Ratio & Proportion – Quick Study Notes
(Madhya Pradesh PSC – Quantitative Aptitude)
1. Simple Explanation
Ratio – a comparison of two numbers using division.
Proportion – an equation that shows two ratios are equal.
Example
8 : 12 is a ratio.
8 : 12 = 4 : 6 (simplified ratio).
8 : 12 = 2 : 3 → 8 / 12 = 2 / 3 → 8 : 12 = 2 : 3 (proportion).
2. Key Points
| Concept | What to remember | Formula / Rule |
|---|---|---|
| Simplifying a ratio | Divide each term by their greatest common divisor (GCD). | a:b = (a/gcd):(b/gcd) |
| Equivalent ratio | If two ratios have the same value, they are equivalent. | a:b = c:d ⇔ a·d = b·c |
| Proportion | Two ratios are equal. | a:b = c:d |
| Finding the fourth term | a:b = c:d → d = (b·c)/a |
|
| Finding the third term | a:b = c:d → c = (a·d)/b |
|
| Finding the second term | a:b = c:d → b = (a·d)/c |
|
| Finding the first term | a:b = c:d → a = (b·c)/d |
|
| Mixed ratio | When one ratio contains a sum or difference, split it into components before solving. | |
| Percentages as ratios | x% of y → (x/100)·y. |
|
| Checking work | Multiply cross‑wise: a·d should equal b·c. |
3. How to Solve a Proportion
- Write the proportion in standard form
x : y = a : b. - Convert to an equation
x·b = y·a. - Solve for the unknown (multiply/divide as needed).
- Check by substituting back into the ratio.
4. Common Problem Types
| Problem | Typical Formula | Quick Strategy |
|---|---|---|
| Finding a missing term | x = (b·c)/a (or variations) |
Use cross‑multiplication. |
| Mixing proportions | x : (total–x) = a : b |
Solve for x using x·b = (total–x)·a. |
| Changing ratio | If ratio changes by a factor k, each term is multiplied/divided by k. |
new a = a·k, new b = b·k. |
| Equivalent ratios | a:b = c:d |
Check using cross‑product. |
| Percentage problems | Convert percent to fraction first. | x% of y = (x/100)·y. |
| Scaling | If a ratio is scaled by n, each term is n times larger. |
new a = a·n, new b = b·n. |
5. Worked Example
Problem
The ratio of boys to girls in a class is 5 : 3. If the class has 48 students, how many boys are there?
Step‑by‑Step Solution
-
Let
- Number of boys =
5k - Number of girls =
3k
(Because the ratio is 5:3.)
- Number of boys =
-
Set up the equation
[ 5k + 3k = 48 \quad\Rightarrow\quad 8k = 48 ] -
Solve for
k
[ k = \frac{48}{8} = 6 ] -
Find the number of boys
[ 5k = 5 \times 6 = 30 ] -
Answer
- Boys = 30
- Girls = (3k = 18) (optional check)
6. Quick Revision Summary
- Ratio – comparison of two numbers (
a:b). - Proportion – equality of two ratios (
a:b = c:d). - Simplify ratios by dividing by GCD.
- Cross‑multiplication (
a·d = b·c) is the quickest way to confirm or solve. - Missing term formulas:
- 4th term:
d = (b·c)/a - 3rd term:
c = (a·d)/b - 2nd term:
b = (a·d)/c - 1st term:
a = (b·c)/d
- 4th term:
- For mixed ratios or percentages, convert everything to a single common form before cross‑multiplying.
- Always check by substituting back into the original ratio.
Practice tip:
- Convert every ratio problem into an equation using cross‑multiplication.
- Keep a small cheat‑sheet of the four missing‑term formulas for quick reference during the exam.
Good luck with your preparation!