A building construction work can be completed by two masons A and B together in 22.5 days. Mason A a

Q: Why can't we just add the days? If A takes $a$ days and B takes $a+24$ days, why isn't the total time $a + (a+24) = 22.5$?

Because time is not additive in this context. Work is additive. If you add time, you imply they are working sequentially, not simultaneously. When they work together, they are both contributing to the same task at the same time, which is why we add their work rates, not their time durations.

Q: Why do we reject the negative root $a = -15$?

In physics and real-world problems, variables like time, mass, and length must be positive. A negative time value is mathematically valid but physically impossible. Since $a$ represents the number of days Mason A takes, it must be a positive value.

Q: Is there a faster way than solving the quadratic equation?

Yes! In competitive exams like JEE, you can often test the given options. If $a=36$, then $B=60$. Check if $\frac{1}{36} + \frac{1}{60} = \frac{1}{22.5}$. This is a great way to verify your answer quickly.

Question

A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Options:

(A) 30 days

(B) 36 days

(C) 24 days

(D) 42 days

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Two masons, $A$ and $B$, are tasked with building a structure. Working together, they finish the job in $22.5$ days.

We are given that Mason $A$ is faster than Mason $B$, completing the job $24$ days earlier. Let $a$ be the number of days Mason $A$ takes to complete the work alone. Consequently, Mason $B$ takes $a + 24$ days to complete the same work.

### The Trap of Additive Time

A common error is to assume that time is additive. If you attempt to set up an equation like $a + (a + 24) = 22.5$, you are incorrectly assuming the masons work sequentially rather than simultaneously.

To solve this correctly, we must shift our perspective to **Work Rate**, which is the fraction of the total job completed in one day. If Mason $A$ takes $a$ days, his rate is $\frac{1}{a}$. Similarly, Mason $B$'s rate is $\frac{1}{a + 24}$.

### The Master Equation

The sum of their individual daily contributions must equal their combined daily work rate. Since they finish the job in $22.5$ days, their combined rate is $\frac{1}{22.5}$.

We simplify the combined rate as follows:
$$ \frac{1}{22.5} = \frac{1}{45/2} = \frac{2}{45} $$

This leads to the fundamental equation:
$$ \frac{1}{a} + \frac{1}{a + 24} = \frac{2}{45} $$

### The Algebraic Odyssey

To solve for $a$, we combine the fractions on the left-hand side:
$$ \frac{(a + 24) + a}{a(a + 24)} = \frac{2a + 24}{a^2 + 24a} = \frac{2}{45} $$

Cross-multiplying to eliminate the fractions yields:
$$ 45(2a + 24) = 2(a^2 + 24a) $$
$$ 90a + 1080 = 2a^2 + 48a $$

Rearranging the terms into a standard quadratic form:
$$ 2a^2 - 42a - 1080 = 0 $$
$$ a^2 - 21a - 540 = 0 $$

### Final Calculation

We factor the quadratic equation by finding two numbers that multiply to $-540$ and add to $-21$. These numbers are $-36$ and $15$:
$$ (a - 36)(a + 15) = 0 $$

This provides two potential roots: $a = 36$ and $a = -15$. Since time cannot be negative, we discard the negative root.

Therefore, the time taken by Mason $A$ to complete the work alone is **$36$ days**.

Animated Solution for Mathematics - Quadratic Equations: A building construction work can be completed by two masons A and B together in 22.5 days. Mason A alone can complete the construction work in 24 days less than mason B alone. Then mason A alone will complete the construction work in :

Select Answer:

Visualized Solution

The Construction Setup

Defining Individual Times

Concept of One-Day Work

Combined Work Rate

Formulating the Equation

Simplifying the Decimal

Combining the Fractions

Cross Multiplication

Expanding the Terms

Forming the Quadratic Equation

Simplifying the Quadratic

Splitting the Middle Term

Finding the Roots

Final Conclusion

The Sigma Insight: Solution of Quadratic Equations

Analyzing the Setup

The Trap of Additive Time

The Master Equation

The Algebraic Odyssey

Final Calculation

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