in stock, and a twentieth
in saving bonds. Then, he put the remaining 21000 in a CD. How much
money did James saved or invested?

**44.** The
motherboards for a desktop computer can be manufactured for 50
dollars each.
The development cost is 250000. The first 20 motherboard are samples and
will not be sold. How many salable motherboards will yield an average cost
of 6325 dollars?

**45.** How
much of a 70% orange juice drink must be mixed with 44 gallons of a
20% orange juice drink to obtain a mixture that is 50% orange juice?

**46.** A
company sells nuts in bulk quantities. When bought in bulk, peanuts
sell for $1.20 per pound, almonds for $ 2.20 per pound, and cashews
for $3.20 per pound. Suppose a specialty shop wants a mixture of 280
pounds that will cost $2.59 per pound. Find the number of pounds of
each type of nut if the sum of the number of pounds of almonds and
cashews is three times the number of pounds of peanuts. Round your
answers to the nearest pound.

**47.** A
Basketball player has successfully made 36 of his last 48 free
throws. Find
the number of consecutive free throws the player needs to increase his
success rate to 80%.

**48.** John
can wash cars 3 times as fast as his son Erick. Working together,
they need to wash 30 cars in 6 hours. How many hours will it takes
each of them working alone?

**49.** In
a college, about 36% of student are under 20 years old and 15% are
over 40 years old. What is the probability that a student chosen at
random is under 20 years old or over 40 years?

**50.** Twice
a number plus the square root of the number is twelve minus the
square root of the number.

**51.** The
light intensity, I , of a light bulb varies inversely as the square
of the
distance from the bulb. A a distance of 3 meters from the bulb, I = 1.5 W/m^2 .
What is the light intensity at a distance of 2 meters from the bulb?

**52.** The
lengths of two sides of a triangle are 2 and 6. find the range of
values for the possible lengths of the third side.

**53.** Find
three consecutive integers such that one half of their sum is between
15 and 21.

**54.** After
you open a book, you notice that the
product of the
two page numbers on
the facing pages is 650. What are the two page numbers?

**55.** Suppose
you start with a number. You multiply the number by 3, add 7, divide
by ½, subtract 5, and then divide by 12. The result is 5. What
number did you start with?

**56.** You
have 156 feet of fencing to enclose a rectangular garden. You want
the garden to be 5 times as long as it is wide. Find the dimensions
of the garden.

**57.** The
amount of water a dripping faucet wastes water varies directly with
the amount of time the faucet drips. If the faucet drips 2 cups of
water every 6 minutes, find out how long it will take the faucet to
drip 10.6465 liters of water.

**58.** A
washer costs 25% more than a dryer. If the store clerk gave a 10% discount
for the dryer and a 20% discount for the washer, how much is the
washer before the discount if you paid 1900 dollars.

**59.** Baking
a tray of blueberry muffins takes 4 cups of milk and 3 cups of wheat
flour. A tray of pumpkin muffins takes 2 cups of milk and 3 cups of
wheat flour. A baker has 16 cups of milk and 15 cups of wheat flour.
You make 3 dollars profit per tray of blueberry muffins and 2 dollars
profit per tray of pumpkin muffins. How many trays of each type of
muffins should you make to maximize profit?

**60.** A
company found that -2p + 1000 models the number of TVs sold per month
where p can be set as low as 200 or as high as 300. How can the
company maximize the revenue?

More hard word problems in algebra

**61.** Your
friends say that he has $2.40 in equal numbers of quarters, dimes,
and nickels. How many of each coin does he have?

**62.** I
am a two-digit number whose digit in the tenth place is 1 less than
twice the digit in the ones place. When the digit in the tenth place
is divided by the digit in the ones place, The quotient is 1 and the
remainder is 4. What number am I?

**63.** A
two-digit number is formed by randomly selecting from the digits 2,
4, 5,
and 7 without replacement. What is the probability that a two-digit number
contains a 2 or a 7?

**64.** Suppose
you interview 30 females and 20 males at your school to find out who
among them are using an electric toothbrush. Your survey revealed
that only 2 males use an electric toothbrush while 6 females use it.
What is the probability that a respondent did not use an electric
toothbrush given that the respondent is a female?

**65.** An
employer pays 15 dollars per hour plus an extra 5 dollars per hour
for every hour worked beyond 8 hours up to a maximum daily wage of
220 dollars. Find
a piecewise function that models this situation.

**66.** Divide
me by 7, the remainder is 5. Divide me by 3, the remainder is 1 and
my quotient is 2 less than 3 times my previous quotient. What number
am I?

**67.** A
company making luggage have these requirements to follow. The length is
15 inches greater than the depth and the sum of length, width, and
depth may not exceed 50 inches. What is the maximum value for the
depth if the manufacturer will only use whole numbers?

**68.** To
make an open box, a man cuts equal squares from each corner of a
sheet of metal that is 12 inches wide and 16 inches long. Find an
expression for the volume in terms of x.

**69.** Ten
candidates are running for president, vice-president, and secretary
in the students government. You may vote for as many as 3 candidates.
In how many ways can you vote for 3 or fewer candidates?

**70.** The
half-life of a medication prescribed by a doctor is 6 hours. How
many mg of this medication is left after 78 hours if the doctor prescribed
100 mg?

**71.** Suppose
you roll a red number cube and a yellow number cube. Find
P(red 2, yellow 2) and the probability to get any matching pairs of
numbers.

**72.** A
movie theater in a small town usually open its doors 3 days in a row
and then closes the next day for maintenance. Another movie theater 3
miles away open 4 days in a row and then closes the next day for the
same reason. Suppose both movie theaters are closed today and today
is Wednesday, when is the next time they will both be closed again on
the same day?

