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100 hard word problems in algebra 

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. Teachers! Feel free to select from this list and give them to your students to see if they have mastered how to solve tough algebra problems. Find out below how you can print these problems. You can also purchase a solution if needed.

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?

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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Latest stories, 20 grade-school math questions so hard you'll wonder how you graduated, seriously, who can do these.

Unless you grew up to be an engineer, a banker, or an accountant, odds are that elementary and middle school math were the bane of your existence. You would study relentlessly for weeks for those silly standardized tests—and yet, come exam day, you'd still somehow have no idea what any of the equations or hard math problems were asking for. Trust us, we get it.

While logic might lead you to believe that your math skills have naturally gotten better as you've aged, the unfortunate reality is that, unless you've been solving algebra and geometry problems on a daily basis, the opposite is more likely the case.

Don't believe us? Then put your number crunching wisdom to the test with these tricky math questions taken straight from grade school tests and homework assignments and see for yourself.

1. Question: What is the number of the parking space covered by the car?

This tricky math problem went viral a few years back after it appeared on an entrance exam in Hong Kong... for six-year-olds. Supposedly the students had just 20 seconds to solve the problem!

Answer: 87.

Believe it or not, this "math" question actually requires no math whatsoever. If you flip the image upside down, you'll see that what you're dealing with is a simple number sequence.

2. Question: Replace the question mark in the above problem with the appropriate number.

This problem shouldn't be too difficult to solve if you play a lot of sudoku.

All of the numbers in every row and column add up to 15! (Also, 6 is the only number not represented out of numbers 1 through 9.)

3. Question: Find the equivalent number.

This problem comes straight from a standardized test given in New York in 2014.

You're forgiven if you don't remember exactly how exponents work. In order to solve this problem, you simply need to subtract the exponents (4-2) and solve for 3 2 , which expands into 3 x 3 and equals 9.

4. Question: How many small dogs are signed up to compete in the dog show?

This question comes directly from a second grader's math homework. Yikes.

Answer: 42.5 dogs.

In order to figure out how many small dogs are competing, you have to subtract 36 from 49 and then divide that answer, 13, by 2, to get 6.5 dogs, or the number of big dogs competing. But you're not done yet! You then have to add 6.5 to 36 to get the number of small dogs competing, which is 42.5. Of course, it's not actually possible for half a dog to compete in a dog show, but for the sake of this math problem let's assume that it is.

5. Question: Find the area of the red triangle.

This question was used in China to identify gifted 5th graders. Supposedly, some of the smart students were able to solve this in less than one minute.

In order to solve this problem, you need to understand how the area of a parallelogram works. If you already know how the area of a parallelogram and the area of a triangle are related, then adding 79 and 10 and subsequently subtracting 72 and 8 to get 9 should make sense—but if you're still confused, then check out this YouTube video for a more in-depth explanation.

6. Question: How tall is the table?

YouTuber MindYourDecisions adapted this mind-boggling math question from a similar one found on an elementary school student's homework in China.

Answer: 150 cm.

Since one measurement includes the cat's height and subtracts the turtle's and the other does the opposite, you can essentially just act like the two animals aren't there. Therefore, all you have to do is add the two measurements—170 cm and 130 cm—together and divided them by 2 to get the table's height, 150 cm.

7. Question: If the cost of a bat and a baseball combined is $1.10 and the bat costs $1.00 more than the ball, how much is the ball?

This problem, mathematically speaking, is very similar to one of the other ones on this list.

Answer: $0.05.

Think back to that problem about the dogs at the dog show and use the same logic to solve this problem. All you have to do is subtract $1.00 from $1.10 and then divide that answer, $0.10 by 2, to get your final answer, $0.05.

8. Question: When is Cheryl's birthday?

If you're having trouble reading that, see here:

"Albert and Bernard just became friends with Cheryl, and they want to know when her birthday is. Cheryl gives them a list of 10 possible dates.

May 15 May 16 May 19

June 17 June 18

July 14 July 16

August 14 August 15 August 17

Cheryl then tells Albert and Bernard separately the month and the day of her birthday respectively.

Albert: I don't know when Cheryl's birthday is, but I know that Bernard doesn't not know too.

Bernard: At first I don't know when Cheryl's birthday is, but I know now.

Albert: Then I also know when Cheryl's birthday is.

So when is Cheryl's birthday?"

It's unclear why Cheryl couldn't just tell both Albert and Bernard the month and day she was born, but that's irrelevant to solving this problem.

Answer: July 16.

Confused about how one could possibly find any answer to this question? Don't worry, so was most of the world when this question, taken from a Singapore and Asian Schools Math Olympiad competition, went viral a few years ago. Thankfully, though, the New York Times explains step-by-step how to get to July 16, and you can read their detailed deduction here.

9. Question: Find the missing letter.

This one comes from a first grader's homework.

