ASVAB Arithmetic Reasoning — Percentages, Interest, and Discount Problems

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).

Let P = original principal. Simple interest formula: A = P(1 + rt), where A is the final amount, r is the rate, t is the time. $1,080 = P(1 + 0.04 × 2) = P(1.08). Solve: P = $1,080/1.08 = $1,000. Verification: interest = $1,000 × 0.04 × 2 = $80; total = $1,000 + $80 = $1,080 ✓. This 'reverse' interest problem asks for the principal given the ending amount, rate, and time. Strategy: compute the total growth factor (1 + rt) = (1 + 0.08) = 1.08; divide the final amount by this factor to find the original. Same approach works for tax problems ('what was the pre-tax price if final price was $X?'), discount problems ('what was the original price if discount was X%?'), and compound interest problems (more complex factor). Always identify what you're solving for and set up the equation accordingly.

Multiplication problem: 250 × 24 = 6,000 meals. Calculation strategies: (1) Standard multiplication: 250 × 24 = 250 × 20 + 250 × 4 = 5,000 + 1,000 = 6,000. (2) Doubling: 250 × 24 = 500 × 12 = 1000 × 6 = 6,000 (doubling one factor and halving the other gives the same product). (3) Mental math: 250 × 24 = 25 × 24 × 10 = 600 × 10 = 6,000. Multi-step word problems on the ASVAB often require setting up the equation and computing the answer. Identify: what are the quantities? What operation links them? Cases × meals per case = total meals — multiplication. Verify reasonableness: 250 × 24 should be in the thousands; 6,000 fits. If the answer choices include both 1,200 (250 × 4 or wrong place value) and 6,000, the units and operations matter. Always double-check by estimating: 250 × 25 = 6,250; 6,000 is just slightly less than that, which is reasonable for 250 × 24.

Sum = average × count = 30 × 5 = 150. This is the basic rearrangement of the average formula: average = sum/count, therefore sum = average × count. This relationship is the key to many average problems. Variations: (1) 'Average of 4 numbers is 25; if one number is removed and the new average is 23, what was the removed number?' Original sum: 25 × 4 = 100; new sum: 23 × 3 = 69; removed number: 100 - 69 = 31. (2) 'A class of 30 has an average grade of 85; another class of 20 has an average of 75. What is the combined average?' Combined sum: (30 × 85) + (20 × 75) = 2,550 + 1,500 = 4,050; combined count: 50; combined average: 4,050/50 = 81. (3) Common ASVAB variations of these problems test understanding that you cannot simply average the averages when the groups have different sizes — you must use weighted averages or compute from sums.

Multi-step problem: first find the area, then multiply by the unit cost. Area = 12 × 15 = 180 square feet. Cost = 180 × $4 = $720. Common mistake: computing the perimeter (2 × 12 + 2 × 15 = 54 feet) instead of the area — but carpet covers the floor area, not just the perimeter. The perimeter would be relevant for baseboards or molding. Always identify whether the problem asks for: linear measurement (perimeter, length, distance) — uses regular units; area measurement (floor coverage, paint coverage, land area) — uses square units; volume measurement (water, air, soil) — uses cubic units. Real-world applications: flooring/carpet (area × cost per sq ft); paint (area × cost per sq ft, with coverage adjustments for second coats); concrete (volume × cost per cubic yard); gravel (volume × cost per ton). Step-by-step approach to multi-step problems: identify all required calculations, solve each in turn, combine.

420 ÷ 28 = 15 miles per gallon. Division word problem: divide total miles by total gallons to find miles per gallon.

Set up the proportion: 3/2 = x/8. Cross-multiply: 2x = 24, so x = 12 cups of flour.

Area = length × width = 15 × 12 = 180 square feet.

15% of 240 = 0.15 × 240 = 36. Convert percent to decimal (÷100) then multiply.

Total deductions = $360 + $120 = $480. Take-home = $2,400 - $480 = $1,920.