ASVAB Arithmetic Reasoning — Rates, Ratios, and Proportions

6 × 4.5 = 27 tons total cargo weight.

3.5 × 50 = 175 miles. Scale proportion: inches × miles-per-inch = actual miles.

This is a division word problem. Divide 156 by 12: 12 × 13 = 156 exactly, so 156 ÷ 12 = 13 with no remainder. That means 13 full boxes can be packed with 0 rations left over. The key skill in arithmetic reasoning is translating words into the correct operation — 'pack into boxes that each hold' signals division, and the remainder tells you the leftover amount. Always check by multiplying the quotient back by the divisor (13 × 12 = 156) to confirm. Here the division comes out even, so the answer is 13 boxes with nothing left over.

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.

Miles per gallon is total miles divided by gallons used: 165 ÷ 5.5 = 30 mpg. Rate problems like this are solved by dividing the quantity (miles) by the unit you want it per (gallons). To divide by 5.5, you can multiply both numbers by 2 to clear the decimal: 330 ÷ 11 = 30. The car gets 30 miles per gallon. Keeping track of which quantity goes on top (the thing being measured, miles) and which on the bottom (the per-unit, gallons) is the key to setting up rate calculations correctly.

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.

First compute gross pay: $18 × 7.5 hours = $135. Then subtract the withholding: $135 − $24 = $111 take-home pay. Multi-step word problems require doing the operations in the right order: earnings first (rate × hours), then deductions. To multiply $18 by 7.5, note 18 × 7 = 126 and 18 × 0.5 = 9, so 126 + 9 = $135. After taxes the worker keeps $111. The $135 distractor is the gross pay before the tax is subtracted, so read carefully for the final 'take-home' value.

Area = π r² = 3.14 × 7² = 3.14 × 49 ≈ 153.86 square inches.

Arrange in order: 4, 5, 6, 7, 8. The median (middle value) is 6 miles.