Answer: The missing letter is J.

When you add together the values given for S, B, and G, the sum comes out to 40, and making the missing letter J (which has a value of 14) makes the other diagonal's sum the same.

10. Question: Solve the equation.

This problem might look easy, but a surprising number of adults are unable to solve it correctly.

Start by solving the division part of the equation. In order to do that, in case you forgot, you have to flip the fraction and switch from division to multiplication, thus getting 3 x 3 = 9. Now you have 9 - 9 + 1, and from there you can simply work from left to right and get your final answer: 1.

11. Question: Where should a line be drawn to make the below equation accurate?

5 + 5 + 5 + 5 = 555.

Answer: A line should be drawn on a "+" sign.

When you draw a slanted line in the upper left quadrant of a "+," it becomes the number 4 and the equation thusly becomes 5 + 545 + 5 = 555.

12. Question: Solve the unfinished equation.

Try to figure out what all of the equations have in common.

Answer: 4 = 256.

The formula used in each equation is 4 x = Y. So, 4 1 = 4, 4 2 = 16, 4 3 = 64, and 4 4 = 256.

13. Question: How many triangles are in the image above?

When Best Life first wrote about this deceiving question, we had to ask a mathematician to explain the answer!

Answer: 18.

Some people get stumped by the triangles hiding inside of the triangles and others forget to include the giant triangle housing all of the others. Either way, very few individuals—even math teachers—have been able to find the correct answer to this problem. And for more questions that will put your former education to the test, check out these 30 Questions You'd Need to Ace to Pass 6th Grade Geography.

14. Question: Add 8.563 and 4.8292.

Adding two decimals together is easier than it looks.

Answer: 13.3922.

Don't let the fact that 8.563 has fewer numberrs than 4.8292 trip you up. All you have to do is add a 0 to the end of 8.563 and then add like you normally would.

15. Question: There is a patch of lily pads on a lake. Every day, the patch doubles in size...

... If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Answer: 47 days.

Most people automatically assume that half of the lake would be covered in half the time, but this assumption is wrong. Since the patch of pads doubles in size every day, the lake would be half covered just one day before it was covered entirely.

16. Question: How many feet are in a mile?

This elementary school-level problem is a little less problem solving and a little more memorization.

Answer: 5,280.

This was one of the questions featured on the popular show Are You Smarter Than a 5th Grader?

17. Question: What value of "x" makes the equation below true?

-15 + (-5x) = 0

Answer: -3.

You'd be forgiven for thinking that the answer was 3. However, since the number alongside x is negative, we need x to be negative as well in order to get to 0. Therefore, x has to be -3.

18. Question: What is 1.92 divided by 3?

You might need to ask your kids for help on this one.

Answer: 0.64.

In order to solve this seemingly simple problem, you need to remove the decimal from 1.92 and act like it isn't there. Once you've divided 192 by 3 to get 64, you can put the decimal place back where it belongs and get your final answer of 0.64.

19. Question: Solve the math equation above.

Don't forget about PEMDAS!

Using PEMDAS (an acronym laying out the order in which you solve it: "parenthesis, exponents, multiplication, division, addition, subtraction"), you would first solve the addition inside of the parentheses (1 + 2 = 3), and from there finish the equation as it's written from left to right.

20. Question: How many zombies are there?

Finding the answer to this final question will require using fractions.

Answer: 34.

Since we know that there are two zombies for every three humans and that 2 + 3 = 5, we can divide 85 by 5 to figure out that in total, there are 17 groups of humans and zombies. From there, we can then multiply 17 by 2 and 3 and learn that there are 34 zombies and 51 humans respectively. Not too bad, right?

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Home / Blog / Math / Hardest Math Questions That Are Surprisingly Easy To Solve!

Hardest Math Questions That Are Surprisingly Easy To Solve!

Have you ever struggled enough with a math problem only to realize that it was too easy to solve? Many students will definitely vouch for it—a math question looked quite simple when they sat with it, but within minutes they understood that they had no idea how to solve it!

Sometimes you come across some of the most difficult math questions that are both challenging and brain teasers. Even if you are well-prepared for your exam, you will encounter difficult math questions. But here’s the thing: They can be solved in a flash, regardless of how ridiculously difficult they appear.

Yes, you heard that right! If you understand basic math skills and concepts, all you need are some math tricks and practice to solve the problems in a jiffy.

Check out These 10 Hardest Math Questions That will Test Your Logic and Problem-solving Skills

Interestingly, these hard math problems can be solved easily with basic math operators. Ready to test your math skills?

• A bat and a ball cost one dollar and ten cents in total. The bat costs a dollar more than the ball. How much does the ball cost?

Answer: The ball costs 5 cents. How did that happen? If you were thinking of some other answers to this problem, then this explanation might help you. Well, to simplify the question, the difference between $1 and 10 cents is 90 cents, not $1. The only way for the bat to cost a dollar more than the ball while still having the total cost equal to $1.10 is for the baseball bat to cost $1.05 and the ball to cost 5 cents.

