1.1 Arithmetic, Fractions, Decimals & Percentages

Arithmetic is the foundation of every NAPT math question, and it resurfaces constantly once you reach algebra, physics, and chemistry calculations later in this guide. Fractions, decimals, and percentages are not an isolated topic — they are the toolkit you use to simplify equations, interpret circuit and reactor values, and solve percent-based word problems under time pressure. Getting comfortable with these mechanics now pays off in every chapter that follows.

Working with Fractions

A fraction expresses a part of a whole using a numerator (top number) and a denominator (bottom number). Before you can add or subtract fractions, the denominators must match — this shared denominator is called the common denominator, and the smallest one that works is the least common denominator (LCD).

Adding and Subtracting Fractions

To add or subtract fractions with different denominators:

1. Find the LCD of the denominators.
2. Convert each fraction to an equivalent fraction with that LCD.
3. Add or subtract the numerators and keep the denominator.
4. Simplify the result if possible.

Worked example: Add 3/8 + 5/12.

The denominators are 8 and 12. The LCD is 24 (8 × 3 = 24, and 12 × 2 = 24).

• 3/8 = 9/24 (multiply top and bottom by 3)
• 5/12 = 10/24 (multiply top and bottom by 2)
• 9/24 + 10/24 = 19/24

19/24 is already in lowest terms, so that is the final answer.

Multiplying Fractions

Multiplying fractions is more direct than adding them — there is no need for a common denominator. Multiply the numerators together, multiply the denominators together, then simplify.

Worked example: Multiply 2/3 × 5/6.

• Numerators: 2 × 5 = 10
• Denominators: 3 × 6 = 18
• 10/18 simplifies to 5/9 (divide top and bottom by their greatest common factor, 2)

Dividing Fractions

To divide by a fraction, use the "keep, change, flip" rule: keep the first fraction as-is, change the division sign to multiplication, and flip (take the reciprocal of) the second fraction.

Worked example: Divide 3/4 ÷ 2/5.

• Keep 3/4, change ÷ to ×, flip 2/5 to 5/2
• 3/4 × 5/2 = 15/8
• 15/8 converts to the mixed number 1 7/8

Mixed Numbers and Improper Fractions

A mixed number combines a whole number and a fraction (such as 3 1/4). An improper fraction has a numerator larger than its denominator (such as 13/4) and represents the exact same value. You will convert between the two forms constantly when adding, multiplying, or dividing fractions that started out as mixed numbers.

To convert a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and keep the same denominator.

Worked example: Convert 3 1/4 to an improper fraction.

• Multiply: 3 × 4 = 12
• Add the numerator: 12 + 1 = 13
• Keep the denominator: 13/4

To convert an improper fraction back to a mixed number: divide the numerator by the denominator; the whole-number quotient becomes the whole number, and the remainder becomes the new numerator over the same denominator. This is exactly how 15/8 became 1 7/8 above (15 ÷ 8 = 1, remainder 7).

---

Decimals and Percentages

A decimal and a percent are just alternate ways of writing a fraction. A percent literally means "per hundred" — 62.5% means 62.5 out of every 100. Converting between the three forms is a skill you will use on nearly every NAPT arithmetic and word-problem item.

To convert a fraction to a decimal: divide the numerator by the denominator (7/8 = 7 ÷ 8 = 0.875).

To convert a decimal to a percent: multiply by 100 and add the % sign (0.875 × 100 = 87.5%).

To convert a percent to a decimal: divide by 100 (87.5% = 0.875).

Percent Change

Percent-change problems ask how much a value increased or decreased relative to its starting point. The formula is:

Percent change = [(New Value − Original Value) ÷ Original Value] × 100%

A positive result is a percent increase; a negative result is a percent decrease.

Worked example (decrease): A ship's fuel tank holds 4,000 gallons before a voyage. After the voyage, only 3,400 gallons remain. What is the percent decrease?

• Change = 3,400 − 4,000 = −600 gallons
• Percent change = (−600 ÷ 4,000) × 100% = −15%
• The tank's fuel decreased by 15%.

Worked example (increase): A component's rated resistance is upgraded from 120 ohms to 150 ohms. What is the percent increase?

• Change = 150 − 120 = 30 ohms
• Percent change = (30 ÷ 120) × 100% = 25%
• The resistance increased by 25%.

Common Traps

• Always divide by the original value, not the new value, when computing percent change — dividing by the new value gives a different (wrong) percentage.
• "Percent of" problems (e.g., "what is 15% of 80?") multiply directly: 0.15 × 80 = 12. Do not confuse a "percent of" question, which asks for a portion of one single value, with a percent-change question, which always compares an original value to a new value.
• When adding fractions, never add denominators together — only numerators, and only once the denominators already match.