Arithmetic Reasoning is described by the ASVAB program as the ability to solve basic arithmetic word problems. On the computer form it is 15 scored questions in 55 minutes — the longest clock of the four AFQT subtests, and still the one people misjudge.
The four shapes that keep coming back
- Rate problems. Distance, speed and time; work done per hour; cost per unit. Almost always one of the three is missing.
- Percent problems. Discounts, markups, tax, increase and decrease. The trap is working from the wrong base.
- Proportions. If this many do that much, how much do those many do. Set it up as a ratio and the arithmetic falls out.
- Multi-step money problems. Several purchases, a total, then change. Nothing hard, but each step is a place to drop a digit.
Ten worked examples
1. A truck travels 240 miles in 4 hours. What is its average speed?
Speed is distance divided by time: 240 ÷ 4 = 60 miles per hour. Whenever two of distance, speed and time are given, the third is one division or one multiplication away.
2. A jacket costs $24 after a 20% discount. What was the original price?
This is the classic wrong-base trap. Adding 20% back to $24 gives $28.80, which is not the answer. A 20% discount leaves 80% of the original, so 0.80 × original = 24, and 24 ÷ 0.8 = $30.
3. A recipe for 4 people uses 6 eggs. How many eggs for 10 people?
Find the per-person figure first: 6 ÷ 4 = 1.5 eggs each, and 1.5 × 10 = 15 eggs.
4. A worker earns $18 an hour for 40 hours and time and a half beyond that. What is the pay for 46 hours?
Regular: 40 × 18 = $720. Overtime rate: 18 × 1.5 = $27, for 6 hours = $162. Total $882. Compute the overtime rate before the overtime hours, not after.
5. Eight pumps drain a pool in 6 hours. How long do twelve take?
Convert to pump-hours: 8 × 6 = 48. Then 48 ÷ 12 = 4 hours. Every "more workers, less time" question is this same conversion.
6. A car uses 12 gallons to travel 336 miles. How far on 5 gallons?
336 ÷ 12 = 28 miles per gallon, and 28 × 5 = 140 miles.
7. A coat priced at $120 is marked down 25%, then 10% more at the register. What is the final price?
The two discounts do not add to 35%. After 25% off: $90. Then 90% of $90 = $81. Successive percentages multiply.
8. Two trains leave the same station in opposite directions at 50 and 70 miles per hour. How far apart after 3 hours?
They separate at 50 + 70 = 120 miles per hour, so 120 × 3 = 360 miles. Opposite directions add; same direction subtracts.
9. A mixture is 3 parts water to 2 parts concentrate. How much concentrate is in 45 gallons?
Five parts in all, so one part is 45 ÷ 5 = 9 gallons. Concentrate is two parts: 18 gallons. Count the total parts before doing anything else.
10. A worker finishes 3/5 of a job in 9 hours. How long is the whole job?
If three fifths take 9 hours, one fifth takes 3, so five fifths take 15 hours.
Fifty-five minutes for fifteen questions is about three and a half minutes each — generous. The mistake is not slowness, it is setting the problem up the wrong way and then doing careful arithmetic on it. Read the question twice and write the equation before touching a number.
Why there is no calculator
The ASVAB program says plainly that you cannot use one, and its own FAQ gives the reason: the ability to do hand calculations may be required on the job. So every number in our bank is chosen to close with pencil and paper. If you are practicing on a site whose answers need a calculator, you are training the wrong skill.
Ninety more arithmetic reasoning questions
Every one with the method written out. First five free, no account, no card.