• The classic PEMDAS problem: 6 ÷ 2(1+2) = ?

Answer: 9 Every PEMDAS problem leads to a huge disruption of opinions, splitting the masses into various answers while everyone believes that they have the correct answer. So, how was this solved and how did they arrive at “9” as the solution? If you remember the order of operations of PEMDAS from grade school, one generally works through the problem by solving parentheses, then the exponents, multiplication, and division, followed by addition and subtraction. Sometimes, some people interpret PEMDAS differently, and that is when the disagreement begins. According to the PEMDAS rule, one should solve anything inside parentheses, then exponents, and then all multiplication and division, starting from the left to the right, as per the operations.

• Replace the question mark in the above diagram with an appropriate number.

Answer: 6 If you enjoy playing Sudoku, then this hard math problem would have been quite a breeze for you! The rows and columns all add up to 15, and that is how the answer turned out to be 6.

• Solve the unfinished equation:

Answer: 4 = 256 Have a close look at the equation again, and find out the common thing among the equations. There is a formula that has been used for these equations. 4^x = Y. So, 4^1 = 4, 4^2 = 16, 4^3 = 64, and 4^4 = 256.

• The Lily Pad problem: A lake has a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long will it take for the patch to cover half of the lake?

Answer: It will take 47 days for the patch to cover half of the lake. If you assumed that half of the lake would be covered in half of the time, then your assumption could be incorrect. Here, in the math problem, it says that the patch “doubles” in size, which means, given any day, the lily pad was half the size the day before. So if the patch covers up the entire lake on the 48th day, it means the lily pad was half the size of the lake on day 47.

• When you add five to nine, you get two. The answer is correct, but how?

Answer: Well, here’s the trick. When it is 9 a.m., you add five hours to it. Voila, you get 2 p.m. as your answer! Getting smarter every minute, aren’t we?

• A group of students was standing in the sun facing due west on a march past event. The captain shouted at them: Right turn! About turn! Left turn! At the end of these commands, which direction are the students facing now?

Answer: East. Well, this one might seem tricky, but the answer is quite simple. The students will first turn 90 degrees in a right turn and then 180 degrees in an about-turn. The students will finally turn 90 degrees in a left turn. Therefore, the students are now facing east.

• Guess how many triangles are there in the above picture?

Answer: 18 This is another tricky math question where different people have come up with different answers. While some forget to count the hidden triangles, others forget to count the giant triangle. Why don’t you concentrate on the triangle again and find all the triangles?

• How to make the following equation precise using three of these four mathematical symbols: + – × ÷ without following any order as such?

Equation: 21 _ 3 _ 18 _ 6 = 6

Answer: 21 – 3 + 18 ÷ 6 = 6 This is more of a symbol math problem than a PEMDAS problem. You can keep testing with the math operators until you find the correct answer.

• John was asked to paint numbers outside 100 apartments. This means he will have to paint numbers one through 100. How many times will he have to paint the number eight?

Answer: 20 times If you carefully read the question, the answer is within the question itself. It’s a tricky one, but this is how it goes: John paints on apartment 8, then he paints on apartment 18 (where the 8 digit comes up again) and, similarly, 28th, 38th, 48th, 58th, 68th, 78th, 80th, 81st, 82nd, 83rd, 84th, 85th, 86th, 87th, 88th, 89th, and 98th apartments.

These were some of the hardest math questions, which probably left your mind boggled. You can always keep improving your logical and math skills with these kinds of math problems that might seem difficult but are easier than you had thought. So, keep testing yourself with more tricky math problems and get your logical juices flowing.

To read more about versatile and new online techniques to improve your math concepts and boost your confidence in solving difficult problems, visit Byju’s FutureSchool blog.

References: 

• Jamie Spafford on Twitter: “Here’s a riddle for you: If a baseball bat and a baseball together cost $1.10 and the bat costs a dollar more than the ball, how much does the ball cost? (H/T @ScottMonty for including it in his really great article: https://t.co/foD6RdOUgy)” / Twitter . (n.d.). Retrieved April 22, 2022, from https://twitter.com/jamiespafford/status/1290641293891317770 
• 20 Tricky But Fun Grade-School Math Questions – Hard Math Problems . (n.d.). Retrieved April 22, 2022, from https://bestlifeonline.com/tricky-math-questions/ 
• 34 Hard Math Riddles and Word Problems for Future Geniuses | Fatherly . (n.d.). Retrieved April 22, 2022, from https://www.fatherly.com/play/hard-math-riddles-for-kids-with-answers/ 
• Southern Math Riddles Jeopardy . (n.d.). Retrieved April 22, 2022, from https://jeopardylabs.com/print/math-riddles-jeopardy 

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