Ratio & Proportion – Quantitative Aptitude (RHCE)
Why it matters
In the RHCE exam you’ll often encounter questions that involve comparing two or more quantities, scaling numbers, or solving for an unknown that keeps a relationship constant. Ratio and proportion are the mathematical language of such problems.
1. Simple Explanation
| Term | What it means | Example |
|---|---|---|
| Ratio | A comparison of two numbers as a fraction. | 3 : 4 = 3/4 |
| Proportion | An equation that says two ratios are equal. | a:b = c:d |
| Direct proportion | As one quantity increases, the other increases proportionally. | y = kx |
| Inverse proportion | As one quantity increases, the other decreases proportionally. | y = k/x |
Key Idea:
If two ratios are equal, the corresponding cross‑products are equal.
2. Key Points & Formulae
| Concept | Formula / Method | When to Use |
|---|---|---|
| Cross‑Multiplication | ( \frac{a}{b} = \frac{c}{d} ; \Rightarrow ; a \times d = b \times c ) | Solving proportions |
| Direct Proportion | ( \frac{y}{x} = \frac{m}{n} \Rightarrow y = x \times \frac{m}{n} ) | “If 5 apples cost $10, how much 8 apples cost?” |
| Inverse Proportion | ( x \times y = k ) | “If 4 workers finish a job in 6 days, how many days for 6 workers?” |
| Ratio Simplification | Divide both parts by GCD | 12 : 18 → 2 : 3 |
| Percentage of a Number | ( \text{Part} = \frac{\text{Ratio part}}{\text{Sum of ratio parts}} \times 100% ) | “What percent is 3 in 3:5?” |
| Unit Rate | ( \text{Unit rate} = \frac{\text{Total quantity}}{\text{Total unit}} ) | “$240 for 60 km → $4 per km.” |
3. Worked Example
Problem
A computer lab has 5 teachers and 15 students. If the ratio of teachers to students is kept the same, how many students will there be when the number of teachers increases to 8?
Step‑by‑Step Solution
-
Identify the given ratio.
Teachers : Students = 5 : 15
Simplify → 1 : 3 (divide by 5) -
Set up the proportion.
[ \frac{8}{x} = \frac{1}{3} ] where (x) = number of students for 8 teachers. -
Cross‑multiply.
[ 8 \times 3 = 1 \times x ;\Rightarrow; 24 = x ] -
Answer
[ \boxed{24\ \text{students}} ]
Quick Check
- Ratio 8 : 24 simplifies to 1 : 3 → matches original ratio. ✔️
4. Quick Revision Summary
| Topic | Quick Tip |
|---|---|
| Ratio | Think of it as a “part‑of‑whole” comparison. |
| Proportion | Two ratios equal → cross‑products equal. |
| Direct Proportion | Multiply the known value by the same factor. |
| Inverse Proportion | Keep the product constant. |
| Simplify Ratios | Reduce using GCD to avoid large numbers. |
| Common Tricks | • Convert percentages to ratios (e.g., 25 % → 1 : 4). • Use unit rates to convert between units. |
Mini‑Practice Problems (Try before checking)
- If 4 cars travel 200 km in 2 hours, how long will 6 cars travel the same distance?
- A recipe calls for 3 kg flour and 1 kg sugar. If you want to make 10 kg of dough, how much sugar is needed?
- The ratio of boys to girls in a class is 5 : 7. If there are 48 girls, how many boys are there?
Answers
- 5 hours (inverse proportion).
- 3.33 kg sugar (direct proportion).
- 30 boys (cross‑multiplication).
5. Final Takeaway
- Always check if the ratio can be simplified before setting up the proportion.
- Use cross‑multiplication to solve for unknowns quickly.
- Remember the difference between direct and inverse proportions; one keeps the ratio constant, the other keeps the product constant.
Master these tools, and solving ratio‑based quantitative problems will become quick and error‑free!