Ratios, Rates, Proportional Relationships, and Units Flashcards

A farmer wants to blend two fertilizers: Brand A contains 40% nitrogen by weight, and Brand B contains 10% nitrogen by weight. If the farmer needs a final blend that is exactly 22% nitrogen, what is the required ratio of Brand A to Brand B by weight?

Answer: 2 : 3

Use the alligation method: the number of parts of each component equals how far the OTHER component's concentration is from the target. Parts of A = 22 − 10 = 12. Parts of B = 40 − 22 = 18. So A : B = 12 : 18 = 2 : 3. Verify: (2 × 40 + 3 × 10) ÷ 5 = (80 + 30) ÷ 5 = 110 ÷ 5 = 22%. The key insight is that the ratio is the INVERSE of the distance each concentration is from the target, not the direct distance.

Cyclist 1 rides from Town P to Town Q at 15 mph, then immediately returns along the same route at 10 mph. Cyclist 2 covers the identical round trip at a constant speed of 12 mph throughout. Which cyclist completes the round trip first?

Answer: They finish at exactly the same time; both average 12 mph over the round trip

When the same distance is traveled at two different speeds, the correct average speed for the whole trip is the harmonic mean — NOT the arithmetic mean. Harmonic mean = 2(15)(10) ÷ (15 + 10) = 300 ÷ 25 = 12 mph. Since Cyclist 1's effective average speed is exactly 12 mph (same as Cyclist 2's constant speed), they tie. The common trap is averaging arithmetically: (15 + 10) ÷ 2 = 12.5 mph, which overstates Cyclist 1's speed because they spend more time at the slower pace.

The variable p varies directly as the square of q and inversely as r. When q = 3 and r = 2, the value of p is 18. What is the value of p when q = 5 and r = 10?

Answer: 10

Write the combined variation equation: p = k·q²/r. Substitute the known values to find k: 18 = k·(3²)/2 = k·9/2, so k = 18·2/9 = 4. Now apply the formula with the new values: p = 4·(5²)/10 = 4·25/10 = 100/10 = 10. A common error is forgetting to square q first before substituting, or misidentifying whether the variation is direct or inverse for each variable.

In a chemical reaction, the rate at which reactant R is consumed is proportional to the square of its current concentration. Initially, the concentration is 0.80 mol/L and the consumption rate is 0.192 mol/(L·s). What is the consumption rate when the concentration falls to 0.50 mol/L?

Answer: 0.075 mol/(L·s)

Since rate = k·[R]², first find k: 0.192 = k·(0.80)² = k·0.64, so k = 0.192 ÷ 0.64 = 0.300. Then at [R] = 0.50: rate = 0.300·(0.50)² = 0.300·0.25 = 0.075 mol/(L·s). Alternatively, use the ratio shortcut: rate₂/rate₁ = ([R]₂/[R]₁)² = (0.50/0.80)² = (0.625)² = 0.390625; rate₂ = 0.192 × 0.390625 = 0.075. Choice B (0.120) is the trap answer for students who use a simple linear proportion instead of squaring.

A European car's fuel efficiency is rated at 6.5 liters per 100 kilometers. A comparable American car achieves 38 miles per gallon. Which car is more fuel-efficient, and by approximately what percentage? (Use: 1 mile = 1.609 km; 1 gallon = 3.785 liters.)

Answer: The American car is more efficient, by approximately 5%

Convert the European rating to mpg: 100 km ÷ 1.609 km/mi ≈ 62.15 miles per 6.5 liters; 6.5 L ÷ 3.785 L/gal ≈ 1.717 gallons. So the European car gets 62.15 ÷ 1.717 ≈ 36.2 mpg. The American car gets 38 mpg. The American car is more efficient by (38 − 36.2) ÷ 36.2 × 100 ≈ 5.0%. The trap answers exploit unit confusion — dividing in the wrong order or skipping the two-step conversion gives wildly different results.

A map has a scale of 1 cm : 5 km. A rectangular forest preserve on the map measures 2.4 cm by 3.5 cm. The government plans to plant trees at a uniform density of 120 trees per hectare (1 hectare = 10,000 m²; 1 km = 1,000 m). How many trees will be planted in the entire preserve?

Answer: 2,520,000

Step 1 — find actual dimensions: 2.4 cm × 5 km/cm = 12 km; 3.5 cm × 5 km/cm = 17.5 km. Step 2 — find area: 12 × 17.5 = 210 km². Step 3 — convert to hectares: 1 km² = 100 hectares (since 1 km = 1,000 m → 1 km² = 10⁶ m² = 100 hectares), so 210 km² = 21,000 hectares. Step 4 — count trees: 21,000 × 120 = 2,520,000. The most common errors are applying the scale factor only once (to one dimension) instead of to both, and confusing the km² → hectare conversion factor.