Basic Operations and Word Problems: Practice Questions & Explanations

This is a basic division problem: 12 slices ÷ 4 people = 3 slices per person. Word problems on the ASVAB require translating the situation into a math operation. Look for keywords: 'share equally' or 'divided equally' indicates division; 'total' or 'altogether' indicates addition; 'how much more' or 'difference' indicates subtraction; 'each' or 'per' often indicates division or multiplication. The strategy is to identify what is being asked, set up the operation, and verify the answer makes sense in context. Here, 3 × 4 = 12 confirms the answer. The Arithmetic Reasoning section emphasizes setting up the problem correctly more than complex calculations.

Average speed = total distance ÷ total time = 240 miles ÷ 4 hours = 60 miles per hour. This is the basic rate formula: rate × time = distance, which rearranges to rate = distance ÷ time. The same formula handles many ASVAB problems: speed, work rate, flow rate, production rate. Three-variable problems can be set up: if you know any two of (distance, time, speed), you can find the third. Examples: at 60 mph, how long to drive 180 miles? Time = 180/60 = 3 hours. If you drove 2.5 hours at 50 mph, how far did you go? Distance = 50 × 2.5 = 125 miles. Memorize the relationship as a triangle: distance on top, rate and time on bottom — cover the one you want to find to see the operation.

Division problem: 24 soldiers ÷ 4 squads = 6 soldiers per squad. Verification: 6 × 4 = 24 ✓. Military-themed word problems are common on the ASVAB. Key skill: translating context to operation. 'Divide equally' or 'split into equal groups' is division. Variations might ask: How many squads of 8 can she form from 24 soldiers? (24 ÷ 8 = 3). How many soldiers are left if she forms 5 squads of 4? (5 × 4 = 20 in squads; 24 - 20 = 4 left over — i.e., 24 ÷ 5 = 4 remainder 4). The remainder concept appears in problems like: 'If you have 30 people and each truck holds 4, how many trucks do you need?' (30 ÷ 4 = 7 remainder 2; you need 8 trucks because the leftover 2 still need transport — this is the 'ceiling' division case where you round up).

Method 1 (subtract the reduction): 1/3 of 45 = 15 pounds reduction; 45 - 15 = 30 pounds new weight. Method 2 (multiply by the fraction remaining): if 1/3 is removed, 2/3 remains; 2/3 × 45 = 30 pounds. Both give the same answer. Fractional reduction is similar to percent reduction. Useful shortcuts: 1/2 reduction = 1/2 remaining (cut in half); 1/3 reduction = 2/3 remaining; 1/4 reduction = 3/4 remaining; 2/3 reduction = 1/3 remaining. Real-world applications: weight reduction, scaling recipes down, reducing dosage by a fraction, lightening loads. Quick mental math: half of 45 is 22.5; one-third would be slightly less, around 15 — that's the reduction; the new total is approximately 30. The estimation supports the precise calculation.

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.

420 ÷ 28 = 15 miles per gallon. Division word problem: divide total miles by total gallons to find miles per gallon. Why the others are wrong: 12 mpg would need 35 gallons for 420 miles; 18 mpg would need about 23 gallons; 21 mpg would need 20 gallons. Quick check: 15 × 28 = 420 ✓. On the ASVAB, 'per' in a rate question always means divide — the quantity before 'per' goes on top.

Total deductions = $360 + $120 = $480. Take-home = $2,400 - $480 = $1,920. Take-home (net) pay is gross pay minus every deduction listed. The distractor $2,040 is what you get if you subtract only the tax ($2,400 − $360), $2,160 subtracts only $240, and $1,800 subtracts $600 — too much. Add all deductions first, then subtract once.

6 × 4.5 = 27 tons total cargo weight. Repeated addition of a decimal: 6 × 4 = 24 and 6 × 0.5 = 3, so 24 + 3 = 27. The distractor 24 tons drops the half-ton on each truck, 30 rounds each load up to 5 tons, and 25 is a near-miss meant to catch arithmetic slips. Breaking the decimal into whole and fractional parts avoids all three.

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.

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.

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.