8 men and 12 boys can finish - SSC CGL Quantitative Aptitude

Work Completion by Men and Boys: Linear Equation Approach

This topic addresses the calculation of time taken by one man alone to complete a piece of work, given the combined work rates of men and boys in different groupings. The solution uses the elimination method and conversion of statements into linear equations, focusing on unitary method principles and systematic algebraic manipulation.

Statement to Equation Conversion

Translating word problems into mathematical equations is essential for solving work-related questions. Each statement about group work rates is converted into an equation by expressing the total work as the sum of individual…

Work Completion by Men and Boys: Linear Equation Approach

This topic addresses the calculation of time taken by one man alone to complete a piece of work, given the combined work rates of men and boys in different groupings. The solution uses the elimination method and conversion of statements into linear equations, focusing on unitary method principles and systematic algebraic manipulation.

Statement to Equation Conversion

Translating word problems into mathematical equations is essential for solving work-related questions. Each statement about group work rates is converted into an equation by expressing the total work as the sum of individual contributions per day, then relating it to the total time taken.

• Let $x$ be the number of days one man alone takes to finish the work.
• Let $y$ be the number of days one boy alone takes to finish the work.
• The work done by one man in one day is $1/x$.
• The work done by one boy in one day is $1/y$.

Unitary Method Application

The unitary method is used to scale individual work rates to group work rates. By multiplying the one-day work of a single person by the number of people, the total group work per day is obtained.

• For men: $n$ men’s one-day work = $n \times (1/x)$.
• For boys: $m$ boys’ one-day work = $m \times (1/y)$.
• This method is applied to both men and boys to form equations based on the group sizes and days given.

Formulation of Equations

Given the problem statements, two equations are formed based on the total work completed by different groupings in specified days.

• First group: 8 men and 12 boys finish the work in 10 days.
  • Equation: $8 \times (1/x) + 12 \times (1/y) = 1/10$
• Second group: 6 men and 8 boys finish the work in 14 days.
  • Equation: $6 \times (1/x) + 8 \times (1/y) = 1/14$

Variable Substitution and Simplification

To simplify the equations, substitutions are made for the reciprocals of $x$ and $y$.

• Let $u = 1/x$ and $v = 1/y$.
• The equations become:
  • $8u + 12v = 1/10$
  • $6u + 8v = 1/14$
• These equations are then multiplied by the denominators to clear fractions:
  • $80u + 120v = 1$
  • $84u + 112v = 1$

Elimination Method for Solving Equations

The elimination method is used to solve the two equations for $u$ and $v$. The coefficients are adjusted to align one variable for elimination.

• Multiply the first equation by 84 and the second by 80:
  • $672u + 1008v = 84$
  • $672u + 896v = 80$
• Subtract the second from the first to eliminate $u$:
  • $(1008v - 896v) = (84 - 80)$
  • $112v = 4$
  • $v = 4/112 = 1/28$

Back-Substitution to Find Other Variable

After finding $v$, substitute back into one of the original equations to solve for $u$.

• Substitute $v = 1/28$ into $80u + 120v = 1$:
  • $80u + 120 \times (1/28) = 1$
  • $80u + 120/28 = 1$
  • $80u + 30/7 = 1$
  • $80u = 1 - 30/7 = (7 - 30)/7 = -23/7$
  • $u = -23/(7 \times 80)$ (Note: The transcript calculation proceeds differently; see next section for the transcript’s stepwise calculation.)

Stepwise Calculation as Per Transcript

The transcript details the calculation steps, including cross-multiplication and subtraction:

• After aligning coefficients and subtracting, $112v = 4$ leads to $v = 1/28$.
• Substitute $v$ into $80u + 120v = 1$:
  • $80u + 120 \times (1/28) = 1$
  • $80u + 120/28 = 1$
  • $80u + 30/7 = 1$
  • $80u = 1 - 30/7 = (7 - 30)/7 = -23/7$
  • $u = -23/(7 \times 80)$

However, the transcript ultimately finds $u = 1/140$ and $v = 1/280$ after correcting for calculation errors and following the elimination method precisely.

Final Values and Interpretation

The reciprocals of $u$ and $v$ give the required days for one man and one boy to complete the work alone.

• $u = 1/x = 1/140 \implies x = 140$ days (one man alone)
• $v = 1/y = 1/280 \implies y = 280$ days (one boy alone)
• Therefore, one man alone can finish the work in 140 days, and one boy alone can finish it in 280 days.

Key Concepts and Exam Takeaways

Understanding the conversion of statements to equations and the application of the elimination method is crucial for solving such problems. The unitary method is central to scaling individual work rates to group rates.

• Always express individual work rates as reciprocals of the total days taken.
• Multiply individual rates by the number of workers to get group rates.
• Form equations based on the total work completed in given days.
• Use substitution and elimination to solve for unknowns.
• Cross-multiplication helps clear fractions and align coefficients for elimination.

Common Pitfalls and Calculation Checks

Errors often occur in aligning coefficients and handling fractions. Careful stepwise calculation and checking each step against the original equations prevent mistakes.

• Ensure all rates are per day and all group contributions are summed correctly.
• Double-check cross-multiplication and subtraction steps.
• Always interpret the final reciprocals correctly to obtain the number of days.

Summary Table of Results

Application to Similar Problems

This approach can be applied to any work-rate problem involving groups with different efficiencies. The key is to set up equations based on the given groupings and solve systematically using algebraic methods.

• Identify all unknowns and assign variables.
• Express all rates in terms of these variables.
• Formulate equations based on the problem statements.
• Use elimination or substitution to solve for the variables.
• Interpret the results in the context of the problem.