Ratio & Proportion – Study Notes
(For the Madhya Pradesh Patwari / Lekhpal Quantitative Aptitude Exam)
1. Simple Explanation
Ratio compares two quantities.
Proportion is an equation that states that two ratios are equal.
Example
If the ratio of boys to girls in a class is 3 : 4 and there are 42 boys, how many girls are there?
[ \frac{3}{4}=\frac{42}{x}\quad\Longrightarrow\quad x=\frac{4\times42}{3}=56 ]
2. Key Points
| Concept | What It Means | How to Use |
|---|---|---|
| Direct Proportion | When one quantity increases, the other increases proportionally. | If (a:b = c:d) then (a) is directly proportional to (c). |
| Inverse (Reciprocal) Proportion | When one quantity increases, the other decreases proportionally. | If (a:b = c:d) and (a) is inversely proportional to (c), then (a\times c = k) (constant). |
| Cross‑Multiplication | (\frac{a}{b}=\frac{c}{d}) ⇔ (ad = bc). | Quick check for equality of ratios. |
| Proportion with Three Variables | (\frac{a}{b} = \frac{c}{d}) ⇒ (ad = bc). | Solve for the missing term. |
| Percent Change | (\text{percent change}= \frac{\text{new} - \text{old}}{\text{old}}\times100%). | Useful in “growth” problems that involve ratio. |
| Unit Rate | (\frac{\text{quantity}}{\text{unit}}). | Convert any ratio into a per‑unit measure. |
| Compound Ratio | Sequential ratios: (a:b:c) can be treated as ((a:b)) and ((b:c)). | Useful in multi‑stage proportional problems. |
3. Worked Example
Problem
A survey found that the ratio of the number of villages in three districts A, B, and C is 5 : 7 : 9.
If the total number of villages in the three districts is 240, find the number of villages in each district.
Step‑by‑Step Solution
-
Set up the proportion
Let the number of villages in A, B, C be (5x), (7x), (9x) respectively.
The ratio 5 : 7 : 9 is represented by the common multiplier (x). -
Use the total
[ 5x + 7x + 9x = 240 \quad\Longrightarrow\quad 21x = 240 ] -
Solve for (x)
[ x = \frac{240}{21} = \frac{80}{7}\approx 11.43 ] -
Find each district
- A: (5x = 5 \times \frac{80}{7} = \frac{400}{7}\approx 57.14)
- B: (7x = 7 \times \frac{80}{7} = 80)
- C: (9x = 9 \times \frac{80}{7} = \frac{720}{7}\approx 102.86)
-
Round to nearest whole number (since villages are whole units)
- A ≈ 57 villages
- B = 80 villages
- C ≈ 103 villages
Answer: District A – 57 villages, District B – 80 villages, District C – 103 villages.
(Check: 57 + 80 + 103 = 240; ratio ≈ 5 : 7 : 9.)
4. Quick Revision Summary
| Topic | Formula / Rule | Quick Tip |
|---|---|---|
| Direct Proportion | (a:b = c:d) ⇔ (ad = bc) | Cross‑multiply to solve. |
| Inverse Proportion | (a:b = c:d) with (a \propto \frac{1}{c}) ⇒ (a\times c = k) | Multiply the two variables; result is constant. |
| Compound Ratio | (a:b:c) → treat as ((a:b)) & ((b:c)) | Break into two proportions. |
| Percent Change | (\frac{\text{new} - \text{old}}{\text{old}}\times100%) | Handy for “increase by X%” or “decrease by X%” problems. |
| Unit Rate | (\frac{\text{quantity}}{\text{unit}}) | Convert any ratio to a per‑unit measure for easier comparison. |
| Cross‑Multiplication | (\frac{a}{b}=\frac{c}{d}) ⇔ (ad = bc) | Quick equality check. |
| Common Multiplier (x) | For ratio (m:n), numbers are (mx) & (nx). | Solve for (x) using any given sum/product. |
Final Tips for the Exam
- Read the question carefully – identify whether it’s a direct or inverse proportion.
- Set up the proportion first, then use the given total or other numeric condition.
- Use cross‑multiplication to avoid fractions unless the problem specifically asks for a percentage or unit rate.
- Always check your answer by plugging back into the original ratio.
- Practice time‑bound problems to build speed, especially for multi‑step ratio problems.
Good luck with your preparation!