Ratios, Proportions, and Rates: Practice Questions & Explanations

Set up a proportion: 5 pounds / $4 = 15 pounds / x. Cross-multiply: 5x = 60. Divide: x = $12. Alternative thinking: 15 pounds is 3 times 5 pounds, so the cost is also 3 times: 3 × $4 = $12. Either method works. Proportions are central to many ASVAB problems: cooking (recipe scaling), maps (distance scale), construction (mixing ratios), and any 'if X equals Y, then how much for Z' situation. Set up as fraction equal to fraction with corresponding units in the same position. Cross-multiply to solve: (a/b) = (c/d) means ad = bc. The unit-rate method also works: $4 ÷ 5 pounds = $0.80 per pound; $0.80 × 15 pounds = $12. Use whichever method clicks for the problem. Verify the answer is reasonable — 15 pounds should cost roughly three times 5 pounds.

Set up the ratio as a proportion. Cats/dogs = 3/5 = 24/x. Cross-multiply: 3x = 120. Solve: x = 40 dogs. Alternative: cats are 3 'parts', and there are 24 actual cats; so each 'part' = 24/3 = 8. Dogs are 5 parts; 5 × 8 = 40 dogs. Total animals would be 24 + 40 = 64, with 3:5 ratio confirmed (24:40 reduces to 3:5). Ratios appear in many ASVAB problems. Key skills: (1) Setting up the proportion correctly with matching units; (2) Solving by cross-multiplication; (3) Understanding part-to-part vs part-to-whole ratios. Part-to-part is the actual ratio (3:5); part-to-whole compares one part to the total (3/8 for cats, 5/8 for dogs). Be careful which form the problem asks for. Inverse ratios (when one variable goes up as another goes down) are different — for example, more workers means less time per worker.

Method 1 (proportion): 200 miles / 8 gallons = x miles / 12 gallons; cross-multiply: 8x = 2400; x = 300 miles. Method 2 (unit rate): 200 miles ÷ 8 gallons = 25 miles per gallon; 25 × 12 = 300 miles. Both methods give the same answer. Fuel efficiency problems are common on the ASVAB and in real military contexts. Related calculations: (1) Cost per mile = total fuel cost ÷ miles driven; (2) Fuel needed for a trip = trip distance ÷ MPG; (3) Cost of a trip = trip distance × cost per mile. Example: trip of 450 miles in a vehicle that gets 25 MPG at $4/gallon: gallons needed = 450/25 = 18; cost = 18 × $4 = $72. The unit-rate method (find the per-unit value, then multiply) is often the most efficient approach to proportion problems and works for many real-world situations.

This is an inverse proportion problem — more workers means less time, not more. Total work = workers × time = 3 × 8 = 24 worker-hours. With 4 workers: time = 24/4 = 6 hours. The product 'workers × time' stays constant; more workers reduces time proportionally. Direct proportion (more of one means more of the other): cost vs quantity, distance vs time at constant speed. Inverse proportion (more of one means less of the other): workers vs time, speed vs time over fixed distance, pipes filling a tank, machines producing widgets. Distinguishing the two types is essential. Setting up: if workers and time are inversely proportional, then workers₁ × time₁ = workers₂ × time₂. So 3 × 8 = 4 × t; t = 6 hours. Common ASVAB version: 'If 5 pumps can fill a tank in 12 hours, how long for 4 pumps?' Inverse: 5×12 = 4×t; t = 15 hours (fewer pumps takes longer).

Set up the proportion: 3/2 = x/8. Cross-multiply: 2x = 24, so x = 12 cups of flour. Think of it as a scale factor: 8 cups of sugar is 4 times the 2 cups in the recipe, so the flour also scales by 4: 3 × 4 = 12. Checking the distractors: 10 and 11 don't keep the 3:2 ratio (10:8 = 5:4, 11:8 is not a whole ratio), and 14 would be the answer to 3/2 = x/9.33, not x/8.

Inverse proportion: 5 workers × 8 days = 10 workers × d days. d = 40 ÷ 10 = 4 days. This is an inverse (indirect) proportion: more workers means fewer days, so the product workers × days stays constant (40 worker-days). Doubling the crew halves the time. The distractor 16 days is the direct-proportion trap (doubling workers and doubling days), which would make a bigger crew slower — a sign the relationship was set up backwards.

3.5 × 50 = 175 miles. Scale proportion: inches × miles-per-inch = actual miles. Map-scale problems are direct proportions: 1 inch : 50 miles, so 3.5 inches : 3.5 × 50 miles. Split the decimal to do it mentally — 3 × 50 = 150 plus 0.5 × 50 = 25 gives 175. The distractor 150 miles ignores the half inch, 200 rounds 3.5 up to 4, and 225 uses 4.5 inches.

Set up a PROPORTION: flour/sugar = 3/2 = 9/x. Cross-multiply: 3x = 2 × 9 = 18, so x = 6 cups of sugar. ASVAB Arithmetic Reasoning tests ratios/proportions.

Shortcut: 9 cups of flour is 3 times the original 3 cups (9 ÷ 3 = 3), so multiply the sugar by 3 too: 2 × 3 = 6 cups. Strategy: keep the ratio consistent — whatever you multiply one part by, multiply the other part by the same factor. Check: 9/6 reduces to 3/2. ✓

Set up a proportion using the scale: 1 inch / 25 miles = 4.5 inches / x miles. Cross-multiply: 1 × x = 25 × 4.5, so x = 112.5 miles. Scale and proportion problems are solved by writing the known ratio equal to the unknown ratio and cross-multiplying. Here each inch is 25 miles, so 4.5 inches is simply 4.5 × 25 = 112.5 miles. Always keep the units consistent (inches with inches, miles with miles) when setting up the proportion, and the cross-multiplication gives the missing value directly.

This is an inverse proportion: more workers means less time. The total work is 3 workers × 8 hours = 24 worker-hours. With 4 workers, time = 24 worker-hours ÷ 4 workers = 6 hours. In inverse-proportion (work-rate) problems, the product of workers and time stays constant for the same job. Compute the total worker-hours, then divide by the new number of workers. Four workers finish the job in 6 hours, which makes sense because adding a worker should reduce the time, not increase it.

Keeping the same proportions means the ratio of sides stays constant. Original ratio is 4:6. Set up a proportion with the new short side: 4/6 = 10/x. Cross-multiply: 4x = 60, so x = 15 inches. Proportional scaling problems are solved by matching corresponding sides in a ratio and cross-multiplying. The shorter side went from 4 to 10 (a factor of 2.5), so the longer side scales the same way: 6 × 2.5 = 15 inches. The enlarged photo's longer side is 15 inches.