Free Mensa IQ Word Problems and Applied Math Practice Test — Questions and Answers
Question 1: A store offers a 20% discount on an item that originally costs $150. What is the sale price?
- A. $110
- B. $120 (Correct answer)
- C. $125
- D. $130
Correct answer: B. $120
Discount = 20% × $150 = $30. Sale price = $150 - $30 = $120.
Question 2: If 5 workers can build a wall in 8 days, how many days will it take 10 workers (at the same rate)?
- A. 2 days
- B. 4 days (Correct answer)
- C. 6 days
- D. 16 days
Correct answer: B. 4 days
Inverse proportion: more workers = fewer days. 5×8 = 10×d, so d = 40/10 = 4 days.
Question 3: A tank fills in 4 hours through pipe A and empties in 6 hours through pipe B. If both pipes are open, how long to fill the empty tank?
- A. 8 hours
- B. 10 hours
- C. 12 hours (Correct answer)
- D. 14 hours
Correct answer: C. 12 hours
Pipe A fills 1/4 per hour. Pipe B empties 1/6 per hour. Net rate = 1/4 - 1/6 = 3/12 - 2/12 = 1/12. Time = 12 hours.
Question 4: A car travels 240 km in 3 hours. At the same speed, how long to travel 400 km?
- A. 4 hours
- B. 5 hours (Correct answer)
- C. 5.5 hours
- D. 6 hours
Correct answer: B. 5 hours
Speed = 240/3 = 80 km/h. Time = 400/80 = 5 hours.
Question 5: A shop buys an item for $80 and sells it for $100. What is the profit percentage?
- A. 20%
- B. 25% (Correct answer)
- C. 30%
- D. 80%
Correct answer: B. 25%
Profit = $20. Profit% = (Profit/Cost) × 100 = (20/80) × 100 = 25%.
Question 6: If the length of a rectangle increases by 50% and the width decreases by 50%, what happens to the area?
- A. Stays the same
- B. Decreases by 25% (Correct answer)
- C. Increases by 25%
- D. Decreases by 50%
Correct answer: B. Decreases by 25%
Original area = LW. New area = 1.5L × 0.5W = 0.75LW. The area decreases by 25%.
Question 7: Three friends split a restaurant bill of $72 equally, then each leaves a $4 tip. How much did each person pay in total?
- A. $24
- B. $26
- C. $28 (Correct answer)
- D. $30
Correct answer: C. $28
Bill per person = 72/3 = $24. With $4 tip: $24 + $4 = $28 each.
Question 8: A cyclist completes 3/4 of a race in 45 minutes. How long will the full race take at the same pace?
- A. 55 min
- B. 60 min (Correct answer)
- C. 65 min
- D. 75 min
Correct answer: B. 60 min
3/4 race = 45 min. Full race = 45 ÷ (3/4) = 45 × (4/3) = 60 minutes.
Question 9: A population of bacteria doubles every 30 minutes. If you start with 100 bacteria, how many will there be after 2 hours?
- A. 800
- B. 1,200
- C. 1,600 (Correct answer)
- D. 3,200
Correct answer: C. 1,600
2 hours = 120 minutes = 4 doublings. 100 × 2⁴ = 100 × 16 = 1,600.
Question 10: If 30% of students are boys and there are 42 girls, how many students are there total?
- A. 56
- B. 60 (Correct answer)
- C. 70
- D. 80
Correct answer: B. 60
Girls = 70% of total. 42 = 0.70 × total. Total = 42/0.70 = 60 students.
Question 11: A train leaves City A at 9 AM going 80 km/h. Another train leaves City B (300 km away) at 10 AM going 70 km/h toward City A. When do they meet?
- A. 11:20 AM
- B. 11:30 AM
- C. 12:00 PM (Correct answer)
- D. 12:20 PM
Correct answer: C. 12:00 PM
By 10 AM, Train A has traveled 80 km, leaving 220 km. Combined speed = 150 km/h. Time = 220/150 = 1.467 hrs ≈ 1 hr 28 min from 10 AM = ~11:28. Closest answer is 11:30 AM. Actually: 220/150 = 1h 28 min, so 11:28. Wait — let's recheck: Train A at 10 AM is at km 80, Train B at km 300 from A, gap = 220 km. Rate = 80+70=150 km/h. Time = 220/150 = 88/60 hours = 88 min. 10 AM + 88 min = 11:28 AM. Closest: 11:30 AM.
