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 , which means .
- 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 , written as .
- Extremes: a and d; Means: b and c.
- Rule: Product of means = Product of extremes
3. Types of Proportion
- Third Proportion: If , then c is the third proportion to a and b.
- Fourth Proportion: If , then d is the fourth proportion to a, b, c.
- Mean Proportion: Mean proportional between a and b is .
4. Compounded Ratio
- Multiply antecedents and consequents of two or more ratios.
- For , compounded ratio is .
5. Key Formulas Cheatsheet
Concept | Formula/Rule |
---|---|
Ratio | |
Proportion | |
Third Proportion | |
Fourth Proportion | |
Mean Proportion | |
Compounded Ratio |
6. Solved Examples
Q1. Combine Two Ratios
If and , find .
Solution:
- Make B common: , Multiply by 5, by 3: , So,
Q2. Compounded Ratio
Find the compound ratio of 17:23, 115:153, 18:25.
Solution:
- Multiply antecedents:
- Multiply consequents:
Q3. Fourth Proportion
If
Solution:
Q4. Third Proportion
What is the third proportion to 17.9 and 16.8?
Solution:
Q5. Mean Proportion
Find the mean proportional between 14 and 15.
Solution:
Q6. Mixed Proportion Problem
Mean proportional of 4 and 36 is ; third proportional of 18 and is . Find the fourth proportional of .
Solution:
- Mean proportional of 4 and 36:
- Third proportional of 18 and 12:
- Fourth proportional of 8, 12, 14:
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:
- 50 paise coins:
- 25 paise coins:
- Total value:
Q8. Variable Ratio Problem
If and , what is the value of ?
Solution:
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 such that
- Total shares = 13 parts; 2nd share =
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 , then the lengths are proportional.
9. Quick Reference Table
Concept | Formula/Rule |
---|---|
Ratio | |
Proportion | |
Mean Proportion | |
Third Proportion | |
Fourth Proportion | |
Compounded Ratio |
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, youโre ready to master Ratio and Proportion for any exam!