Ratios, Proportions and Percentages

Ratios

A ratio compares two or more quantities. The ratio of $a$ to $b$ can be written as $a:b$, $\frac{a}{b}$, or "$a$ to $b$." All three notations mean the same thing.

Key idea: Ratios describe relative size, not actual size. If the ratio of boys to girls in a class is $3:5$, there could be 6 boys and 10 girls, or 30 boys and 50 girls, or 300 boys and 500 girls. The actual numbers are multiples of the ratio parts.

Finding actual values from a ratio and total:

If the ratio of $a$ to $b$ is $3:5$ and the total is 40:

• Total parts: $3 + 5 = 8$
• Value of $a$: $\frac{3}{8} \times 40 = 15$
• Value of $b$: $\frac{5}{8} \times 40 = 25$
• Check: $15 + 25 = 40$ ✓

Three-part ratios work the same way. If the ratio is $2:3:5$ and the total is 100:

• Total parts: $2 + 3 + 5 = 10$
• Each quantity: $\frac{2}{10} \times 100 = 20$, $\frac{3}{10} \times 100 = 30$, $\frac{5}{10} \times 100 = 50$

Proportions

A proportion is an equation stating that two ratios are equal:

$\frac{a}{b} = \frac{c}{d}$

To solve a proportion, cross-multiply: $a \cdot d = b \cdot c$. This is the single most important technique for proportion problems and works every time.

Example: If $\frac{3}{8} = \frac{x}{24}$, cross-multiply: $3 \times 24 = 8 \times x$, so $72 = 8x$ and $x = 9$.

Setting up proportions correctly: Make sure corresponding quantities are in the same position. If you write $\frac{\text{boys}}{\text{girls}}$ on one side, you must write $\frac{\text{boys}}{\text{girls}}$ on the other side — never mix the order.

Direct and Inverse Proportion

Direct proportion: As one quantity increases, the other increases at the same rate. The equation is $y = kx$ where $k$ is a constant. Doubling $x$ doubles $y$.

Example: If 3 pounds of apples cost \6$, then 9 pounds cost$$18$(same rate of$$2$ per pound).

Inverse proportion: As one quantity increases, the other decreases proportionally. The equation is $xy = k$ where $k$ is a constant. Doubling $x$ halves $y$.

Example: If 4 workers can complete a job in 6 hours, how long would 8 workers take? The total work is constant: $4 \times 6 = 24$ worker-hours. So $8 \times t = 24$, giving $t = 3$ hours. More workers means less time.

Identifying which type: Ask yourself: "If one quantity goes up, does the other go up (direct) or down (inverse)?" More speed → less time (inverse). More hours worked → more money earned (direct).

Percentages — Core Calculations

A percentage is a ratio expressed out of 100. The word "percent" literally means "per hundred."

Converting between forms:

• Fraction to percent: divide, then multiply by 100. $\frac{3}{8} = 0.375 = 37.5\%$
• Percent to decimal: divide by 100 (move decimal two places left). $45\% = 0.45$
• Decimal to percent: multiply by 100 (move decimal two places right). $0.72 = 72\%$

Three fundamental percentage calculations:

1. Finding a percentage of a number: "$x\%$ of $n$" means $\frac{x}{100} \times n$. Example: $30\%$ of $80 = 0.30 \times 80 = 24$.

2. Finding what percent one number is of another: "What percent is $a$ of $b$?" means $\frac{a}{b} \times 100\%$. Example: 15 is what percent of 60? $\frac{15}{60} \times 100 = 25\%$.

3. Finding the whole from a part and a percent: "25% of what number is 30?" means $0.25 \times n = 30$, so $n = \frac{30}{0.25} = 120$.

Percent Increase and Decrease

The percent change formula is one of the most commonly tested concepts on the ACT:

$\text{Percent Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100\%$

• If the result is positive, it is a percent increase.
• If negative, it is a percent decrease.

The critical detail: The denominator is always the original value — the value before the change occurred. This is where many students make errors. If a price goes from \50$to$$65$, the percent increase is$\frac{65-50}{50} \times 100 = 30%$, not$\frac{65-50}{65} \times 100 \approx 23%$.

Multiplier shortcut: To increase a number by $p\%$, multiply by $(1 + \frac{p}{100})$. To decrease by $p\%$, multiply by $(1 - \frac{p}{100})$.

Examples:

• $200$ increased by $15\% = 200 \times 1.15 = 230$
• $200$ decreased by $15\% = 200 \times 0.85 = 170$

Successive Percent Changes

When a quantity changes by $a\%$ and then by $b\%$, you cannot simply add the percentages. Instead, you must multiply the individual factors:

$\text{Final Value} = \text{Original} \times \left(1 + \frac{a}{100}\right) \times \left(1 + \frac{b}{100}\right)$

Classic ACT trap: A price increases by 20% and then decreases by 20%. Is the final price the same as the original?

$\text{Final} = P \times 1.20 \times 0.80 = 0.96P$

No! The price actually decreased by 4%. This is because the 20% decrease applies to the larger (increased) amount, so the decrease is bigger in absolute terms than the increase was.

Scale Factors, Maps, and Unit Conversion

Ratios are the foundation of scale problems and unit conversions:

Map scales: If a map scale is $1 \text{ inch} : 50 \text{ miles}$ and two cities are $3.5$ inches apart on the map, the actual distance is $3.5 \times 50 = 175$ miles.

Unit conversion: Convert 5 miles to feet. Since 1 mile = 5,280 feet: $5 \times 5{,}280 = 26{,}400$ feet. This is a proportion: $\frac{1 \text{ mile}}{5{,}280 \text{ feet}} = \frac{5 \text{ miles}}{x \text{ feet}}$.

Model/scale problems: A model car is built at a $1:24$ scale. If the model is 7.5 inches long, the actual car is $7.5 \times 24 = 180$ inches = 15 feet.

Tip 1: Use Cross-Multiplication for Every Proportion

Whenever you see two fractions set equal (or can set up a proportion from the problem), cross-multiply immediately. This is the fastest, most reliable method and virtually eliminates errors.

Tip 2: Translate Percent Problems Using Keywords

Use this pattern to convert English to math: "is" = $=$, "of" = $\times$, "what" = $x$, "percent" = $\div 100$. So "what is 40% of 250?" becomes $x = 0.40 \times 250 = 100$.

Tip 3: Use Multipliers Instead of Two-Step Calculations

Instead of computing the increase/decrease amount and then adding/subtracting, multiply directly by the appropriate factor. Increase of 25%? Multiply by $1.25$. Decrease of 10%? Multiply by $0.90$. This is faster and less error-prone.

Tip 4: Set Up Part-to-Whole Ratios Carefully

When a problem gives a ratio of parts (like $2:3:5$), the total number of parts is $2 + 3 + 5 = 10$. Each quantity is its fraction of the total. This approach handles money division, mixture problems, and group allocation.

Tip 5: Sanity-Check with Quick Estimation

Before finalizing your answer, estimate to verify reasonableness. If a \50$item is discounted by 30$$35$. If your answer is$$65$, something is wrong. Estimation catches errors that algebra alone misses.

Tip 6: Watch for the "Percent OF" vs. "Percent MORE" Distinction

"A is 30% of B" means $A = 0.30B$. "A is 30% more than B" means $A = 1.30B$. These are very different, and the ACT exploits this distinction.