ACT Math: Ratios, Proportions, and Percents

Mastering ratios, proportions, and percents is non-negotiable for a strong ACT Math score. These concepts form the backbone of a significant number of problems, from straightforward calculations to complex, multi-step word problems. Your ability to quickly and accurately set up relationships between quantities will save you precious time and unlock points across the entire section. This guide will build your understanding from the ground up, ensuring you can tackle any variation the test throws at you.

Understanding Ratios as Comparative Relationships

A ratio is a comparative relationship between two or more quantities, expressing how much of one thing exists relative to another. Ratios can be written in three equivalent forms: using the word "to" (3 to 4), as a fraction ($\frac{3}{4}$), or with a colon (3:4). On the ACT, you must be comfortable manipulating ratios in all these forms.

A critical skill is handling part-to-part versus part-to-whole ratios. If a mixture has a sugar-to-flour ratio of 1:3, that is a part-to-part ratio. The part-to-whole ratio of sugar to the total mixture is 1:4 (because 1 part sugar + 3 parts flour = 4 total parts). Confusing these two is a common trap. For example, if the ratio of boys to girls in a class is 2:3, the ratio of boys to the total class is 2:5. You will often use a ratio multiplier, $k$, to convert ratios into actual quantities. If the ratio of apples to oranges is 5:2, then the number of apples is $5k$ and oranges is $2k$, where $k$ is the multiplier.

Setting Up and Solving Proportions

A proportion is a statement that two ratios are equal. It is your primary tool for solving problems where quantities maintain a constant relationship. The fundamental setup is $\frac{a}{b}=\frac{c}{d}$, which can be solved by cross-multiplication: $a\cdot d=b\cdot c$.

Your main task on the ACT is correctly identifying proportional situations from word problems. A classic example involves scale models: "On a map, 1 inch represents 20 miles. If two towns are 3.5 inches apart on the map, what is the actual distance?" You set up the proportion by keeping consistent units in the same position: $\frac{1\text{ inch}}{20\text{ miles}}=\frac{3.5\text{ inches}}{x\text{ miles}}$. Cross-multiplying gives $1\cdot x=20\cdot 3.5$, so $x=70$ miles. Always check that your units align horizontally or vertically in your fraction setup to avoid errors.

Calculating Percents, Change, and Applications

A percent is a ratio whose denominator is 100. The most fundamental equation is: $\text{Percent}=\frac{\text{Part}}{\text{Whole}}\cdot 100\%$. More commonly, you'll use its algebraic forms: $\text{Part}=\frac{\text{Percent}}{100}\cdot \text{Whole}$ or $\text{Whole}=\frac{\text{Part}}{\text{Percent}/100}$.

Percent change is a frequent ACT topic. The formula is: $\text{Percent Change}=\frac{\text{New Value}-\text{Original Value}}{\text{Original Value}}\cdot 100\%$. A positive result is an increase; a negative result is a decrease. Crucially, the denominator is always the original value. For successive percent changes, you cannot simply add the percentages. A price increased by 20% and then decreased by 20% does not return to the original price. You must multiply by the respective multipliers: New Price = Original Price $\cdot (1.20)\cdot (0.80)=\text{Original Price}\cdot 0.96$, a net 4% decrease.

Solving Multi-Step Problems: Mixtures, Rates, and Finance

The ACT often combines these concepts into layered problems. Your strategy is to break them down into manageable steps.

• Mixtures: For problems combining elements (like two types of nuts or acid solutions), define variables for the amounts of each component. The key equation often involves the total amount of a substance. For example: "How many liters of a 20% saline solution must be added to 10 liters of a 50% saline solution to create a 30% solution?" Let $x$ be the liters of 20% solution. The amount of pure salt from the first is $0.20x$, from the second is $0.50(10)=5$, and in the final mixture is $0.30(10+x)$. The equation is: $0.20x+5=0.30(10+x)$. Solve for $x$.
• Rates: A rate is a ratio comparing two different units (e.g., miles per hour). The core formula is $\text{Rate}\cdot \text{Time}=\text{Distance}$ (or $\text{Rate}\cdot \text{Time}=\text{Work}$). For combined rates, like two people working together, add their individual rates.
• Financial Applications: These are percent applications in disguise.
• Discount/Markup: Sale Price = Original Price $\cdot (1-\text{Discount Percent})$. Markup Price = Wholesale Price $\cdot (1+\text{Markup Percent})$.
• Commission/Sales Tax: These are calculated as a percent of a sale amount and are typically added to (commission for a worker) or added onto (tax for a buyer) a base amount.
• Simple Interest: Use the formula $I=P\cdot r\cdot t$, where $I$ is interest, $P$ is principal, $r$ is the annual rate (as a decimal), and $t$ is time in years.

Common Pitfalls

1. Misreading Ratios as Numbers: Treating a ratio like 3:4 as the fraction $\frac{3}{4}$ is correct, but forgetting it represents a relationship, not necessarily the quantities themselves. If the ratio of blue to red marbles is 3:4, there could be 6 blue and 8 red marbles (using a multiplier of 2).
2. Inverting Proportions: Setting up a proportion backwards is easy. Ask yourself: "As quantity A increases, does quantity B increase or decrease?" If it increases (direct variation), keep fractions oriented the same way. If it decreases (inverse variation), you must invert one ratio.
3. Percent Change vs. Percentage Point Change: An increase from 10% to 15% is a 5 percentage point increase, but a 50% percent change because $\frac{15-10}{10}\cdot 100\%=50\%$. The ACT will test if you know the difference.
4. Forgetting the Distributive Property in Mixtures: In the saline solution example, a common error is writing $0.30(10)+x$ instead of $0.30(10+x)$ on the right side of the equation. The percent applies to the entire final volume.

Summary

• Ratios compare parts to parts or parts to a whole. Use a multiplier $k$ to convert ratio relationships into workable numbers.
• Proportions, equations of equal ratios, are solved via cross-multiplication. Consistently align corresponding units when setting them up.
• Master the core percent equation and its forms. Percent change is always calculated relative to the original value.
• Tackle complex word problems by defining variables and writing equations step-by-step, especially for mixtures (balance the "pure" substance), rates (use $R\cdot T=D$), and financial applications (apply multipliers like $(1+\text{percent})$).
• Stay vigilant for common traps like inverting proportions, confusing percent with percentage points, and misapplying percents in multi-step contexts.