Ratio and Proportion

This guide breaks down the Ratio and Proportion PDF into simple explanations, key formulas, solved examples, a cheatsheet, practice questions (with answers), and visual aids for better understanding.

1. Ratio: Definition and Basics

• Ratio compares two quantities by division.
• Written as $ a : b $, which means $ \frac{a}{b} $.
• Antecedent: First term (a), Consequent: Second term (b).

Example: Ratio 5:9 means antecedent = 5, consequent = 9.

Visual: Ratio Bar

|-----5-----|-----9-----|
   a (5)       b (9)

2. Proportion: Definition

• Proportion is the equality of two ratios.
• If $ a : b = c : d $, written as $ a : b :: c : d $.
• Extremes: a and d; Means: b and c.
• Rule: Product of means = Product of extremes $ b \times c = a \times d $

3. Types of Proportion

• Third Proportion: If $ a : b = b : c $, then c is the third proportion to a and b.
• Fourth Proportion: If $ a : b = c : d $, then d is the fourth proportion to a, b, c.
• Mean Proportion: Mean proportional between a and b is $ \sqrt{ab} $.

4. Compounded Ratio

• Multiply antecedents and consequents of two or more ratios.
• For $ (a:b), (c:d), (e:f) $, compounded ratio is $ ace : bdf $.

5. Key Formulas Cheatsheet

6. Solved Examples

Q1. Combine Two Ratios

If $ A:B = 2:3 $ and $ B:C = 5:7 $, find $ A:B:C $.

Solution:

• Make B common: $ A:B = 2:3 $, $ B:C = 5:7 $ Multiply $ A:B $ by 5, $ B:C $ by 3: $ A:B = 10:15 $, $ B:C = 15:21 $ So, $ A:B:C = 10:15:21 $

Q2. Compounded Ratio

Find the compound ratio of 17:23, 115:153, 18:25.

Solution:

• Multiply antecedents: $ 17 \times 115 \times 18 $
• Multiply consequents: $ 23 \times 153 \times 25 $
• $ = (17 \times 115 \times 18) : (23 \times 153 \times 25) = 2:5 $

Q3. Fourth Proportion

If $ 3:27 :: 5:? $

Solution:

• $ 3/27 = 5/x \implies x = (5 \times 27)/3 = 45 $

Q4. Third Proportion

What is the third proportion to 17.9 and 16.8?

Solution:

• $ 17.9:x = 16.8: x \implies x = \frac{16.8^2}{17.9} = 15.76 $

Q5. Mean Proportion

Find the mean proportional between 14 and 15.

Solution:

• $ \sqrt{14 \times 15} = \sqrt{210} \approx 14.5 $

Q6. Mixed Proportion Problem

Mean proportional of 4 and 36 is $ a $; third proportional of 18 and $ a $ is $ b $. Find the fourth proportional of $ b, 12, 14 $.

Solution:

• Mean proportional of 4 and 36: $ a = \sqrt{4 \times 36} = 12 $
• Third proportional of 18 and 12: $ b = \frac{12^2}{18} = 8 $
• Fourth proportional of 8, 12, 14: $ 8/12 = 14/x \implies x = 21 $

7. Practice Questions with Answers

Q7. Coin Ratio Problem

A bag has coins of Rs. 1, 50 paise, and 25 paise in the ratio 5:9:4. If the total number of coins is 72, what is the worth of the bag?

Solution:

• Rs. 1 coins: $ 5/18 \times 72 = 20 $
• 50 paise coins: $ 9/18 \times 72 = 36 $
• 25 paise coins: $ 4/18 \times 72 = 16 $
• Total value: $ (20 \times 1) + (36 \times 0.5) + (16 \times 0.25) = 20 + 18 + 4 = Rs. 42 $

Q8. Variable Ratio Problem

If $ 18:13.5:16:x $ and $ (x+y):y:18:10 $, what is the value of $ y $?

Solution:

• $ x = (16 \times 13.5)/18 = 12 $
• $ (12+y):y = 9:5 \implies 5(12+y) = 9y \implies 4y = 60 \implies y = 15 $

Q9. Share Division

Mr. Raj divides Rs. 1573 so that 4 times the 1st share, 3 times the 2nd, and 2 times the 3rd are equal. Find the value of the 2nd share.

Solution:

• Let shares be $ A, B, C $ such that $ 4A = 3B = 2C $
• $ A:B:C = 1/4:1/3:1/2 = 3:4:6 $
• Total shares = 13 parts; 2nd share = $ (4/13) \times 1573 = Rs. 484 $

8. Visuals & Graphics

A. Ratio Pie Chart Example

If A:B:C = 2:3:5

[ 20% ] [ 30% ] [ 50% ]
   A        B       C

B. Proportion Bar

|---a---|---b---| = |---c---|---d---|

If $ a:b = c:d $, then the lengths are proportional.

9. Quick Reference Table

10. Tips & Tricks

• To combine ratios, make the common term equal.
• For mean proportion, always use the square root.
• For compounded ratio, multiply all antecedents and all consequents.
• For share division, convert conditions into ratios.

With these explanations, formulas, solved examples, cheatsheet, practice questions, and visuals, you’re ready to master Ratio and Proportion for any exam!