Ratio & Proportion – Study Notes
(For the Jammu & Kashmir Patwari / Lekhpal Quantitative Aptitude Paper)
1. Simple Explanation
| Concept | What it is | How it looks | Example |
|---|---|---|---|
| Ratio | A comparison of two quantities by division. | a : b or a/b |
Lengths of two roads 12 km : 18 km = 2 : 3 |
| Proportion | An equation showing that two ratios are equal. | a : b = c : d or a/b = c/d |
If 5 apples cost ₹15, 8 apples cost ₹? → 5:8 = 15:?. |
2. Key Points
| # | Topic | Important Facts | Quick Formula |
|---|---|---|---|
| 1 | Simplifying a Ratio | Divide both terms by their greatest common divisor (GCD). | a : b → a/GCD : b/GCD |
| 2 | Direct Proportion | When one quantity increases, the other increases at a constant rate. | a : b = c : d → b/c = a/d |
| 3 | Inverse Proportion | When one quantity increases, the other decreases. | a × b = k (constant) |
| 4 | Cross‑Multiplication | For a : b = c : d, cross‑multiply: a·d = b·c. |
ad = bc |
| 5 | Proportionate Parts | If x : y = a : b, then x = (a/(a+b))·(x+y). |
x = a/(a+b) × total |
| 6 | Mixture & Weighted Average | Sum of parts divided by sum of weights. | Total = Σ (part_i × weight_i) / Σ weight_i |
| 7 | Age Problems | Ages are always in direct proportion to a given ratio. | age1 = (ratio1/(ratio1+ratio2)) × total age |
| 8 | Percent Ratio | Ratio expressed as a percentage. | ratio = (part/whole) × 100 |
| 9 | Ratio in Money | Cost per unit is inversely proportional to quantity. | price × quantity = constant |
| 10 | Working with units | Keep units consistent; cancel them when simplifying. |
3. Worked Example
Problem
The ratio of the ages of two brothers, A and B, is 3 : 5.
The sum of their ages is 48 years.
Find the ages of A and B.
Step‑by‑Step Solution
-
Set up the proportion
Let the ages be (3x) and (5x) (because the ratio is 3:5).
-
Use the given sum
[ 3x + 5x = 48 \quad\Rightarrow\quad 8x = 48 ]
-
Solve for (x)
[ x = \frac{48}{8} = 6 ]
-
Find the ages
[ \begin{aligned} \text{Age of A} &= 3x = 3 \times 6 = 18 \text{ years} \ \text{Age of B} &= 5x = 5 \times 6 = 30 \text{ years} \end{aligned} ]
Quick Check
- Ratio (18 : 30 = 3 : 5) ✔️
- Sum (18 + 30 = 48) ✔️
Answer: A is 18 years old, B is 30 years old.
Another Typical Question (Inverse Proportion)
Question:
If 5 machines can complete a job in 12 days, how many days will 8 machines take, if all machines work at the same rate?
Solution
- Let the required days be (d).
- Inverse proportion: ( \text{machines} \times \text{days} = \text{constant} ).
- (5 \times 12 = 8 \times d).
- (60 = 8d \Rightarrow d = \frac{60}{8} = 7.5) days.
Answer: 8 machines will finish the job in 7.5 days.
4. Quick Revision Summary
| Concept | Formula | How to Use |
|---|---|---|
| Ratio Simplification | a:b → divide by GCD |
Speed up calculations |
| Direct Proportion | a:b = c:d → ad = bc |
Solve for any missing part |
| Inverse Proportion | a × b = k |
Compute missing variable |