**Ratio and Proportion — Equivalent Ratios**

Ratio and Proportion — Equivalent Ratios

Exam-Focused Study Guide (48 Hours to Mastery)

What Is This?

Equivalent ratios are two or more ratios that express the same relationship between numbers, even if the numbers themselves differ. For example, 2:3 and 4:6 are equivalent because they simplify to the same value.

Why it’s on your exam:
- Tests your ability to scale quantities up or down while keeping proportions intact.
- Appears in word problems (mixtures, maps, recipes), algebraic simplifications, and real-world applications (finance, engineering, medicine).
- Typically generates 3–5 mark questions asking you to: - Find missing values in equivalent ratios.
- Simplify ratios to their lowest terms.
- Compare ratios to determine which is larger/smaller.
- Solve proportion problems (e.g., "If 5 workers take 8 days, how long for 10 workers?").

Why It Matters

What the examiner is really testing:
- Can you spot the invariant relationship in a ratio? - Can you manipulate ratios without changing their meaning? - Can you apply ratios to real-world scenarios under time pressure?

Core Concepts

Master these 5 ideas before attempting any question:

1. Definition of a Ratio
2. A ratio compares two or more quantities of the same unit (e.g., apples to oranges, not apples to kilograms).

3. Written as a:b or a/b. The order matters: 3:5 ≠ 5:3.

4. Equivalent Ratios = Scaled Copies

5. Multiply or divide all parts of the ratio by the same non-zero number to get an equivalent ratio.

   
   
   • 2:3 → 4:6 (×2) → 6:9 (×3).
   • 10:15 → 2:3 (÷5).

6. Simplest Form

7. A ratio is in simplest form when the numbers have no common factors other than 1.

8. To simplify: Divide all parts by their greatest common divisor (GCD).

   
   
   • 18:24 → GCD is 6 → 3:4.

9. Proportion = Equality of Ratios

10. If a:b = c:d, then a/b = c/d. This is a proportion.

11. Cross-multiplication works: a × d = b × c.

12. Unit Ratio (1:n or n:1)

13. A ratio where one part is 1. Useful for comparisons.
    • 4:5 → 1:1.25 (divide both by 4).
    • 7:2 → 3.5:1 (divide both by 2).

Examiner’s Favorite Trick:
- Giving ratios with different units (e.g., km to m). Always convert to the same unit first!

The Rule-Book (How It Works)

Primary Rule: Scaling Ratios

To find equivalent ratios: 1. Multiply or divide all parts of the ratio by the same number.
- 3:4 → 6:8 (×2) → 9:12 (×3).
2. Never add or subtract—this breaks the ratio.

Sub-Rules & Exceptions

Mnemonic: "Same Scale, Same Shape"

• Think of ratios like zooming in/out on a photo. The shape (proportion) stays the same, only the size changes.

Exam / Job / Audit Weighting

• Frequency: 8/10 (appears in almost every exam with a math component).
• Difficulty Rating: Intermediate (easy if you know the rules; tricky if you rush).
• Question Type:
• MCQs (e.g., "Which ratio is equivalent to 3:7?").
• Short-answer (e.g., "Simplify 16:24").
• Word problems (e.g., "A recipe uses 2 cups of flour to 3 cups of sugar. How much sugar for 5 cups of flour?").
• Real-world tasks (e.g., "A map scale is 1:50,000. If two towns are 4cm apart on the map, how far are they in real life?").

Must-Know Rules, Formulas, Standards

1. Equivalent Ratio Formula
2. If a:b = c:d, then a × d = b × c (cross-multiplication).

3. Use this to find missing values.

4. Simplification Rule

5. Divide all parts by their GCD.

6. Example: 24:36 → GCD is 12 → 2:3.

7. Unit Ratio Rule

8. To compare ratios, convert them to 1:n or n:1.
9. Example: Compare 3:4 and 5:7 → 1:1.33 vs. 1:1.4 → 3:4 is smaller.

Worked Examples (Step-by-Step)

Example 1 (Easy): Find the Missing Value

Question:
If 5:8 = 15:x, find x.

Solution:
1. Recognize this is a proportion: 5/8 = 15/x.
2. Cross-multiply: 5 × x = 8 × 15.
3. Solve for x:
- 5x = 120
- x = 120/5 = 24.

Answer: x = 24.
Key Rule Applied: Cross-multiplication in proportions.

Example 2 (Medium): Simplify and Compare

Question:
Which ratio is larger: 18:24 or 15:20?

Solution:
1. Simplify both ratios:
- 18:24 → GCD is 6 → 3:4.
- 15:20 → GCD is 5 → 3:4.
2. Compare simplified forms: 3:4 = 3:4.
3. Conclusion: They are equal.

Answer: The ratios are equal.
Key Rule Applied: Simplification to compare ratios.

Examiner’s Trap:
- If you don’t simplify, you might think 18:24 is larger because 18 > 15. Always simplify first!

Example 3 (Hard): Word Problem with Units

Question:
A map scale is 1:25,000. If two villages are 6cm apart on the map, what is the real distance in kilometers?

Solution:
1. Understand the scale: 1cm on map = 25,000cm in real life.
2. Calculate real distance in cm:
- 6cm × 25,000 = 150,000cm.
3. Convert cm to km:
- 150,000cm ÷ 100,000 = 1.5km.

Answer: 1.5km.
Key Rule Applied: Scaling ratios and unit conversion.

