Mathematics

একজন একটি দ্রব্য 25% লাভে বিক্রয় করে। সে যদি দ্রব্যটি 20% কম দাম দিয়ে কিনতো এবং তা যদি 10.50 টাকা কমে নিত্রনা করতো, তবে সে ঐ দ্রব্যটি বিক্রয় করে 30% লাভ করতো।  দ্রব্যটির ক্রয়মূল্য কত?

Let, cost price of the article is Tk. x 

At, 25% profit. Then first selling price is Tk. $\left(125\% of x\right)=TK.\frac{125x}{100}=TK. \frac{5x}{4}$

At, 20% less price, New cost price = Tk. $\left(80\% of x\right)= TK. \frac{80x}{100}=TK.\frac{4x}{5}$

At 30% profit, New selling price = TK. $\left(130\% of \frac{4x}{5}\right)=Tk.\frac{130\times 4x}{5\times 100}=Tk.\frac{26x}{25}$

According to question,

$$\begin{gathered} \frac{5x}{4}-\frac{26x}{25}=10.50 \\ \Rightarrow \frac{125x-104x}{100}=10.50 \\ \Rightarrow 21x=10.50\times 100 \\ \therefore x=50 \end{gathered}$$

So, cost price of the article is Tk. 50

A ও B একটি কাজ 18 দিনে, B ও C 24 দিনে এবং A ও C 36 দিনে করতে পারে। ঐ কাজটি তারা একত্রে আলাদাভাবে কয়দিনে করবে?

(A+B) take 18 days to complete= 1 work

(A + B) take 1 day to complete = ($\frac{1}{18}$) part of work

Similarly, (B+C) take 1 day to complete =$\left(\frac{1}{24}\right)$ part of work

And (A+C) take 1 day to complete = $\left(\frac{1}{36}\right)$ part of work

On adding, [(A+B)+(B+C)+(A+C)] take 1 day to complete

=$\left(\frac{1}{18}+\frac{1}{24}+\frac{1}{36}\right)=\left(\frac{4+3+2}{72}\right)=\left(\frac{9}{72}\right)$ Part of work

2(A+B+C) take 1 day to complete =$\left(\frac{1}{8}\right)$ Part of work

$\Rightarrow (A+B+C) take 1 day to complete =\left(\frac{1}{8\times 2}\right)=\left(\frac{1}{16}\right) Part of work$

$\frac{1}{16}$ Part of work is done by (A+B+C) in 1 day

$\therefore$ 1 or complete part of work is done by (A+B+C) in =$\frac{1\times 16}{1}=16$ days

Only A can do the work (A+B+C)-(B+C)

Or, $\frac{1}{16}-\frac{1}{36}=\frac{5}{144}$ part in 1 day.

So, A can complete the whole work in 48 days

Now, Only B can do the work (A+B+C)-(A+C)

$or, \frac{1}{16}-\frac{1}{18}=\frac{5}{144}$ 

So, B can complete the whole work in 28.8 or 29 days.

And only C can do the work (A+B+C)-(A+B)

$or, \frac{1}{16}-\frac{1}{18}=\frac{1}{144}$ part in I day. 

So, C can complete the whole work in 144 days

ans. Working together A, B and C finish the work in 16 days and separately A, B and C finish the we in 48 days, 29 days & 144 days respectively