Question

During our campaign against child labour we have found that in three glass making factories \(A, B\) and \(C\) there were total 33 children aged below 18 were involved. The ratio of male to female in \(A, B\) and \(C\) was \(4: 3,3: 2\) and \(5: 4\) respectively. If the no. of female children working in the factories \(B\) and \(C\) be equal then find the no. of female children working in factory \(A:\) \(\begin{array}{ll}A: & \text { (b) } 2\end{array}\) (a) 5 (b) 2 (c) 8 (d) 6

Short answer

(a) 5 (b) 2 (c) 8 (d) 6 Given the available answer choices and provided information, it is not possible to conclusively determine the number of female children working in factory A, as none of the answer choices result in an integer value for the number of females in factory B. Some additional information may be required to find the correct answer.

Step-by-step solution

Set up the equations for the number of children in factories A, B, and C

Let the number of male and female children in the factories be denoted by M and F, respectively. Using the given ratios, we can write the following equations for the factories A, B, and C: Factory A: \(4/3 = M_A/F_A\) Factory B: \(3/2 = M_B/F_B\) Factory C: \(5/4 = M_C/F_C\) We are also given that the number of female children in factories B and C is equal: \(F_B = F_C\).

Find the total number of children in each factory

Now, let's express the total number of children in each factory in terms of the male and female children: Factory A: \(T_A = M_A + F_A\) Factory B: \(T_B = M_B + F_B\) Factory C: \(T_C = M_C + F_C\) The total number of children in all three factories is 33, so \(T_A + T_B + T_C = 33\).

Substitute the ratios and find the common ratio

Using the equations for the ratios of male to female children in the factories, let's substitute the male children in terms of female children: Factory A: \(M_A = 4/3 * F_A\) Factory B: \(M_B = 3/2 * F_B\) Factory C: \(M_C = 5/4 * F_C\) Now, substituting these expressions back into the equations for the total number of children in each factory: Factory A: \(T_A = 4/3 * F_A + F_A = (7/3) * F_A\) Factory B: \(T_B = 3/2 * F_B + F_B = (5/2) * F_B\) Factory C: \(T_C = 5/4 * F_C + F_C = (9/4) * F_C\) The sum of the total number of children is 33, so: \((7/3) * F_A + (5/2) * F_B + (9/4) * F_C = 33\) Since \(F_B = F_C\), we can substitute \(F_B\) for \(F_C\): \((7/3) * F_A + (5/2 + 9/4) * F_B = 33\)

Solve the equation for the number of female children in factory A

Now, we will solve the equation for F_A: \((7/3) * F_A + (17/4) * F_B = 33\) Let's clear the fractions by multiplying the equation by 12: \(28 * F_A + 51 * F_B = 396\) Since we do not have enough information to solve for both \(F_A\) and \(F_B\), we will test the available answer choices for \(F_A\). We will test if \(F_A\) can be 2, 5, 8, or 6. (a) 5: \(28 * 5 + 51 * F_B = 396\) \(140 + 51 * F_B = 396\) This doesn't lead to an integer value for \(F_B\). (b) 2 \(28 * 2 + 51 * F_B = 396\) \(56 + 51 * F_B = 396\) When we solve for \(F_B\), we get: \(51 * F_B = 340\) \(F_B = 20/3\) This doesn't lead to an integer value for \(F_B\). (c) 8 \(28 * 8 + 51 * F_B = 396\) \(224 + 51 * F_B = 396\) This doesn't lead to an integer value for \(F_B\). (d) 6 \(28 * 6 + 51 * F_B = 396\) \(168 + 51 * F_B = 396\) When we solve for \(F_B\), we get: \(51 * F_B = 228\) \(F_B = 76/17\) This doesn't lead to an integer value for \(F_B\). None of the answer choices result in an integer value for \(F_B\). Therefore, it seems there is an error in the given answer choices or some additional information is missing.

Key concepts

Quantitative Aptitude

Quantitative aptitude refers to the ability of an individual to perform calculations and solve numerical problems. It is a critical skill for various competitive exams and real-world problem-solving scenarios. In the context of the given problem, quantitative aptitude involves understanding the problem statement, identifying the ratio and proportion of male to female children in factories, and applying algebraic equations to determine the number of female children working in factory A.

To enhance one's quantitative aptitude, it is important to practice a variety of problems that require logical reasoning and numerical calculations. The given exercise involves not just basic arithmetic but also the application of ratios and algebra to solve for an unknown quantity. By carefully setting up equations and manipulating them, we delve deeper into the essence of quantitative aptitude, which is a blend of comprehension, strategy, and computational skills.

Ratio and Proportion

The concept of ratio and proportion is fundamental to understanding relationships between different quantities. A ratio is a comparison of two quantities by division, and a proportion is an equation stating that two ratios are equal. In our exercise, the ratios of male to female children working in different factories are given, and these ratios are pivotal to setting up the initial mathematical relationships needed for solving the problem.

Improving Comprehension of Ratios

To better comprehend the ratios involved, one can visualize the ratios with simple drawings or models, representing males and females as separate entities. These visual aids can help students bridge the gap between abstract numbers and a more tangible understanding of the problem's context. Knowing how to translate these ratios into algebraic equations is a key step in moving towards the solution.

Algebraic Equations

Algebraic equations are a central tool in expressing and solving mathematical problems where an unknown variable is involved. In our problem, we use algebraic equations to represent the total number of children in each factory in terms of the number of male and female children. By doing so, we are able to introduce variables for unknown quantities and manipulate these equations to solve for the desired variable.

Navigating through Algebraic Equations

When improving one's ability to handle algebraic equations, it is critical to master the art of 'balancing' the equation—what you do to one side of the equation, you must do to the other to maintain equality. Simplifying complex equations by combining like terms and clearing denominators, as in the step where we multiplied the equation by 12 to eliminate the fractions, is an essential skill. Additionally, a crucial aspect in problem-solving is to check the answer choices provided, which may guide us towards the correct solution or indicate potential errors or missing information within the problem itself.