ASVAB Study Guide

Comprehensive preparation for the Armed Services Vocational Aptitude Battery

Arithmetic Reasoning

Solving word problems using basic arithmetic operations

Core Concepts

ASVAB Format: - CAT-ASVAB: 15 questions in 55 minutes
- P&P-ASVAB: 30 questions in 36 minutes
- Contributes to AFQT score for enlistment eligibility

Basic Arithmetic Operations

Addition & Subtraction

  • Combine quantities or find differences
  • Align decimals and pay attention to signs
  • For fractions, find common denominator first

Multiplication & Division

  • Repeated addition or equal grouping
  • Sign rules: (-) × (-) = (+), (-) × (+) = (-)
  • For decimals, multiply as whole numbers then place decimal

Order of Operations (PEMDAS)

  1. Parentheses
  2. Exponents
  3. Multiplication & Division (left to right)
  4. Addition & Subtraction (left to right)

Fractions, Decimals, and Percentages

Concept Key Operations Example
Fractions Addition/Subtraction: Common denominator
Multiplication: Multiply numerators & denominators
Division: Multiply by reciprocal 		1/4 + 3/8 = 2/8 + 3/8 = 5/8
Decimals Add/Subtract: Align decimals
Multiply: Multiply as whole numbers, then place decimal
Divide: Move decimal in divisor 		3.14 + 2.5 = 5.64
Percentages To decimal: Divide by 100
To fraction: Write over 100, simplify
Percentage of number: Multiply by decimal equivalent
Percentage change: (Change ÷ Original) × 100 		15% of 80 = 0.15 × 80 = 12

Ratios and Proportions

Ratios

  • Comparison of quantities (3:4, 3 to 4, or 3/4)
  • Should be simplified to lowest terms
  • Can be scaled up or down proportionally

Proportions

  • Equation stating two ratios are equal
  • Solved using cross-multiplication: a/b = c/d → ad = bc
  • Direct proportion: Both quantities increase together
  • Inverse proportion: One increases as other decreases

Word Problems

Problem Type Formula/Approach Example
Rate Problems Rate × Time = Distance 60 mph × 2.5 h = 150 miles
Work Problems Time = (a × b)/(a + b) John (3h) + Mary (2h) = (3×2)/(3+2) = 1.2h
Mixture Problems (Q1 × V1) + (Q2 × V2) = (Qtotal × Vavg) Mix 20% and 60% solution to get 30%
Age Problems Set up equations with current and future ages John = Mary + 5; in 3 years, sum = 41
Money Problems Sale price = Original - Discount
Discount = Original × Rate		 $45 shirt, 20% off: $45 - $9 = $36
Interest Problems Simple: I = Prt
Compound: A = P(1 + r)^t		 $1000 at 5% for 3 years: $150 interest

Geometry Basics

Area & Perimeter

  • Rectangle: A = l×w, P = 2(l+w)
  • Square: A = s², P = 4s
  • Triangle: A = ½bh, P = sum of sides
  • Circle: A = πr², C = 2πr

Volume

  • Rectangular Prism: V = l×w×h
  • Cube: V = s³
  • Cylinder: V = πr²h
  • Sphere: V = ⁴⁄₃πr³

Algebraic Concepts

Linear Equations

  • Simplify both sides
  • Isolate variable terms
  • Solve using inverse operations
  • Example: 3x + 5 = 2x - 7 → x = -12

Word Problems to Equations

  • Identify unknowns and assign variables
  • Write equations based on relationships
  • Solve and check in context
  • Example: Consecutive integers summing to 35

Data Interpretation

Tables & Graphs

  • Read values accurately
  • Compare data points
  • Calculate averages and percentages
  • Identify trends and patterns

Statistics Basics

  • Mean: Sum of values ÷ count
  • Median: Middle value in ordered list
  • Mode: Most frequent value
  • Range: Max - Min

Examples and Solved Problems

Calculate: 4 + 3 × 5 - 2²

Solution:

1. Follow PEMDAS: Exponents first → 2² = 4

2. Multiplication: 3 × 5 = 15

3. Addition and subtraction left to right: 4 + 15 - 4 = 15

Answer: 15

A store offers 15% discount on a $80 jacket. What is the sale price?

Solution:

1. Discount amount = 15% of $80 = 0.15 × 80 = $12

2. Sale price = Original price - Discount = $80 - $12 = $68

Answer: $68

The ratio of boys to girls is 3:5. If there are 24 boys, how many girls are there?

Solution:

1. Set up proportion: 3/5 = 24/x

2. Cross multiply: 3x = 5 × 24

3. Solve: 3x = 120 → x = 40

Answer: 40 girls

A car travels at 65 mph. How far in 3 hours 45 minutes?

Solution:

1. Convert time: 3h 45m = 3 + 45/60 = 3.75 hours

2. Distance = Rate × Time = 65 × 3.75 = 243.75 miles

Answer: 243.75 miles

John paints a fence in 6 hours, Mary in 4 hours. How long together?