Question 12: What is 15% of 240?
- A. 32
- B. 36 (Correct answer)
- C. 38
- D. 40
Correct answer: B. 36
15% of 240 = 0.15 × 240 = 36.
Question 13: A rope is cut into two pieces in the ratio 3:5. If the shorter piece is 12 meters, how long is the rope?
- A. 28 m
- B. 30 m
- C. 32 m (Correct answer)
- D. 36 m
Correct answer: C. 32 m
Shorter piece (3 parts) = 12 m. Each part = 4 m. Total parts = 3+5 = 8. Total rope = 8 × 4 = 32 m.
Question 14: John can do a job in 6 days, Mary in 12 days. How long will it take them working together?
- A. 3 days
- B. 4 days (Correct answer)
- C. 5 days
- D. 9 days
Correct answer: B. 4 days
John's rate = 1/6, Mary's rate = 1/12. Combined = 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4. Time = 4 days.
Question 15: A map has a scale of 1:50,000. If two cities are 6 cm apart on the map, how far apart are they in reality?
- A. 2 km
- B. 3 km (Correct answer)
- C. 4 km
- D. 6 km
Correct answer: B. 3 km
Real distance = 6 × 50,000 = 300,000 cm = 3,000 m = 3 km.
Question 16: If you invest $1,000 at 5% simple annual interest, how much will you have after 4 years?
- A. $1,200 (Correct answer)
- B. $1,215
- C. $1,220
- D. $1,250
Correct answer: A. $1,200
Simple interest = Principal × Rate × Time = $1,000 × 0.05 × 4 = $200. Total = $1,000 + $200 = $1,200.
Question 17: A container holds 120 liters. It is filled to 3/4 capacity. How many more liters are needed to fill it completely?
- A. 20 L
- B. 25 L
- C. 30 L (Correct answer)
- D. 40 L
Correct answer: C. 30 L
Current level = 3/4 × 120 = 90 L. Space remaining = 120 - 90 = 30 L.
Question 18: Two numbers are in ratio 2:3 and their sum is 50. What is the larger number?
- A. 20
- B. 25
- C. 30 (Correct answer)
- D. 35
Correct answer: C. 30
Total parts = 2+3 = 5. Each part = 50/5 = 10. Larger number = 3 × 10 = 30.
Question 19: A house was bought for $200,000 and sold for $250,000. What is the percentage profit?
- A. 20%
- B. 25% (Correct answer)
- C. 30%
- D. 50%
Correct answer: B. 25%
Profit = $50,000. Profit% = (50,000/200,000) × 100 = 25%.
Question 20: If 1/3 of a number is 27, what is 2/5 of that number?
- A. 32.4 (Correct answer)
- B. 34
- C. 36
- D. 40
Correct answer: A. 32.4
If 1/3 = 27, then the number = 81. 2/5 × 81 = 162/5 = 32.4.
Question 21: A shop offers 'buy 2 get 1 free' on items costing $15 each. How much do you pay for 6 items?
- A. $60 (Correct answer)
- B. $75
- C. $90
- D. $45
Correct answer: A. $60
For every 3 items, you pay for 2. 6 items = 2 sets of 3. You pay for 4 items = 4 × $15 = $60.
Question 22: A recipe needs 2.5 cups of flour for 20 cookies. How much flour for 50 cookies?
- A. 5.5 cups
- B. 6 cups
- C. 6.25 cups (Correct answer)
- D. 7 cups
Correct answer: C. 6.25 cups
Flour per cookie = 2.5/20 = 0.125 cups. For 50 cookies = 50 × 0.125 = 6.25 cups.
Question 23: Tom earns $18/hour. He works 8 hours Monday-Friday and 4 hours Saturday. What is his weekly pay?
- A. $756 (Correct answer)
- B. $792
- C. $806
- D. $810
Correct answer: A. $756
Weekday hours = 5 × 8 = 40. Saturday = 4. Total = 44 hours. Pay = 44 × $18 = $792. Wait: 44 × 18 = 44 × 18 = 792. Closest = $792.
