Why Arithmetic Reasoning Matters
Arithmetic Reasoning (AR) is one of the four sections that make up your AFQT score — the core enlistment eligibility score. It's also a component in multiple Army line scores (GT, ST, FA, CO, SC) and Navy and Marine composites. A strong AR score opens more doors than almost any other section.
The CAT-ASVAB has 16 AR questions. The paper version has 30. Every question is a word problem — no pure equations. You'll read a scenario and have to figure out what math operation is needed.
The Universal Word Problem Strategy
Before doing any arithmetic, go through these four steps:
- Read the question last first. Read the final question before the problem setup. This tells you what you're solving for, so you know what to pay attention to.
- Identify the numbers and units. Underline every number and label what it represents (miles, hours, dollars, etc.)
- Write the equation before calculating. Resist the urge to calculate in your head. Write it out.
- Check your units. If the question asks for miles per hour and your answer is just miles, you're not done.
Core Topics Covered on AR
Fractions, Decimals, and Percentages
These appear in nearly every AR question in some form. Know these cold:
- To find X% of Y: multiply Y × (X ÷ 100)
- To find what percent X is of Y: (X ÷ Y) × 100
- Percent increase: [(New − Old) ÷ Old] × 100
- Converting fractions to decimals: divide the numerator by the denominator
Example: A soldier uses 3/8 of his 40-liter water supply. How many liters remain?
3/8 × 40 = 15 liters used. 40 − 15 = 25 liters remain.
Ratios and Proportions
Set up as fractions and cross-multiply:
Example: A vehicle travels 120 miles in 3 hours. At the same rate, how far will it travel in 5 hours?
120/3 = x/5 → 3x = 600 → x = 200 miles.
Distance, Rate, and Time
Formula: Distance = Rate × Time (or D = RT)
Rearrangements: R = D/T, T = D/R
Example: A convoy leaves base at 45 mph. How long to travel 270 miles?
T = 270 ÷ 45 = 6 hours.
Work Rate Problems
Formula: If person A completes a job in X hours and person B in Y hours, working together they complete it in:
Time = (X × Y) ÷ (X + Y)
Example: One soldier can unload a truck in 4 hours; another in 6 hours. Together?
(4 × 6) ÷ (4 + 6) = 24 ÷ 10 = 2.4 hours.
Averages
Average = Sum of all values ÷ Number of values
A common twist: finding a missing value when you know the average.
Example: A Marine scored 78, 85, and 92 on three tests. What must he score on the fourth to average 85?
Total needed: 85 × 4 = 340. Current total: 78 + 85 + 92 = 255. Missing score: 340 − 255 = 85.
Simple and Compound Interest
- Simple interest: I = P × R × T (Principal × Rate × Time)
- Compound interest: A = P(1 + r/n)^(nt) — rarely tested in depth on ASVAB
Geometry Basics
- Area of rectangle: l × w
- Area of triangle: (base × height) ÷ 2
- Area of circle: π × r²
- Volume of rectangular box: l × w × h
- Perimeter: sum of all sides
Common Traps to Avoid
- Unit mismatches: If one speed is in mph and another in minutes, convert before calculating.
- Reading past the question: The problem might give you extra information not needed for the answer — ignore it.
- Rounding too early: Keep decimals through intermediate steps; only round at the final answer.
- Confusing "of" and "off": "30% of $80" = $24. "30% off $80" = $80 − $24 = $56.
Practice Strategy: Work Backwards
On multiple choice questions, you can sometimes plug answer choices back into the problem and check which one works. This is slower but can save you when you can't figure out the setup.
Start with the middle value (usually choice B or C). If the result is too high, try a lower answer. If too low, try higher. This process of elimination often leads to the right answer in 2–3 tries.
Timed Practice is Essential
The CAT-ASVAB gives you about 39 minutes for 16 AR questions — roughly 2.4 minutes per question. Most people don't struggle with the math itself; they struggle with the clock. Train timed from day one. Take section drills under strict time limits, not open-ended review sessions.
If you're running out of time on a question, write down your best estimate, flag it mentally, and move on. Spending 5 minutes on one question and getting it right still costs you two other questions.
Use the AR section drills in our practice test section to build speed under real time pressure.