Ratios, Proportions, and Percentages Flashcards

A solution is 40% alcohol by volume. If 15 mL of pure alcohol is added to 60 mL of this solution, what is the new concentration of alcohol?

Answer: 49.3%

The original solution contains 40% × 60 mL = 24 mL of alcohol. After adding 15 mL of pure alcohol, total alcohol = 24 + 15 = 39 mL. Total volume = 60 + 15 = 75 mL. Concentration = 39/75 = 0.52 — wait, that gives 52%. Let me recheck: 39/75 = 0.52 = 52%. The correct answer is 52.0%.

The ratio of men to women in a company is 5:4. After hiring 18 new employees — all women — the ratio becomes 5:7. How many employees did the company have originally?

Answer: 36

Let original men = 5x and women = 4x. After hiring, men stay at 5x and women become 4x + 18. New ratio: 5x/(4x + 18) = 5/7. Cross-multiplying: 35x = 20x + 90 → 15x = 90 → x = 6. Original total = 5(6) + 4(6) = 30 + 24 = 54. Wait — that gives 54. Checking: 30 men, 24 women → ratio 30:24 = 5:4 ✓. After hiring: 30 men, 42 women → ratio 30:42 = 5:7 ✓. Original total = 54.

A price increased by 25%, then decreased by 20%. A second price decreased by 20%, then increased by 25%. Which statement is true about the final prices relative to the original?

Answer: Both end at the same value, equal to the original

For any starting price P: Path 1 = P × 1.25 × 0.80 = P × 1.00 = P. Path 2 = P × 0.80 × 1.25 = P × 1.00 = P. Multiplication is commutative, so both paths return to exactly the original price. This is a classic trap — many assume sequential % changes compound differently based on order.

Three quantities are in the ratio 2:3:5. The largest quantity exceeds the smallest by 48. What is 15% of the middle quantity?

Answer: 14.4

Let the quantities be 2x, 3x, and 5x. Largest − Smallest = 5x − 2x = 3x = 48 → x = 16. Middle quantity = 3x = 3 × 16 = 48. 15% of 48 = 0.15 × 48 = 7.2. Hmm, that's not among the answers. Let me re-examine: 15% of 48 = 7.2. The answer choices suggest the middle quantity is 96: 15% × 96 = 14.4. That means 3x = 96 → x = 32, so 3x − 2x = x = 32 ≠ 48. Let me retry: difference between largest and smallest is 5x − 2x = 3x = 48 → x = 16, middle = 48, 15% = 7.2. The correct answer is 14.4 if middle = 96. Re-checking with ratio 2:3:5, difference = 48, x=16, middle=48, 15%=7.2. Answer: 7.2.

A tank is 60% full. After removing 45 liters, it is 45% full. If 30 liters are then added, what percentage of the tank is full?

Answer: 55%

The difference in fullness = 60% − 45% = 15%, which corresponds to 45 liters. So 1% of the tank = 45/15 = 3 liters, and total capacity = 300 liters. After removing 45 L the tank holds 45% × 300 = 135 L. Adding 30 L gives 135 + 30 = 165 L. Percentage = 165/300 = 55%.

If a : b = 3 : 4 and b : c = 6 : 7, what is the value of (a + c) : (a + b + c)?

Answer: 23:38

First unify the ratios. a:b = 3:4 and b:c = 6:7. To combine, make b the same: a:b = 9:12 and b:c = 12:14. So a:b:c = 9:12:14. Then a + c = 9 + 14 = 23 and a + b + c = 9 + 12 + 14 = 35. Wait — 23:35 is not listed. Let me redo: a:b = 3:4, b:c = 6:7. LCM of b-values (4 and 6) = 12. a:b = 9:12, b:c = 12:14. So a:b:c = 9:12:14. (a+c):(a+b+c) = 23:35. The answer 23:38 is not correct either. Correct answer is 23:35.