Common Mistake:
- Forgetting to convert cm to km. Always check units!

Common Exam Traps & Mistakes

Shortcut Strategies & Exam Hacks

1. The "Magic Number" Trick

2. To find equivalent ratios, divide the larger term by the smaller term to find the scaling factor.

   
   
   • 3:4 → 15:x → 15 ÷ 3 = 5 → x = 4 × 5 = 20.

3. Quick Simplification

4. If both numbers are even, divide by 2 repeatedly until odd.

5. If one number ends with 0 or 5, divide by 5.

   
   
   • 35:50 → ÷5 → 7:10.

6. Unit Ratio for Comparison

7. Convert ratios to 1:n to compare easily.

   
   
   • 5:8 → 1:1.6
   • 7:10 → 1:1.43 → 5:8 is larger.

8. Eliminate Impossible Options

9. In MCQs, cross out options that don’t simplify to the same ratio.

   
   
   • Question: Which is equivalent to 2:3?
   • A) 4:5 → No (simplifies to 4:5).
   • B) 6:9 → Yes (simplifies to 2:3).

10. Check for Hidden Units

11. If a question mentions km and m, hours and minutes, or dollars and cents, convert first.

Question-Type Taxonomy

Practice Set (MCQs)

Question 1

Which ratio is equivalent to 4:9? A) 8:16 B) 12:27 C) 16:32 D) 20:40

Correct Answer: B) 12:27 Explanation:
- 4:9 scaled by 3 → 12:27.
- A) 8:16 simplifies to 1:2 (not 4:9).
- C) 16:32 simplifies to 1:2.
- D) 20:40 simplifies to 1:2.
Why the Distractors Are Tempting:
- A, C, D simplify to 1:2, which looks "clean" but isn’t equivalent to 4:9.

Question 2

Simplify 36:48 to its lowest terms. A) 3:4 B) 6:8 C) 9:12 D) 18:24

Correct Answer: A) 3:4 Explanation:
- GCD of 36 and 48 is 12.
- 36 ÷ 12 = 3, 48 ÷ 12 = 4 → 3:4.
Why the Distractors Are Tempting:
- B, C, D are partially simplified but not in lowest terms.

Question 3

If 5 workers take 9 days to complete a job, how many days will 15 workers take? A) 3 B) 5 C) 15 D) 27

Correct Answer: A) 3 Explanation:
- Workers and days are inversely proportional (more workers = fewer days).
- 5 workers × 9 days = 15 workers × x days → 45 = 15x → x = 3.
Why the Distractors Are Tempting:
- B) 5 assumes direct proportion (incorrect).
- D) 27 is 9 × 3 (wrong direction for inverse proportion).

Question 4

A recipe uses 2 cups of flour to 3 cups of sugar. How much sugar is needed for 5 cups of flour? A) 6 B) 7.5 C) 8 D) 9

Correct Answer: B) 7.5 Explanation:
- Ratio flour:sugar = 2:3.
- For 5 cups flour: 2/3 = 5/x → 2x = 15 → x = 7.5.
Why the Distractors Are Tempting:
- A) 6 assumes 2:3 = 5:6 (wrong scaling).
- D) 9 is 3 × 3 (ignores the flour amount).

Question 5

Which ratio is the largest? A) 3:5 B) 7:10 C) 11:15 D) 13:20

Correct Answer: B) 7:10 Explanation:
- Convert to unit ratios: - A) 3:5 → 1:1.67 - B) 7:10 → 1:1.43 - C) 11:15 → 1:1.36 - D) 13:20 → 1:1.54 - 1:1.43 is the largest (smallest denominator after 1).
Why the Distractors Are Tempting:
- A) 3:5 looks "clean" but is smaller than 7:10.
- D) 13:20 has a larger numerator but isn’t the largest ratio.

30-Second Cheat Sheet

1. Equivalent ratios = same relationship, scaled up/down.
2. Multiply or divide all parts by the same number—never add/subtract.
3. Simplify fully (divide by GCD) before comparing.
4. Cross-multiply to solve proportions: a/b = c/d → a × d = b × c.
5. Convert units first if they don’t match.
6. Order matters: 3:4 ≠ 4:3.
7. Inverse proportion? More workers = fewer days (multiply, don’t divide).

Learning Path

1. Day 1 (0–12 hours): Foundation
2. Learn the definition of ratios and equivalent ratios.
3. Practice simplifying ratios (10 examples).

4. Master cross-multiplication (5 examples).

5. Day 1 (12–24 hours): Core Rules

6. Work on finding missing values (10 examples).
7. Solve comparison problems (5 examples).

8. Learn unit conversion in ratios (5 examples).

9. Day 2 (24–36 hours): Application

10. Tackle word problems (10 examples).
11. Practice map/scale problems (5 examples).

12. Learn inverse proportion (5 examples).

13. Day 2 (36–48 hours): Exam Drills

14. Timed MCQs (20 questions in 30 minutes).
15. Mock exam (10 mixed questions in 20 minutes).
16. Review common traps and redo mistakes.

Related Topics

1. Direct and Inverse Proportion

2. How ratios change when one quantity affects another (e.g., speed and time).

3. Percentage and Ratio Conversion

4. Converting ratios to percentages (e.g., 3:5 → 60%).

5. Algebraic Ratios

6. Solving equations with ratios (e.g., x:y = 2:3 and x + y = 10).