ARArithmetic Reasoning

Ratios & Proportions

Set up equal fractions and cross-multiply to solve for an unknown — the bread-and-butter of ASVAB Arithmetic Reasoning.

Formula Reference

  • Proportion: a/b = c/d → cross-multiply: a×d = b×c
  • Unit rate: divide the quantity by the total count (e.g. $4.50 ÷ 3 cans = $1.50/can)
  • Scaling up: unit rate × new count = answer
  • Part-to-whole: ratio A:B means A out of every (A+B) total

What the ASVAB is actually testing

Ratio and proportion questions look like word problems, but they're almost always just one operation: find a rate, then scale it. The Army wants to know if you can figure out how many gallons a vehicle needs on a longer convoy if you know its fuel consumption over a shorter distance. That's it.

The test doesn't throw trick math at you. The trap is in the setup — reading too fast and flipping numbers around.

The two patterns you'll see

Pattern 1 — Direct proportion (more of X means more of Y)

A truck uses 8 gallons for every 100 miles. How much fuel for 350 miles?

Set it up so matching units are in the same position on both sides:

8 gal / 100 mi = x gal / 350 mi

Cross-multiply: 8 × 350 = 100x → x = 2800/100 = 28 gallons

Pattern 2 — Part-to-part vs. part-to-whole

"The ratio of mechanics to drivers is 1:4" means for every 1 mechanic there are 4 drivers — 5 people total in every group. If there are 20 drivers, there are 20/4 = 5 mechanics, and the total crew is 25.

Watch for the question asking about the total, not just one part.

Speed trick: unit rate first

For most ratio problems, the fastest path is:

  1. Find the unit rate (cost per 1 item, miles per 1 gallon, etc.)
  2. Multiply by the new count

You skip setting up a proportion entirely. On a timed test, that matters.

What trips people up

The most common mistake: a problem says "ratio of boys to girls is 2:3" and someone writes 2/3 of the total as boys instead of 2/5. Remember — a ratio compares two groups. The whole is the sum of both parts.

The second most common mistake: inconsistent units. If one side of your proportion is miles per gallon, the other side must also be miles per gallon, not gallons per mile.

Practice strategy

These questions reward setup over arithmetic. If you're getting them wrong, slow down on the first 15 seconds — just make sure you know what ratio you're writing down and what you're solving for. The math after that is usually straightforward multiplication or division.

Common Pitfalls

  • Confusing 'ratio of A to B' (A/B) with 'A as a fraction of the total' (A/(A+B))
  • Forgetting to keep units consistent on both sides of the proportion
  • Setting up the proportion upside-down — always put the same unit in the same position (top or bottom) on both sides

Worked Examples

Q1: If 3 cans of soup cost $4.50, how much do 7 cans cost?

Answer: Unit rate: $4.50 ÷ 3 = $1.50/can. Then 7 × $1.50 = $10.50. Or as a proportion: 3/4.50 = 7/x → x = (4.50 × 7)/3 = $10.50

Q2: A recipe calls for 2 cups of flour for every 3 cups of oats. If you use 9 cups of oats, how much flour do you need?

Answer: Set up: 2/3 = x/9 → x = (2 × 9)/3 = 6 cups of flour

Q3: In a class of 30 students, the ratio of boys to girls is 2:3. How many girls are there?

Answer: Total parts = 2+3 = 5. Girls = (3/5) × 30 = 18 girls

Loading practice questions...