A question of arithmetic regarding erection cost of a structure.

Question

The current erection cost of a structure is Rs. $13,200$. If the labour wages per day increase by $\frac 1 5$ of the current wages and the working hours decrease by $\frac 1 {24}$ of the current period, then the new cost of erection in Rs. is

$(A)\ 16,500$.

$(B)\ 15,180$.

$(C)\ 11,000$.

$(D)\ 10,120$.

I have got an answer different from the answer which is approximately equal to $16528$. Which is very close to option $(A)$. Is it correct? Please help me in this regard.

Thank you very much.

Attempt:

Labour wages increment is proportional to erection cost and working hours decrement is reverse proportional to the erection cost.

So the required erection cost is $13200×\frac{\frac65}{\frac{23}{24}}$ which simplifies to $16528$ (approx.)

Answer 1

Suppose we assuming that the total number of hours needed remain the same, then indeed your solution is correct, that is the answer is

$$13200 \times \frac65 \times \frac{24}{23} \approx 16528$$

which unfortunately not one of the option.

However, suppose for some reason, their efficiency improves and the days needed remains the same, then the answer is $$13200 \times \frac65 \times \frac{23}{24}=15180.$$

This is a badly framed question where the setting is not clear.

Just FYI, $13200 \times \frac65 \times \frac{25}{24}$ gives you the first option but it is not correct.

I am aware of the background of the question which is from GATE, the answer key is $B$, hence they have assumed that the number of days required remain the same.

Answer 2

As you said, the increase in erection cost is proportional to the increase in wages, so the new cost after wages increase is:

$$13,200\cdot \left(1+\frac{1}{5}\right)=15,840$$

Now, the decrease in working hours is actually proportional to the decrease in wages because if the working hours decrease, then since the workers are on an hourly wages, the amount you need to pay the workers also decreases, so the erection cost decreases. Thus, the new cost after wages increase and hours decrease is:

$$13,200\cdot \left(1+\frac{1}{5}\right)\cdot (1-\frac{1}{24})=13,200\cdot\frac{6}{5}\cdot\frac{23}{24} =15,180$$