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MathematicsPrevious Year Paper 2024

If 330 persons can complete the construction of a shopping complex in 50 days, how many persons are required to complete the same work in 30 days?

$550$

$450$

$198$

$505$

Correct Answer

📐 MA • Math Inverse ProportionMathematicsPrevious Year Paper 2024

Option A

$550$

Quick Summary:

Use the inverse ratio of days directly: Days are reduced in the ratio $50: 30 = 5: 3$ , so persons must increase in the ratio $3: 5$ . Thus, required persons $= 330 \times \frac{5}{3} = 550$ .

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📐MAMath SolutionInverse Proportion

📋 Given

Initial number of persons $M_1 = 330$ , Initial time period $D_1 = 50\text{ days}$ , Target time period $D_2 = 30\text{ days}$ .

🔢 Formula Used

M1D1=M2D2M_1 D_1 = M_2 D_2M1​D1​=M2​D2​

⚡ Exam Hall Shortcut / Speed Trick

Use the inverse ratio of days directly: Days are reduced in the ratio $50: 30 = 5: 3$ , so persons must increase in the ratio $3: 5$ . Thus, required persons $= 330 \times \frac{5}{3} = 550$ .

⚠️ Common Student Trap / Pitfall

Confusing direct and inverse proportion and incorrectly multiplying $330 \times \frac{30}{50} = 198$ , which yields incorrect Option C.

📊 Diagram / Illustration

🔢 Step-by-Step Solution

Identify the relationship between work force and time

Since total work remains constant, the number of persons ( $M$ ) is inversely proportional to the number of days ( $D$ ) required to finish the work.

W=M×D=constantW = M \times D = \text{constant}W=M×D=constant

Set up the work equivalence equation

Equating the total man-days required for the same work gives the formula relating initial and final conditions.

M1D1=M2D2M_1 D_1 = M_2 D_2M1​D1​=M2​D2​

Substitute given values into the equation

Substitute $M_1 = 330$ , $D_1 = 50$ , and $D_2 = 30$ to solve for the required number of persons $M_2$ .

330×50=M2×30330 \times 50 = M_2 \times 30330×50=M2​×30

Solve for required persons $M_2$

Divide total man-days by the target number of days to find $M_2$ .

M2=330×5030=11×50=550M_2 = \frac{330 \times 50}{30} = 11 \times 50 = 550M2​=30330×50​=11×50=550

✅

A is correct because $550$ persons are required to complete the shopping complex construction in $30$ days.

Core Concepts Used

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Time and Work Inverse Proportion Man-Days Concept

💡 EXAM TIP

This concept directly connects to 'Pipes & Cisterns' where total capacity equals efficiency multiplied by time ( $C = E \times T$ ).

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