Questions & explanations
1. In a survey, the ratio of people who prefer coffee to tea is 5:3. If 40% of the people prefer coffee, what percent prefer tea?
- 20%
- 24%
- 30%
- 40%
- 60%
Answer: 24%
Ratio coffee:tea = 5:3, so total parts = 8. Coffee fraction = 5/8 = 62.5%, but given 40% prefer coffee, the ratio is not directly the percent. However, the ratio implies that for every 5 coffee, there are 3 tea. If coffee is 40%, then tea = (3/5) × 40% = 24%. Alternatively, let total = 100, coffee = 40, then tea = (3/5)×40 = 24, so 24%.
2. A jar contains only red and blue marbles. Initially, the ratio of red marbles to blue marbles is 3:5. After removing 10 red marbles and adding 14 blue marbles, the ratio becomes 1:3. What is the original total number of marbles in the jar?
- 40
- 48
- 56
- 64
- 72
Answer: 48
Let initial red = 3x, blue = 5x. After removing 10 red and adding 14 blue, red = 3x - 10, blue = 5x + 14. The new ratio (red:blue) is 1:3, so (3x-10)/(5x+14) = 1/3. Cross-multiply: 3(3x-10) = 5x+14 → 9x-30 = 5x+14 → 4x = 44 → x = 11. Original total = 3x+5x = 8x = 88.
3. Two cars travel the same distance. The ratio of their speeds is 3:4. If the slower car takes 8 hours, how many hours does the faster car take?
- 4
- 5
- 6
- 7
- 8
Answer: 6
Distance = speed × time. For same distance, speed and time are inversely proportional. Speed ratio 3:4 implies time ratio 4:3 (inverse). Slower car time = 8 hours corresponds to 4 parts, so 1 part = 2 hours. Faster car time = 3 parts = 6 hours.
4. A tank contains 20 liters of a mixture that is 40% milk and 60% water. If 5 liters of the mixture are removed and replaced with pure milk, what is the percentage of milk in the new mixture?
- 45%
- 50%
- 55%
- 60%
- 65%
Answer: 55%
Initial milk: 0.4*20=8 liters. Remove 5 liters of mixture: milk removed = 0.4*5=2 liters, so milk left = 8-2=6 liters. Add 5 liters pure milk: milk becomes 6+5=11 liters. Total volume remains 20 liters. New percentage = (11/20)*100% = 55%.
5. In a class, the ratio of boys to girls is 3:2. If 1/4 of the boys and 1/3 of the girls are left-handed, what fraction of the class is left-handed?
- 17/60
- 7/20
- 5/12
- 1/4
- 1/3
Answer: 17/60
Let total students = 5k (boys=3k, girls=2k). Left-handed boys = (1/4)*3k = 3k/4. Left-handed girls = (1/3)*2k = 2k/3. Total left-handed = 3k/4 + 2k/3 = (9k+8k)/12 = 17k/12. Fraction = (17k/12)/(5k) = 17/60.
6. Working alone, machine A can complete a task in 6 hours, and machine B can complete the same task in 4 hours. Machine A starts working alone, and after 2 hours, machine B joins. How many additional hours will it take for both machines to finish the task together?
- 1.2 hours
- 1.6 hours
- 2 hours
- 2.4 hours
- 3 hours
Answer: 1.6 hours
A's rate = 1/6 per hour, B's rate = 1/4 per hour. In 2 hours, A completes 2*(1/6)=1/3 of task. Remaining = 2/3. Combined rate = 1/6+1/4=5/12 per hour. Time = (2/3) / (5/12) = (2/3)*(12/5)=24/15=1.6 hours.
7. A jar contains only red and blue marbles in the ratio 3:5. If 10 red marbles are added and 6 blue marbles are removed, the new ratio becomes 5:7. How many red marbles were originally in the jar?
- 30
- 45
- 50
- 60
- 75
Answer: 45
Let original red = 3k, blue = 5k. After changes: red = 3k+10, blue = 5k-6. New ratio (3k+10)/(5k-6) = 5/7. Cross-multiply: 7(3k+10)=5(5k-6) => 21k+70=25k-30 => 100=4k => k=25. Original red = 3*25=75.
8. A car travels from town A to town B at a speed of 60 mph and returns from B to A at a speed of 40 mph. What is the average speed for the entire round trip?
- 48 mph
- 50 mph
- 52 mph
- 45 mph
- 55 mph
Answer: 48 mph
Let distance one way = d. Time A to B = d/60, time B to A = d/40. Total distance = 2d, total time = d/60 + d/40 = (2d/120 + 3d/120) = 5d/120 = d/24. Average speed = 2d / (d/24) = 48 mph.
9. In a certain mixture, the ratio of sand to cement is 3:2 by weight, and the ratio of cement to gravel is 4:5. What is the ratio of sand to gravel in the mixture?
- 3:5
- 5:6
- 6:5
- 12:10
- 12:5
Answer: 6:5
Sand:Cement = 3:2, Cement:Gravel = 4:5. Make cement common: LCM of 2 and 4 is 4. Multiply first ratio by 2: Sand:Cement = 6:4. Then Sand:Cement:Gravel = 6:4:5. So Sand:Gravel = 6:5.
10. On a map, a distance of 3 inches represents an actual distance of 45 miles. What is the scale of the map in inches per mile?
- 1:15
- 1:12
- 1:10
- 1:5
- 1:3
Answer: 1:15
The scale is the ratio of map distance to actual distance. 3 inches represents 45 miles, so the ratio is 3:45, which simplifies to 1:15. Thus, 1 inch represents 15 miles.
11. A chemist has two solutions: Solution X is 20% acid and 80% water; Solution Y is 50% acid and 50% water. How many liters of Solution Y must be added to 10 liters of Solution X to obtain a mixture that is 30% acid?
- 2
- 3
- 4
- 5
- 6
Answer: 5
Let y be liters of Y. Acid from X: 0.2*10=2 liters. Acid from Y: 0.5y. Total acid: 2+0.5y. Total mixture: 10+y. Set (2+0.5y)/(10+y)=0.3 => 2+0.5y=3+0.3y => 0.2y=1 => y=5.
12. A chemist mixes a 20% acid solution with a 50% acid solution to obtain 10 liters of a 30% acid solution. How many liters of the 20% solution should be used?
- 6.67 liters
- 5 liters
- 4 liters
- 3.33 liters
- 7 liters
Answer: 6.67 liters
Let x = liters of 20% solution, then (10-x) liters of 50% solution. Acid equation: 0.2x + 0.5(10-x) = 0.3*10 → 0.2x + 5 - 0.5x = 3 → -0.3x = -2 → x = 20/3 ≈ 6.67 liters.