If 6 men and 8 boys can do a piece of work... | গণিত Multiple Choice Question

<html lang="en"> <head> <meta charset="UTF-8"> <meta name="viewport" content="width=device-width, initial-scale=1.0"> <title>Detailed Explanation for MCQ</title> <style> body { font-family: Arial, sans-serif; line-height: 1.6; margin: 20px; } .math { font-family: "Times New Roman", Times, serif; font-style: italic; } </style> </head> <body> <h1>Detailed Explanation for the MCQ</h1>

Question: If 6 men and 8 boys can do a piece of work in 10 days and 26 men and 48 boys can do the same in 2 days, the time taken by 15 men and 20 boys to do the same type of work will be:

Choices: ['4 days', '5 days', '6 days', '7 days']

Correct Answer: '4 days'

<h2>Explanation:</h2>

To solve this problem, we need to determine the combined work rates of men and boys. Let's denote the work rate of one man as <span class="math">M</span> and the work rate of one boy as <span class="math">B</span>.

<h3>Step 1: Establish Two Equations</h3>

From the question, we know:

6 men and 8 boys can complete the work in 10 days.
26 men and 48 boys can complete the work in 2 days.

The amount of work done is the same in both cases. Since work rate (work done per day) is the reciprocal of time, we can write the following equations:

6M + 8B = \frac{1}{10}

26M + 48B = \frac{1}{2}

<h3>Step 2: Solve the Equations</h3>

Let's solve these equations simultaneously. First, multiply both sides of the first equation by 10 to get rid of the fraction:

10(6M + 8B) = 1

60M + 80B = 1 \quad \text{(Equation 1)}

Next, multiply both sides of the second equation by 2:

2(26M + 48B) = 1

52M + 96B = 1 \quad \text{(Equation 2)}

<h3>Step 3: Eliminate One Variable</h3>

To eliminate one variable, we can subtract Equation 2 from Equation 1:

(60M + 80B) - (52M + 96B) = 0

8M - 16B = 0

Simplifying, we get:

8M = 16B

M = 2B

<h3>Step 4: Substitute M</h3>

Substitute <span class="math">M = 2B</span> into one of the original equations, say Equation 1:

60M + 80B = 1

60(2B) + 80B = 1

120B + 80B = 1

200B = 1

B = \frac{1}{200}

Using <span class="math">B = \frac{1}{200}</span> to find <span class="math">M</span>:

M = 2B = 2 \times \frac{1}{200} = \frac{1}{100}

<h3>Step 5: Determine Work Rate of 15 Men and 20 Boys</h3>

The combined work rate of 15 men and 20 boys is:

15M + 20B

Substituting <span class="math">M = \frac{1}{100}</span> and <span class="math">B = \frac{1}{200}</span>:

15 \left( \frac{1}{100} \right) + 20 \left( \frac{1}{200} \right)

= \frac{15}{100} + \frac{20}{200}

= \frac{15}{100} + \frac{10}{100}

= \frac{25}{100}

= \frac{1}{4}

Thus, the time taken by 15 men and 20 boys to complete the work is:

1 \div \frac{1}{4} = 4 \, \text{days}

<h3>Conclusion</h3>

The correct answer is '4 days'. This solution involves setting up and solving equations to find the individual work rates and then combining these rates to find the total work rate for 15 men and 20 boys.

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If 6 men and 8 boys can do a piece of work in 10 days and 26 men and 48 boys can do the same in 2 days , the time taken by 15 men and 20 boys to do the same type of work will be

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