Solution:

1. John's rate: 1/6 fence/hour

2. Mary's rate: 1/4 fence/hour

3. Combined rate: 1/6 + 1/4 = 5/12 fence/hour

4. Time = 1 ÷ (5/12) = 12/5 = 2.4 hours (2h 24m)

Answer: 2.4 hours

Mix 25% salt solution with 10L of 40% to get 30% solution. How much 25% solution?

Solution:

1. Let x = amount of 25% solution

2. Equation: 0.25x + 0.40(10) = 0.30(x + 10)

3. Solve: 0.25x + 4 = 0.30x + 3

4. Rearrange: 4 - 3 = 0.30x - 0.25x → 1 = 0.05x → x = 20

Answer: 20 liters

Solve for x: 3(x - 2) = 2x + 7

Solution:

1. Distribute: 3x - 6 = 2x + 7

2. Subtract 2x: x - 6 = 7

3. Add 6: x = 13

Answer: x = 13

Review Section

Problem-Solving Strategies

  1. Read carefully: Understand what the problem is asking
  2. Identify relevant information: Determine what data is provided and needed
  3. Choose appropriate method: Select the right mathematical concept/formula
  4. Set up the problem: Write equation or draw diagram
  5. Solve: Perform calculations carefully
  6. Check: Verify answer makes sense in context

Common Pitfalls to Avoid

Basic Arithmetic

  • Order of operations errors (PEMDAS)
  • Division by zero (undefined)
  • Mishandling negative numbers

Fractions & Decimals

  • Adding fractions without common denominators
  • Decimal place value errors
  • Incorrect fraction comparisons

Percentages

  • Confusing percentage increase/decrease
  • Mishandling consecutive percentage changes
  • Using wrong base value for calculations

Word Problems

  • Misinterpreting the problem
  • Unit conversion errors
  • Setting up incorrect equations

Memory Aids and Tricks

PEMDAS

"Please Excuse My Dear Aunt Sally"

  • Parentheses
  • Exponents
  • Multiplication/Division
  • Addition/Subtraction

Fraction Operations

  • Addition/Subtraction: Common denominator needed
  • Multiplication: Numerator × numerator, denominator × denominator
  • Division: Multiply by reciprocal

Percentage Triangle

Cover the value you want to find:

  • Part = Percentage × Whole ÷ 100
  • Percentage = Part ÷ Whole × 100
  • Whole = Part × 100 ÷ Percentage

Distance Formula Triangle

Cover the value you want to find:

  • Distance = Rate × Time
  • Rate = Distance ÷ Time
  • Time = Distance ÷ Rate

Quick Review Questions

  1. What is 4 × 3 + 2²?
  2. Convert 3/8 to decimal and percentage
  3. If 3 shirts cost $45, how much for 5?
  4. A car travels at 60 mph. How far in 2.5 hours?
  5. John paints room in 4h, Mary in 6h. How long together?
  6. Shirt $50, 20% off. Sale price?
  7. Solve: 2(x + 3) = 3x - 4
  8. Rectangle area: length 8ft, width 5ft
  9. Mean, median, range of: 5,8,12,8,7
  10. $1000 at 5% simple interest for 2 years

Answers to Quick Review

  1. 16
  2. 0.375, 37.5%
  3. $75
  4. 150 miles
  5. 2.4 hours
  6. $40
  7. x = 10
  8. 40 sq ft
  9. Mean: 8, Median: 8, Range: 7
  10. $1,100

Final Tips

  • Time management: Don't spend too long on one problem
  • Estimation: Use to check reasonableness of answers
  • Show work: Helps avoid careless mistakes
  • Double-check: Verify calculations and units
  • Practice regularly: The key to improving arithmetic reasoning skills

Practice Test

Question 1

Calculate: 6 + 3 × 4 - 2²

14
16
20
28

Solution: Following PEMDAS: 2² = 4, 3 × 4 = 12, 6 + 12 = 18, 18 - 4 = 14

Answer: 14

Question 2

What is 15% of 80?

8
12
15
20

Solution: 15% of 80 = 0.15 × 80 = 12

Answer: 12

Question 3

A shirt originally priced at $60 is on sale for 25% off. What is the sale price?

$15
$35
$45
$75

Solution: Discount = 25% of $60 = 0.25 × 60 = $15, Sale price = $60 - $15 = $45

Answer: $45

Question 4

A car travels at a speed of 55 miles per hour. How far will it travel in 3 hours and 20 minutes?

165 miles
176 miles
183.3 miles
198 miles

Solution: Time = 3h 20m = 3.33 hours, Distance = 55 × 3.33 ≈ 183.3 miles

Answer: 183.3 miles

Question 5

John can paint a fence in 5 hours, while Mary can paint the same fence in 3 hours. How long would it take them to paint the fence together?

1.5 hours
1.875 hours
2 hours
4 hours

Solution: John's rate: 1/5, Mary's rate: 1/3, Combined: 1/5 + 1/3 = 8/15, Time = 1 ÷ (8/15) = 15/8 = 1.875 hours

Answer: 1.875 hours

Test Results

0/5 (0%)

Score Interpretation:

  • 5/5: Excellent! Strong understanding
  • 4/5: Good job! Solid foundation
  • 3/5: On the right track, needs more study
  • Below 3: Needs focused study on weak areas

Review the explanations for each question to understand the correct answers.