Ratio And Proportion : Short Tricks and PYQs

 

Mastering Ratio And Proportion :  Here is a detailed note on the topic of Ratio and Proportion, including short tricks, previous year papers, and solutions for CUET, CLAT, and IPMAT exams:

 

What is Ratio?

 

A ratio is a way of comparing two quantities by division. It tells us how many times one quantity is contained in another. A ratio is often written in the form a:b, where a and b are the two quantities being compared.

 

What is Proportion?

 

A proportion is a statement that two ratios are equal. It is often written in the form a:b::c:d, where a, b, c, and d are the quantities being compared.

 

Types of Ratios

 

There are several types of ratios, including:

 

1. Part-to-Part Ratio: This type of ratio compares one part of a quantity to another part of the same quantity.

Example: The ratio of boys to girls in a class is 3:5.

2. Part-to-Whole Ratio: This type of ratio compares one part of a quantity to the entire quantity.

Example: The ratio of students who passed the exam to the total number of students is 3:10.

3. Whole-to-Whole Ratio: This type of ratio compares one entire quantity to another entire quantity.

Example: The ratio of the number of students in School A to the number of students in School B is 5:7.

 

How to Simplify Ratios

 

To simplify a ratio, we need to find the greatest common divisor (GCD) of the two quantities and divide both quantities by the GCD.

 

Example: Simplify the ratio 12:18.

 

GCD of 12 and 18 is 6.

 

Simplified ratio = 12 ÷ 6 : 18 ÷ 6 = 2:3

 

How to Add and Subtract Ratios

 

To add or subtract ratios, we need to have the same denominator.

 

Example: Add the ratios 1:3 and 2:3.

 

Since both ratios have the same denominator (3), we can add the numerators.

 

Resulting ratio = (1 + 2):3 = 3:3

 

How to Multiply and Divide Ratios

 

To multiply ratios, we multiply the numerators and denominators separately.

 

Example: Multiply the ratios 2:3 and 4:5.

 

Resulting ratio = (2 × 4):(3 × 5) = 8:15

 

To divide ratios, we invert the second ratio and multiply.

 

Example: Divide the ratio 2:3 by the ratio 4:5.

 

Invert the second ratio: 5:4

 

Resulting ratio = (2 × 5):(3 × 4) = 10:12

 

Short Tricks:

 

Here are some short tricks to help you solve ratio and proportion problems:

 

1. Use the concept of equivalent ratios: If two ratios are equal, then the product of the means is equal to the product of the extremes.

Example: If a:b::c:d, then ad = bc.

2. Use the concept of proportionality: If two quantities are directly proportional, then the ratio of the two quantities is constant.

Example: If x is directly proportional to y, then x:y::k, where k is a constant.

3. Use the concept of percentage: Percentage is a way of expressing a ratio as a fraction of 100.

Example: 25% of 120 = (25/100) × 120 = 30.

 

Previous Year Papers with Solutions

 

Here are some previous year papers with solutions for CUET, CLAT, and IPMAT exams:

 

CUET 2022:

 

Question: If the ratio of the number of boys to the number of girls in a class is 7:5, and the number of boys is 84, find the number of girls.

 

Solution: Let the number of girls be x. Then, the ratio of boys to girls is 84:x. Since the ratio is given as 7:5, we can set up the proportion 84:x::7:5. Solving for x, we get x = 60.

 

CLAT 2022:

 

Question: A bag contains 120 coins, consisting of one-rupee and two-rupee coins. If the ratio of one-rupee coins to two-rupee coins is 3:5, find the number of two-rupee coins.

 

Solution: Let the number of one-rupee coins be 3x and the number of two-rupee coins be 5x. Then, the total number of coins is 3x + 5x = 120. Solving for x, we get x = 12. Therefore, the number of two-rupee coins is 5x = 60.

 

IPMAT 2022:

 

Question: If the ratio of the number of employees in Company A to the number of employees in Company B is 3:4, and

 

 

IPMAT 2022:

 

Question: If the ratio of the number of employees in Company A to the number of employees in Company B is 3:4, and the number of employees in Company A is 270, find the number of employees in Company B.

 

Solution: Let the number of employees in Company B be x. Then, the ratio of employees in Company A to Company B is 270:x. Since the ratio is given as 3:4, we can set up the proportion 270:x::3:4. Solving for x, we get x = 360.

 

Tips and Tricks

 

1. Use the concept of equivalent ratios: If two ratios are equal, then the product of the means is equal to the product of the extremes.

2. Use the concept of proportionality: If two quantities are directly proportional, then the ratio of the two quantities is constant.

3. Use the concept of percentage: Percentage is a way of expressing a ratio as a fraction of 100.

 

Practice Questions

 

1. If the ratio of the number of boys to the number of girls in a class is 7:5, and the number of boys is 84, find the number of girls.

2. A bag contains 120 coins, consisting of one-rupee and two-rupee coins. If the ratio of one-rupee coins to two-rupee coins is 3:5, find the number of two-rupee coins.

3. If the ratio of the number of employees in Company A to the number of employees in Company B is 3:4, and the number of employees in Company A is 270, find the number of employees in Company B.

 

Solutions

 

1. Let the number of girls be x. Then, the ratio of boys to girls is 84:x. Since the ratio is given as 7:5, we can set up the proportion 84:x::7:5. Solving for x, we get x = 60.

2. Let the number of one-rupee coins be 3x and the number of two-rupee coins be 5x. Then, the total number of coins is 3x + 5x = 120. Solving for x, we get x = 12. Therefore, the number of two-rupee coins is 5x = 60.

3. Let the number of employees in Company B be x. Then, the ratio of employees in Company A to Company B is 270:x. Since the ratio is given as 3:4, we can set up the proportion 270 :x::3:4. Solving for x, we get x = 360.

 

Conclusion

 

Ratio and proportion are important concepts in mathematics and are used extensively in various fields such as business, economics, and science. In this notes, we have covered the basics of ratio and proportion, including the concept of equivalent ratios, proportionality, and percentage. We have also provided tips and tricks for solving ratio and proportion problems, as well as practice questions and solutions.