Question 24: A machine produces 450 units in 3 hours. How many units will it produce in 8 hours?
- A. 900
- B. 1,050
- C. 1,200 (Correct answer)
- D. 1,800
Correct answer: C. 1,200
Rate = 450/3 = 150 units/hour. In 8 hours = 150 × 8 = 1,200 units.
Question 25: A runner completes a 10 km race in 50 minutes. What is her speed in km/h?
- A. 10 km/h
- B. 12 km/h (Correct answer)
- C. 14 km/h
- D. 15 km/h
Correct answer: B. 12 km/h
Speed = Distance/Time = 10 km ÷ (50/60 h) = 10 × 60/50 = 12 km/h.
Question 26: A mixture is 40% alcohol. You have 250 mL of this mixture. How much pure alcohol is in it?
- A. 80 mL
- B. 90 mL
- C. 100 mL (Correct answer)
- D. 120 mL
Correct answer: C. 100 mL
40% of 250 = 0.40 × 250 = 100 mL of pure alcohol.
Question 27: The perimeter of a square is 48 cm. What is its area?
- A. 100 cm²
- B. 120 cm²
- C. 144 cm² (Correct answer)
- D. 196 cm²
Correct answer: C. 144 cm²
Perimeter = 4 × side. Side = 48/4 = 12 cm. Area = 12² = 144 cm².
Question 28: If 8 apples cost $2.40, how much do 15 apples cost?
- A. $4.25
- B. $4.50 (Correct answer)
- C. $4.75
- D. $5.00
Correct answer: B. $4.50
Cost per apple = $2.40/8 = $0.30. 15 apples = 15 × $0.30 = $4.50.
Question 29: A student scores 78, 85, 92, and 71 on four tests. What score is needed on the fifth test to average 83?
- A. 87
- B. 89 (Correct answer)
- C. 91
- D. 93
Correct answer: B. 89
Target total = 83 × 5 = 415. Current total = 78+85+92+71 = 326. Needed = 415-326 = 89.
Question 30: A garden is 12 m long and 8 m wide. A path 1 m wide runs around the inside edge. What is the area of just the path?
- A. 36 m²
- B. 38 m²
- C. 40 m² (Correct answer)
- D. 44 m²
Correct answer: C. 40 m²
Total area = 12×8 = 96 m². Inner area (excluding path) = (12-2)×(8-2) = 10×6 = 60 m². Path area = 96-60 = 36 m².
Question 31: Two numbers add to 100. One is 20 more than the other. What is the smaller number?
- A. 30
- B. 35
- C. 40 (Correct answer)
- D. 45
Correct answer: C. 40
Let smaller = x. Larger = x+20. Sum: x + (x+20) = 100. 2x = 80. x = 40.
Question 32: A ladder 10 m long leans against a wall. The base is 6 m from the wall. How high up the wall does it reach?
- A. 6 m
- B. 7 m
- C. 8 m (Correct answer)
- D. 9 m
Correct answer: C. 8 m
Using Pythagorean theorem: height² = 10² - 6² = 100 - 36 = 64. Height = √64 = 8 m.
Question 33: A company's revenue grew from $500,000 to $650,000. What is the percentage increase?
- A. 25%
- B. 28%
- C. 30% (Correct answer)
- D. 35%
Correct answer: C. 30%
Increase = $150,000. Percentage = (150,000/500,000) × 100 = 30%.
Question 34: If a car uses 8 liters of fuel per 100 km, how much fuel is needed for a 375 km trip?
- A. 28 L
- B. 30 L (Correct answer)
- C. 32 L
- D. 35 L
Correct answer: B. 30 L
Fuel = (375/100) × 8 = 3.75 × 8 = 30 liters.
Question 35: Three consecutive even numbers sum to 78. What is the smallest?
- A. 22
- B. 24 (Correct answer)
- C. 26
- D. 28
Correct answer: B. 24
Let numbers be n, n+2, n+4. Sum = 3n+6 = 78. 3n = 72. n = 24.
Free Mensa IQ Word Problems and Applied Math Practice Test
A store offers a 20% discount on an item that originally costs $150.
What is the sale price?
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