Percentages Ratios and Proportions Flashcards

A mixture contains alcohol and water in the ratio 3:7. If 20 liters of water are added, the ratio becomes 3:11. How many liters of alcohol are in the original mixture?

Answer: 15

Let alcohol = 3x and water = 7x. After adding 20 liters of water: 3x/(7x+20) = 3/11. Cross-multiplying: 33x = 21x + 60, so 12x = 60, x = 5. Alcohol = 3(5) = 15 liters.

A store marks up an item by 40% and then offers a 25% discount on the marked price. What is the net percentage change from the original price?

Answer: 5% increase

Let the original price = 100. After 40% markup: 100 × 1.40 = 140. After 25% discount: 140 × 0.75 = 105. Net change = (105 − 100)/100 = 5% increase. The key insight is 1.40 × 0.75 = 1.05, not 1.15.

If a:b = 2:3 and b:c = 4:5, what percentage of c is a? (Round to the nearest tenth.)

Answer: 53.3%

Make b consistent: a:b = 2:3 = 8:12 and b:c = 4:5 = 12:15. So a:b:c = 8:12:15. a as a percentage of c = 8/15 × 100 ≈ 53.3%.

Population A grows at 20% per year and population B shrinks at 10% per year. If they are equal today, after how many full years will population A first exceed twice population B?

Answer: 3 years

Starting equal, A/B after n years = (1.20/0.90)ⁿ = (4/3)ⁿ. We need (4/3)ⁿ > 2. Year 1: ≈1.33; Year 2: ≈1.78; Year 3: ≈2.37 > 2. The ratio first exceeds 2 after 3 years.

A solution is 15% salt by weight. How many grams of pure salt must be added to 400 grams of this solution to produce a 25% salt solution?

Answer: 53.3 g

Current salt = 0.15 × 400 = 60 g. Adding x grams of pure salt: (60 + x)/(400 + x) = 0.25. Solving: 60 + x = 100 + 0.25x → 0.75x = 40 → x ≈ 53.3 g.

On a map with scale 3 cm : 45 km, a rectangular region measures 7 cm by 4 cm. What is the actual area of this region in square kilometers?

Answer: 6,300 km²

Scale: 3 cm = 45 km, so 1 cm = 15 km. Actual dimensions: 7 × 15 = 105 km and 4 × 15 = 60 km. Area = 105 × 60 = 6,300 km². Equivalently, area scale factor = 15² = 225 km²/cm²; map area = 28 cm²; actual = 28 × 225 = 6,300 km².