How to Solve: ACT Math – Ratios, Proportions, and Rate Problems

How to Solve: ACT Math – Ratios, Proportions, and Rate Problems

Introduction

"Mastering ratios, proportions, and rates unlocks 5–7 questions on the ACT Math section—enough to boost your score by 3–5 points and save you from costly careless errors on test day."

What You Need To Know First

1. Fractions: You must be comfortable simplifying, multiplying, and dividing fractions.
2. Unit Conversion: Know how to convert between units (e.g., hours to minutes, miles to feet).
3. Basic Algebra: Solve for one variable in an equation (e.g., 3x = 12 → x = 4).

Key Vocabulary

Formulas To Know

1. Proportion Formula
2. Formula: a/b = c/d
3. Variables:
   • a, b, c, d = quantities in the ratio.

4. Memorise This.: Used to solve for an unknown in a proportion.

5. Rate Formula

6. Formula: Rate = Distance / Time or Rate = Work / Time
7. Variables:
   • Distance = total distance traveled.
   • Time = time taken.
   • Work = amount of work done.

8. Memorise This.: The foundation for all rate problems.

9. Unit Rate Formula

10. Formula: Unit Rate = Total Quantity / 1 Unit of Time
11. Example: If you drive 150 miles in 3 hours, the unit rate is 150 miles / 3 hours = 50 mph.

12. Memorise This.: Always convert rates to "per 1 unit" for comparisons.

13. Scaling Ratios

14. Formula: New Quantity = Original Quantity × Scaling Factor
15. Example: If a recipe uses 2 cups of flour for 4 servings, scaling to 10 servings: 2 × (10/4) = 5 cups.
16. Memorise This.: Used when increasing or decreasing quantities proportionally.

Step-by-Step Method

Step 1: Identify the Type of Problem

• Ratio Problem? → Compare parts of a whole (e.g., boys to girls).
• Proportion Problem? → Two ratios set equal (e.g., 3/4 = x/8).
• Rate Problem? → Involves time, distance, or work (e.g., speed, wages).

Step 2: Write Down What You Know

• List all given numbers and units.
• Label what you’re solving for (e.g., x = miles per hour).

Step 3: Set Up the Equation

• For ratios: Write as a:b or a/b.
• For proportions: Set two ratios equal (a/b = c/d).
• For rates: Use Rate = Distance / Time or Work / Time.

Step 4: Solve for the Unknown

• Proportions: Cross-multiply and solve for x.
• Rates: Isolate the variable (e.g., x = Distance / Time).
• Ratios: Scale up or down to match given quantities.

Step 5: Check Units and Reasonableness

• Do the units match? (e.g., miles per hour, not miles per minute).
• Does the answer make sense? (e.g., 50 mph is reasonable; 500 mph is not).

Step 6: Box Your Answer

• Circle or box the final answer to avoid mistakes.

Worked Examples

Example 1 – Basic Ratio Problem

Problem: The ratio of cats to dogs in a shelter is 3:5. If there are 15 cats, how many dogs are there?

Step 1: Identify the ratio (cats:dogs = 3:5). Step 2: Write the proportion: 3/5 = 15/x. Step 3: Cross-multiply: 3x = 5 × 15 → 3x = 75. Step 4: Solve for x: x = 75 / 3 = 25. Step 5: Check: 15 cats / 25 dogs = 3/5 (matches the ratio). Answer: 25 dogs.

What we did and why: We used the given ratio to set up a proportion, cross-multiplied, and solved for the unknown. This ensures the ratio stays consistent.

Example 2 – Medium Rate Problem

Problem: A car travels 240 miles in 4 hours. At the same speed, how far will it travel in 7 hours?

Step 1: Find the rate (speed). - Rate = Distance / Time = 240 miles / 4 hours = 60 mph. Step 2: Use the rate to find new distance. - Distance = Rate × Time = 60 mph × 7 hours = 420 miles. Step 3: Check units (miles) and reasonableness (420 miles in 7 hours is faster than 240 in 4, but the speed is constant). Answer: 420 miles.

What we did and why: We first found the unit rate (speed), then used it to calculate the new distance. This avoids setting up a proportion incorrectly.

Example 3 – Exam-Style Proportion Problem

Problem: A map uses a scale of 1 inch = 50 miles. If two cities are 3.5 inches apart on the map, what is the actual distance between them?

Step 1: Write the proportion: 1 inch / 50 miles = 3.5 inches / x miles. Step 2: Cross-multiply: 1 × x = 50 × 3.5 → x = 175. Step 3: Check units (miles) and reasonableness (3.5 inches is larger than 1 inch, so 175 miles > 50 miles). Answer: 175 miles.

What we did and why: We treated the scale as a proportion and solved for the unknown distance. Cross-multiplication ensures accuracy.

Common Mistakes

Exam Traps

1-Minute Recap

"Here’s your 60-second crash course for ACT ratios, proportions, and rates: 1. Ratios compare parts—write them as a:b or a/b. 2. Proportions set two ratios equal—cross-multiply to solve. 3. Rates are ratios with units (like mph)—always find the unit rate first. 4. Check units—if the answer is in hours but the question asks for minutes, convert! 5. Avoid traps—watch for hidden unit changes and part-to-whole questions. You’ve got this. Now go crush those 5–7 questions on test day!