Ratio and Proportion
Ratio and proportion are fundamental concepts in mathematics that deal with comparing quantities and establishing relationships between them. Here’s an explanation of each concept
Ratio
Definition: A ratio is a comparison of two quantities by division. It expresses how much one quantity is there in relation to another.
Notation: Ratios are typically written in the form of a:b, where a and b are numbers.
Example: If there are 3 red balls and 5 blue balls, the ratio of red balls to blue balls is 3:5.
Properties: Ratios can be simplified like fractions, meaning a/b is ka/kb equivalent to for any non-zero k.
Key Concepts of Ratio
- The ratio should exist between the quantities of the same kind
- While comparing two things, the units should be similar
- There should be significant order of terms
- The comparison of two ratios can be performed, if the ratios are equivalent like the fractions
Proportion
Definition: A proportion is an equation stating that two ratios are equal.
Form: Proportions are written in the form a/b=c/d or a:b :: c:d.
Example: If 3 red balls correspond to 5 blue balls, and 6 red balls correspond to xxx blue balls, then 3/5=6/x is a proportion.
Solving Proportions: To solve a proportion, cross-multiply and then solve for the variable.
Key Concepts of Proportion
Direct Proportion: Two quantities are directly proportional if they increase or decrease in the same ratio. For example, if x increases, y also increases proportionally.
Inverse Proportion: Two quantities are inversely proportional if one quantity increases while the other decreases by the same factor. For example, if x increases, y decreases proportionally.
Continued Proportion: A continued proportion, also known as a continued ratio or continued proportionality, refers to a sequence of ratios that are all equal to each other. In other words, it is an extension of the concept of proportion where more than two quantities are involved in maintaining the same ratio throughout the sequence.
Difference Between Ratio and Proportion
Nature: Ratio is a comparison of quantities, while proportion establishes equality between ratios.
Structure: Ratios are simpler expressions comparing two quantities, whereas proportions involve two equal ratios.
Application: Ratios are used for direct comparison, while proportions are used for solving problems involving consistent relationships between quantities.
Fourth, Third and Mean Proportional
Fourth Proportional:
- If a:b=c:d, then d is called the fourth proportional to a,b, and c.
- This means d is the quantity that completes the proportion such that a/b=c/d
Third Proportional:
- If a:b = c:d, then c is called the third proportional to a and b.
- This means c is the quantity that is in the same proportion to b as a is to b.
Mean Proportional:
- The mean proportional (or geometric mean) between two numbers a and b is √ab
- This means √ab is the number that satisfies the proportion
a/√ab=√ab/b
Example
If 2 : 3 = 4 : x, find x.
Here, 2 : 3 = 4 : x implies that x is the fourth proportional to 2,3, and 4. To find x:
- Cross-multiply to solve the proportion:
2 : x = 4 3
2 : x = 12
x = 6
Therefore, x = 6 is the fourth proportional to 2,3 and 4.
Ratio and Proportion Solved Problems
Q1. Are the ratios 3:4 and 6:8 said to be in Proportion?
Solution: Simplify 6:8:
Divide both terms by their GCD, which is 2
6 ÷ 2 = 3
8 ÷ 2 = 4
Simplified Ratio: 3:4
Conclusion: 3 : 4 and 6 : 8 are in proportion.
Previous year Ratio And Proportion Based Questions
Q1: In a bag, the ratio of red balls to blue balls is 2:3. If there are 40 blue balls, how many red balls are there?
Q2: A sum of money is distributed between Babu, Christo and David in the ratio of 5:4:3. How much will Babu receive if Christo receives Rs.119 more from David?
Q3: The ratio of salaries to Hamid, Clement, and Ganesh is 3:57. If Ganesh gets Rs 868 more to Hamid, then how much is Clement’s salary (in Rs.)?
Q4: Rs. 4,200 is divided among Kamal, Dev and Rajat in the ratio of 7:8: 6 respectively. If Rs. 200 is added to the amounts of each of them, find the new ratio of their amounts.
Q5: Ratio between A and B is 6:5, ratio between B and C is 2: 3. What is the ratio between A and C?
| Related Links | |
|---|---|
| Quadratic Equations | Number System |
| Vedic Maths | Pie Chart DI |
| Time and Work | Problem on Ages |
| Flow Chart DI | |
| Arithmetic DI | Permutation and Combination |
| Ratio and Proportion |
|
Ration and proportion√ FAQs
Ans. A ratio is a comparison of two quantities by division. It expresses how many times one quantity is contained within another. For example, if there are 2 red balls and 5 blue balls, the ratio of red to blue balls is 2:5.
Ans. A proportion is an equation stating that two ratios are equal. It involves four quantities in which the ratio of the first pair is equal to the ratio of the second pair. For example, ab=cd.
Ans. To solve problems involving ratios, follow these steps: Understand the given ratios and the relationships between the quantities involved. Use cross-multiplication to find unknown quantities in proportions. Simplify ratios to their lowest terms to ensure accuracy.
Ans. There are two main types of proportion: Direct Proportion: When two quantities increase or decrease in the same ratio. Inverse Proportion: When an increase in one quantity leads to a decrease in another by the same factor.
Ans The mean proportional (or geometric mean) between two numbers a and b is √ab. It is used to find a missing value in a proportion where the relationship remains consistent.