Question

The average earning of each member of the Ambani family is \(20 \%\) less than the average earning of each member of the Sahara family and the total earning of Ambani's family is \(20 \%\) more than the total earning of Saharas's family. The no. of family members in the Sahara is what per cent of the no. of family members of Ambani : (a) \(25 \%\) (b) \(20 \%\) (c) \(66.66 \%\) (d) none of these

Short answer

Answer: 150%

Step-by-step solution

Define Variables

Let's denote the average earning of a member of the Ambani family as A and the average earning of a member of the Sahara family as S. Let x represent the number of members in the Ambani family, and y represent the number of members in the Sahara family.

Setup Equations

Given that the average earning of an Ambani family member is 20% less than a Sahara family member, we can write the equation: A = 0.8S Also, given that the total earning of the Ambani family is 20% more than the total earning of the Sahara family, we can write the equation: Ax = 1.2Sy.

Solve for the ratio x/y

In this step, we will solve the equations to find the ratio x/y. From Step 2, we have: A = 0.8S Now, substitute this value of A into the equation Ax = 1.2Sy, which gives us: (0.8S)x = 1.2Sy Divide both sides by S and rearrange to get the ratio x/y: x/y = 1.2/0.8

Calculate the percentage

To find the percentage of the number of members in the Sahara family compared to the Ambani family, divide x by y and multiply by 100. Percentage = (1.2/0.8) * 100 #= 150\% Thus, the number of family members in the Sahara family is 150% of the number of family members in the Ambani family which corresponds to the option (d) none of these.

Key concepts

Average Earning Comparison

Understanding how to compare the average earning between two or more groups is an important concept in quantitative aptitude and statistics. To compare the average earnings, one must analyze not just the individual amounts but also how the collective earnings of each group relate when considering the number of members in each group.

As seen in the provided exercise, a percentage decrease in individual earnings doesn't directly translate to the same percentage change in total earnings because the size of each group can vary significantly. To effectively compare average earnings:

• Determine the averages for each group separately
• When given a percentage decrease or increase, adjust the average accordingly
• Remember that a change in the average does not imply a proportional change in total earnings unless the group sizes are the same

Variable Definition in Algebra

In algebra, variables are symbols (often letters) that represent numbers or values that can vary. They are the foundation of algebraic expressions and equations. For instance, in the exercise, 'A' represents the average earning of the Ambani family, and 'S' for the Sahara family, with 'x' and 'y' introduced as the representations for the number of members in each family respectively.

Defining variables correctly is critical for forming equations that can solve problems. Here are key points to remember when defining variables:

• Pick letters that help you remember what they stand for (e.g., A for Ambani, S for Sahara)
• Make sure each variable represents a single, specific concept
• Variables should be clearly defined before setting up equations

Solving Ratio Problems

Ratio problems involve finding the relative sizes of two or more values. They are a crucial part of many quantitative reasoning tasks, including comparisons of different quantities and conversions. To solve a ratio problem, you establish a relationship between the quantities and express it in the simplest form.

When solving the ratio problems like the one between the Ambani and Sahara families' members:

• Identify the known ratios or percentages
• Set up an equation that represents the relationship
• Simplify the ratio to find the answer

In our example, the ratio of the number of family members (x/y) was calculated to understand the proportion between the two families.

Quantitative Aptitude

Quantitative aptitude covers a wide range of mathematical abilities and concepts, including number systems, arithmetic operations, algebra, percentages, and ratio and proportion, among others. It is fundamental for problem-solving and decision-making in various fields.

Exercises like the one discussed help develop this aptitude by requiring the application of several mathematical concepts to arrive at a solution. To improve quantitative aptitude:

• Practice different types of math problems regularly
• Understand the underlying principles rather than just memorizing formulas
• Develop a step-by-step approach to breaking down complex problems into simpler parts