Mastering fractions, decimals, and percentages is essential for success on the ASVAB Math Section. These topics frequently appear in Mathematics Knowledge and Arithmetic Reasoning questions. Below is a well-organized study guide with key concepts, formulas, and examples to help you strengthen your math skills.
Fractions
Multiplying Fractions
Rule: Multiply the numerators and multiply the denominators.
Formula:
ab×cd=a×cb×d\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}ba×dc=b×da×c
Example:
23×45=2×43×5=815\frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}32×54=3×52×4=158
Dividing Fractions (Flip It & Multiply)
Rule: Flip the second fraction (reciprocal) and multiply.
Example:
34÷25=34×52=3×54×2=158=178\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{3 \times 5}{4 \times 2} = \frac{15}{8} = 1 \frac{7}{8}43÷52=43×25=4×23×5=815=187
Adding & Subtracting Fractions
With a Common Denominator
Example:
38+58=3+58=88=1\frac{3}{8} + \frac{5}{8} = \frac{3+5}{8} = \frac{8}{8} = 183+85=83+5=88=1
With Different Denominators
Steps:
- Find the Least Common Denominator (LCD).
- Convert fractions to have the same denominator.
- Perform the operation.
Example:
14+16\frac{1}{4} + \frac{1}{6}41+61
LCD = 12
14=312,16=212\frac{1}{4} = \frac{3}{12}, \quad \frac{1}{6} = \frac{2}{12}41=123,61=122 312+212=512\frac{3}{12} + \frac{2}{12} = \frac{5}{12}123+122=125
Finding Common Denominators for Three or More Fractions
To compare or add multiple fractions, find the Least Common Denominator (LCD) and convert each fraction.
Example:
13,25,34\frac{1}{3}, \quad \frac{2}{5}, \quad \frac{3}{4}31,52,43
LCD = 60
13=2060,25=2460,34=4560\frac{1}{3} = \frac{20}{60}, \quad \frac{2}{5} = \frac{24}{60}, \quad \frac{3}{4} = \frac{45}{60}31=6020,52=6024,43=6045
Reducing (Simplifying) Fractions
Rule: Divide both numerator and denominator by their Greatest Common Factor (GCF).
Example:
1824(GCF = 6)\frac{18}{24} \quad \text{(GCF = 6)}2418(GCF = 6) 18÷624÷6=34\frac{18 \div 6}{24 \div 6} = \frac{3}{4}24÷618÷6=43
Mixed Numbers & Improper Fractions
Convert Improper to Mixed
Example:
114=234(11 ÷ 4 = 2 remainder 3)\frac{11}{4} = 2 \frac{3}{4} \quad \text{(11 ÷ 4 = 2 remainder 3)}411=243(11 ÷ 4 = 2 remainder 3)
Convert Mixed to Improper
Example:
234=(2×4)+34=1142 \frac{3}{4} = \frac{(2 \times 4) + 3}{4} = \frac{11}{4}243=4(2×4)+3=411
Decimals
Multiplying Decimals
Ignore decimals while multiplying, then place the decimal point correctly.
Example:
4.2×3.54.2 \times 3.54.2×3.5
Multiply normally: 42 × 35 = 1470
Count two decimal places: 14.70
Dividing Decimals
Move the decimal in the divisor until it’s a whole number, then move it the same amount in the dividend.
Example:
4.8÷0.44.8 \div 0.44.8÷0.4
Move decimal → 48 ÷ 4 = 12
Adding & Subtracting Decimals
Align decimal points before adding or subtracting.
Example:
7.50
+ 3.27
——-
10.77
Percentages
Converting Between Percentages & Decimals
Move decimal two places left for percent → decimal.
Move decimal two places right for decimal → percent.
Examples:
75%=0.75,0.42=42%75\% = 0.75, \quad 0.42 = 42\%75%=0.75,0.42=42%
Percentage Calculations
Formula:
Percentage×Number\text{Percentage} \times \text{Number}Percentage×Number
Example:
30% of 80=0.30×80=2430\% \text{ of } 80 = 0.30 \times 80 = 2430% of 80=0.30×80=24
Real-World Example (Discount Calculation):
A $60 item is on 20% discount.
0.20×60=120.20 \times 60 = 120.20×60=12
New price: $60 – $12 = $48
Ratios & Proportions
Understanding Ratios
Ratios compare two quantities (e.g., 3:2 or 3/2).
Example:
A class has 15 boys and 10 girls.
Ratio of boys to girls:
15:10=3215:10 = \frac{3}{2}15:10=23
Solving Proportions
📌 Cross-multiply to solve for an unknown in a proportion.
✅ Example:
46=x9\frac{4}{6} = \frac{x}{9}64=9x
Cross multiply:
4×9=6x⇒36=6×4 \times 9 = 6x \quad \Rightarrow \quad 36 = 6×4×9=6x⇒36=6x
Solve for x:
x=6x = 6x=6
Rates & Speed Calculations
Speed Formula
Formula:
Speed=DistanceTime\text{Speed} = \frac{\text{Distance}}{\text{Time}}Speed=TimeDistance
Example:
A car travels 120 miles in 3 hours. Find its speed.
1203=40 mph\frac{120}{3} = 40 \text{ mph}3120=40 mph
Final Review
- Multiply/Divide Fractions using “Flip & Multiply” for division
- Convert between Mixed & Improper Fractions
- Find Common Denominators before Adding/Subtracting
- Multiply/Divide Decimals by counting decimal places
- Move decimal two places for Percentage-Decimal conversions
- Ratios compare quantities; Proportions solve for unknowns
- Use the Speed Formula: Distance ÷ Time