Question

The average weight of boys in a class is \(30 \mathrm{~kg}\) and the average weight of girls in the same class is \(20 \mathrm{~kg}\). If the average weight of the whole class is \(23.25 \mathrm{~kg}\), what could be the possible strength of boys and girls respectively in the same class ? (a) 14 and 26 (b) 13 and 27 (c) 17 and 27 (d) none of these

Short answer

Answer: (d) None of these.

Step-by-step solution

Understand the concept of weighted average

A weighted average is an average that takes into account the importance of the components. In this problem, the average weight of the whole class is a weighted average of the average weights of boys and girls, with the importance(modifier) given by the number of boys and girls in the class.

Set up the equation for total weights of boys and girls

Let the number of boys in the class be B and the number of girls be G. The total weight of boys is 30B kg, and the total weight of girls is 20G kg. The total weight of the class is (B+G) multiplied by the average weight of the whole class (23.25 kg): (B+G)(23.25).

Express the total weights in terms of the average

Since the total weights are equal, we set up an equation as follows: 30B = 20G + 23.25(B+G) Now, let us test the options and see which one fits the equation.

Test the options

(a) If there are 14 boys and 26 girls in the class: 30(14) = 20(26) + 23.25(14 + 26) 420 = 520 + 23.25(40), which is 420 = 520 + 930, which is not true. (b) If there are 13 boys and 27 girls in the class: 30(13) = 20(27) + 23.25(13 + 27) 390 = 540 + 23.25(40), which is 390 = 540 + 930, which is also not true. (c) If there are 17 boys and 27 girls in the class: 30(17) = 20(27) + 23.25(17 + 27) 510 = 540 + 23.25(44), which is 510 = 540 + 1023, which is not true as well.

Conclusion

Since none of the options fits the equation, the answer to the problem is (d) none of these.

Key concepts

Quantitative Aptitude

Quantitative aptitude involves the ability to handle numbers and perform mathematical calculations with ease. It's an essential skill for problem-solving in many fields, including education, finance, and engineering. When students work with quantitative aptitude exercises, like the weighted average problem presented here, they sharpen their mathematical reasoning and their proficiency in dealing with numerical data.

In the context of this problem, students must apply their quantitative skills to find a solution by considering the given weights and calculating potential ratios of boys to girls in a class. Quantitative aptitude covers understanding the significance of weighted averages and applying algebraic equations to deduce the answer. By breaking down the problem into clear steps, students can systematically solve the problem, enhancing their quantitative reasoning abilities.

Average Weight Calculation

Calculating average weight can be a straightforward task when dealing with simple averages; however, when different groups have different weights and different counts, as seen in classrooms or data samples, the calculation becomes more complex and is termed as a weighted average problem.

The term 'weighted' refers to the importance assigned to different values based on their frequency or size. In the given exercise, students learn how to calculate the average weight of a mixed group by assigning appropriate weights to the averages of separate groups (boys and girls). Understanding that the weight of a group (like boys and girls in a classroom setting) is the product of the average weight and the number of members in each group is pivotal in solving these types of problems. Recognizing these distinctions is key to obtaining the correct overall average.

Algebraic Equation

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating these symbols; it is one of the foundational tools for solving quantitative problems. An algebraic equation represents the relationship between different variables, and learning to create and solve these equations is a fundamental skill in mathematics.

In the provided exercise, the relationship between the total weights of boys and girls in a class is represented using an algebraic equation; it factors in the number of boys and girls as well as their average weights. Students learn to solve for unknown variables, in this case, the number of boys and girls, by setting up an equation and then testing various possible solutions with given numerical values. An appreciation for the structured approach of algebra allows students to tackle complex problems in a logical and sequential manner, as demonstrated in the steps of this solution.