Ratio & Proportion – Quick Study Notes
(For the Haryana Police Constable Exam – Quantitative Aptitude)
1. What is a Ratio?
- Definition – A ratio compares two numbers.
- Everyday analogy – Think of it like a recipe: if a cake needs 2 cups of flour and 3 cups of sugar, the ratio of flour to sugar is 2 : 3.
- Notation –
A : BorA/B.
Tiny Example
If a classroom has 12 boys and 8 girls, the boy‑to‑girl ratio is 12 : 8 (which can be simplified to 3 : 2).
2. What is a Proportion?
-
Definition – A proportion is an equation saying two ratios are equal.
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Everyday analogy – Imagine two baskets of fruit.
- Basket 1: 4 apples : 6 oranges
- Basket 2: 8 apples : ? oranges
If the two baskets have the same apple‑to‑orange ratio, the missing number makes a proportion.
-
Notation –
A : B = C : DorA/B = C/D.
3. How to Work with Ratios
| Step | What to Do | Example |
|---|---|---|
| 1. Write the ratio in simplest form. | 12 : 8 → 3 : 2 |
|
| 2. Find the common factor if you need the smallest whole‑number parts. | Divide both by 4. | |
| 3. Use the parts to find totals or missing numbers. | If the total number of students = 30, each part = 30 ÷ (3+2) = 6. |
|
| 4. Multiply each part by the part‑size to get actual numbers. | Boys = 3 × 6 = 18; Girls = 2 × 6 = 12. |
4. How to Solve a Proportion
- Set the proportion:
A : B = C : D. - Cross‑multiply (multiply the outer terms and the inner terms).
- Solve for the unknown.
Tiny Example
Find D if 4 : 9 = 12 : D.
| Step | Calculation | Result |
|---|---|---|
| 1. Cross‑multiply | 4 × D = 9 × 12 |
|
| 2. Compute right side | 4D = 108 |
|
| 3. Divide by 4 | D = 27 |
✅ |
5. Quick Tips
| Tip | Why it helps |
|---|---|
| Simplify first – always reduce the ratio to its lowest terms. | Makes calculations easier. |
| Use whole numbers – if the answer must be an integer, check that the denominator divides the numerator in the cross‑multiply step. | Avoids fractions when not needed. |
| Check your work – after solving, substitute back into the proportion to verify equality. | Confirms no arithmetic error. |
6. Practice Problems
- Ratio – The number of police officers and traffic guards at a checkpoint are in the ratio 5 : 3. If there are 48 guards, how many officers are there?
- Proportion – In a batch of 200 tickets, the number of adult tickets to child tickets is 7 : 3. How many child tickets were sold?
- Proportion – If
6 : 8 = x : 10, findx. - Ratio & Proportion – Three villages have a combined population of 12,000. Their populations are in the ratio 4 : 5 : 7. What is the population of the second village?
Solutions
| # | Question | Answer |
|---|---|---|
| 1 | Officers → ? | 48 × (5/3) = 80 officers |
| 2 | Child tickets → ? | 200 × (3/10) = 60 child tickets |
| 3 | x |
x = (6 × 10) / 8 = 7.5 |
| 4 | Second village | Total parts = 4+5+7 = 16. Part size = 12,000 ÷ 16 = 750. Second village = 5 × 750 = 3,750. |
7. Summary
- Ratio compares two numbers (e.g., 3 : 2).
- Proportion states two ratios are equal (e.g.,
3/2 = 6/4). - Simplify first, then use cross‑multiplication for proportions.
- Always check your answer by plugging it back into the original relation.
Keep practicing these steps, and you’ll ace the ratio & proportion section of the exam!