2.2 Fractions, Decimals & Percents

Operations with Fractions

A fraction names a part of a whole; the numerator is on top and the denominator on the bottom. To add or subtract fractions you need a common denominator, ideally the least common denominator (LCD) — the LCM of the denominators.

Worked Example 1. Compute 3/4 + 1/6.

1. The LCD of 4 and 6 is 12.
2. Rewrite: 3/4 = 9/12 and 1/6 = 2/12.
3. Add numerators: 9/12 + 2/12 = 11/12.

To multiply, multiply numerators and denominators straight across, then simplify: 2/3 × 9/10 = 18/30 = 3/5. To divide, use keep-change-flip — keep the first fraction, change ÷ to ×, and flip (take the reciprocal of) the second.

Worked Example 2. Compute 5/8 ÷ 3/4. Keep 5/8, change to ×, flip 3/4 to 4/3: 5/8 × 4/3 = 20/24 = 5/6.

Converting Among Fractions, Decimals, and Percents

These three forms describe the same value, and ALEKS expects fast conversion in every direction.

• Fraction → decimal: divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375.
• Decimal → percent: multiply by 100 (move the point two places right). 0.375 → 37.5%.
• Percent → decimal: divide by 100 (move the point two places left). 8% → 0.08.
• Percent → fraction: write over 100 and simplify. 60% = 60/100 = 3/5.

Worked Example 3. Write 0.6 as a fraction in lowest terms. 0.6 = 6/10, and dividing numerator and denominator by 2 gives 3/5.

The Percent Equation

Most percent questions reduce to one relationship:

part = percent (as a decimal) × whole

Identify which quantity is missing and solve.

Percent of a number (find the part). What is 35% of 80? part = 0.35 × 80 = 28.

Find the whole. 18 is 30% of what number? Here part = 18 and percent = 0.30, so whole = part ÷ percent = 18 ÷ 0.30 = 60.

Find the percent. 42 is what percent of 56? percent = part ÷ whole = 42 ÷ 56 = 0.75 = 75%.

A reliable habit is to translate the English directly: "is" becomes =, "of" becomes ×, and "what" becomes your variable. So "15 is what percent of 60" becomes 15 = p × 60, giving p = 0.25 = 25%.

Before multiplying, always rewrite the percent as a decimal or fraction; mixing a raw percent into the equation is the single most common error. And when an answer choice looks strange, estimate: 35% of 80 should be just over a third of 80, so an answer near 28 is reasonable while 280 or 2.8 clearly is not.

Percent Increase and Decrease

Percent change measures how much a quantity grew or shrank relative to its original value:

percent change = (new − old) ÷ old × 100

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

Worked Example 4 (percent increase). A jacket's price rises from $40 to $50. Change = 50 − 40 = 10. Percent change = 10 ÷ 40 = 0.25 = 25% increase.

Worked Example 5 (percent decrease). A laptop is marked down from $800 to $680. Change = 680 − 800 = −120. Percent change = −120 ÷ 800 = −0.15 = 15% decrease. Equivalently, a 15%-off sale price is 85% of the original: 0.85 × 800 = 680.

Common Mistakes

• Dividing by the new value: percent change always divides by the old (original) amount.
• Forgetting to convert: 25% must become 0.25 before multiplying in the percent equation.
• Adding instead of finding a common denominator: 1/2 + 1/3 is 5/6, not 2/5 — never add numerators and denominators straight across.

Mixed Numbers and Multi-Step Percent Problems

A mixed number like 2 3/4 combines a whole number and a fraction. To compute with mixed numbers, first convert to an improper fraction: multiply the whole number by the denominator and add the numerator. So 2 3/4 = (2 × 4 + 3)/4 = 11/4. Convert back at the end by dividing: 11/4 = 2 remainder 3, which is 2 3/4.

Worked Example 6. Compute 2 1/2 × 1 1/3. Convert: 2 1/2 = 5/2 and 1 1/3 = 4/3. Multiply across: 5/2 × 4/3 = 20/6 = 10/3 = 3 1/3.

Real placement problems often chain percents together. Treat each step as its own percent equation and carry the result forward.

Worked Example 7 (tip and tax). A meal costs $40. Add 8% sales tax, then a 20% tip on the original price. Tax = 0.08 × 40 = $3.20. Tip = 0.20 × 40 = $8.00. Total = 40 + 3.20 + 8.00 = $51.20.

Worked Example 8 (successive discounts). A $200 coat is 25% off, then an extra 10% off the reduced price. After 25% off: 0.75 × 200 = $150. After 10% off $150: 0.90 × 150 = $135. Notice the two discounts are not simply 35% off ($130) — each percent applies to a different base, a trap ALEKS loves to set.