📚 Intelligence Bureau (IB) – ACIO Quantitative Aptitude
📖 Topic: Ratio & Proportion
Ratio & proportion form the backbone of many quantitative questions in the IB ACIO exam. Mastering them will help you solve problems on relative magnitudes, scaling, percentages, and time‑speed‑distance type questions with confidence.
1️⃣ Simple Explanation
- Ratio = “How many parts of one quantity compared to another.”
[ \text{Ratio of } a \text{ to } b = \frac{a}{b} ] - Proportion = “When two ratios are equal.”
[ \frac{a}{b} = \frac{c}{d} ] - Cross‑multiplication is the quickest way to check or solve proportions.
[ a \times d = b \times c ]
Remember: Ratios can also be expressed as “a : b” or “a/b”. Proportions often involve “increasing/decreasing” relationships.
2️⃣ Key Points to Remember
| # | Concept | Key Idea | Quick Tip |
|---|---|---|---|
| 1 | Simplifying Ratios | Divide both terms by their greatest common divisor (GCD). | Use gcd(a, b) before cross‑multiplying. |
| 2 | Finding Missing Term | If a : b = c : ?, then ? = (b × c) / a. |
Always cross‑multiply: a × ? = b × c. |
| 3 | Scaling | When each part of a ratio is multiplied by the same factor, the ratio remains unchanged. | Useful for “increasing” or “decreasing” problems. |
| 4 | Proportion with Percentages | “x% of a” is x/100 × a. |
Convert % to fraction before working out. |
| 5 | Using Ratios for Time–Speed–Distance | speed : time : distance = constant. |
Often the ratio of speeds equals the ratio of distances if times are equal. |
| 6 | Checking for Integer Solutions | If a term in a proportion is not an integer, reconsider simplification or use “least common multiple” (LCM). | LCM helps avoid fractional intermediates. |
| 7 | Quick Mental Maths | Keep ratios in lowest terms; use common factors to cancel early. | Reduces chance of arithmetic errors. |
3️⃣ Worked Example
Question (Typical IB ACIO Format)
The ratio of boys to girls in a class is 3 : 5.
If the number of girls is increased by 20% and the number of boys is decreased by 10%, what is the new ratio of boys to girls?
Step‑by‑Step Solution
-
Assign Variables
Let original number of boys = (3x)
Let original number of girls = (5x) -
Apply Changes
- Boys decreased by 10% → (3x - 0.10(3x) = 3x(1 - 0.10) = 3x \times 0.90 = 2.7x)
- Girls increased by 20% → (5x + 0.20(5x) = 5x(1 + 0.20) = 5x \times 1.20 = 6x)
-
New Ratio
[ \text{New ratio} = \frac{2.7x}{6x} = \frac{2.7}{6} ] -
Simplify
Divide numerator and denominator by 0.3:
[ \frac{2.7}{6} = \frac{9}{20} ] So the new ratio of boys to girls is 9 : 20. -
Check (Cross‑multiplication)
(2.7 \times 20 = 54)
(6 \times 9 = 54) → ✔️
Answer: 9 : 20
4️⃣ Quick Revision Summary
| Topic | What to Check | Quick Recall |
|---|---|---|
| Ratio | Express in simplest form | a : b → divide by GCD |
| Proportion | Cross‑multiply to verify | a/b = c/d → a·d = b·c |
| Missing Term | a:b = c:? → ? = (b·c)/a |
|
| Scaling | Multiply all terms by same factor | Ratio unchanged |
| Percentages | Convert % to fraction first |
x% of a = (x/100)·a |
| Time‑Speed‑Distance | Use speed : time : distance |
|
| Mental Math | Cancel early, use LCM/GCD | |
| Common Pitfalls | Forget to simplify before solving | Simplify → Solve → Verify |
Final Tips for IB ACIO
- Always Simplify First: Reduces the chance of arithmetic errors.
- Cross‑Multiplication: The fastest way to solve most proportion problems.
- Unit Consistency: Keep units same (e.g., all in “kms”, “days”, “percent”).
- Practice Mixed Problems: Ratio + percentage + time‑speed‑distance often appear together.
- Time Management: Spend ≤ 2 minutes on ratio problems; if stuck, move on and return if time permits.
Good luck with your preparation! 🚀