**73.** An
investor invests 5000 dollars at 10% and the rest at 5%. How much was
invested at 5% if the yield is one-fifth of the amount invested at at
10%?

**74.** 20000
students took a standardized math test. The scores on the test are
normally distributed, with a mean score of 85 and a standard
deviation of 5. About how many students scored between 90 and 95?

**75.** A
satellite, located 2400 km above Earth’s surface, is in circular
orbit around the earth. If it takes the satellite 3 hours to complete
1 orbit, how far is the satellite after 1 hour?

**76.** In
a group of 10 people, what is the probability that at least two
people in
the group have the same birthday?

**77.** During
a fundraising for cancer at a gala, everybody shakes hands with
everyone else in the room before the event and after the event is
finished. If
n people attended the gala, how many different handshakes occur?

**78.** Two
cubes have side lengths that are equal to 2x and 4x. How many times greater
than the surface of the small cube is the surface area of the large
cube?

**79.** Suppose
you have a job in a restaurant that pays $8 per hour. You also have
a job at Walmart that pays $10 per hour. You want to earn at least
200 per week. However, you want to work no more than 25 hours per
week . Show 3 different ways you could work at each job.

**80.** Two
company offer tutoring services. Company A realizes that when they
tutor for 3 hours, they make 45 dollars. When they tutor for 7 hours
they make 105 hours. Company B realizes that when they tutor for 2
hours, they make 34 dollars. When they tutor for 6 hours they make
102 hours. Assuming that the number of hours students sign for
tutoring is the same for both company, which company will generate
more revenue?

**81.** You
want to fence a rectangular area for kids in the backyard. To save on
fences, you will use the back of your house as one of the four sides.
Find the possible dimensions if the house is 60 feet wide and you
want to use at least 160 feet of fencing.

**82.** When
a number in increased by 20%, the result is the same when it is
decreased by 10% plus 12. What is the number?

**83.** The
average of three numbers is 47. The biggest number is five more than
twice the smallest. The range is 35. What are the three numbers?

**84.** The
percent of increase of a number from its original amount to 36 is
80%. What
is the original amount of the number?

**85.** When
Peter drives to work, he averages 45 miles per hour because of
traffic. On the way back home, he averages 60 miles per hour because
traffic is not as bad. The total travel time is 2 hours. How far is
Peter’s house from work?

**86.** An
advertising company takes 20% from all revenue that it generates for
its affiliates. If the affiliates were paid 15200 dollars this month,
how much revenue
did the advertising company generate this month?

**87.** A
company’s revenue can be modeled with a quadratic equation. The
company noticed that when they sell either 2 or or 12 items, the
revenue is 0. How much is the revenue when they sell 20 items?

**88.** A
ball bounced 4 times, reaching three-fourths of its previous height
with each bounce. After the fourth bounce, the ball reached a height
of 25 cm. How high was the ball when it was dropped?

**89.** A
rental company charges 40 dollars per day plus $0.30 per mile. You
rent a car and drop it off 4 days later. How many miles did you drive
the car if you paid 325.5 dollars which included a 5% sales tax?

**90.** Two
students leave school at the same time and travel in opposite
directions along the same road. One walk at a rate of 3 mi/h. The
other bikes at a rate of 8.5 mi/h. After how long will they be 23
miles part?

**91.** Brown
has the same number of brothers as sisters. His sister Sylvia as
twice as many brothers as sisters. How many children are in the
family?

**92.** Ethan
has the same number of male classmates as female classmates. His
classmate Olivia has three-fourths as many female classmates as male
classmates. How many students are in the class?

**93.** Noah
wants to share a certain amount of money with 10 people. However, at
the last minute, he is thinking about decreasing the amount by 20 so
he can keep 20 for himself and share the money with only 5 people.
How much money is Noah trying to share if each person still gets the
same amount?

**94.** The
square root of me plus the square root of me is me. Who am I?

**95.** A
cash drawer contains 160 bills, all 10s and 50s. If the total value
of the 10s and 50s is $1,760. How many of each type of bill are in
the drawer?

**96.** You
want to make 28 grams of protein snack mix made with peanuts and
granola. Peanuts contain 7 grams of protein per ounce and granola
contain 3 grams of protein per ounce. How many ounces of granola
should you use for 1 ounce of peanuts?

**97.** The
length of a rectangular prism is quadrupled, the width is doubled,
and the height is cut in half. If V is the volume of the rectangular
prism before the modification, express the volume after the
modification in terms of V.

**98.** A
car rental has CD players in 85% of its cars. The CD players are
randomly distributed throughout the fleet of cars. If a person rents
4 cars, what is the probability that at least 3 of them will have CD
players?

**99.** Jacob’s
hourly wage is 4 times as much as Noah. When Jacob got a raise of 2
dollars, Noah accepted a new position that pays him 2 dollars less
per hour. Jacob now earns 5 times as much money as Noah. How much
money do they make per hour after Jacob got the raise?

**100.**

You own a catering business that makes specialty cakes. Your company has decided to create three types of cakes. To create these cakes, it takes a team that consists of a decorator, a baker, and a design consultant. Cake A takes the decorator 9 hours, the baker 6 hours, and the design consultant 1 hour to complete. Cake B takes the decorator 10 hours, the baker 4 hours, and the design consultant 2 hours. Cake C takes the decorator 12 hours, the baker 4 hour, and the design consultant 1 hour. Without hiring additional employees, there are 398 decorator hours available, 164 baker hours available, and 58 design consultant hours available. How many of each type of cake can be created?

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