Solve the following mathematics problems:

Let, the cost price of the goods is 100 Tk.

By this 100 Tk. he actually bought goods @ 15% profit = (100+15% of 100) Tk. = 100+15=115 tk. 

At 10% loss,

Selling price of the goods be= $\left(115-10\% of 115\right)=\left(115-\frac{10\times 115}{100}\right)=115-11.5=103.5$ Tk. 

His overall gain = 103.5-100=3.5 Tk. 

When gain is 3.5 Tk. Cost price = $\frac{100\times 3500}{3.5}$

ans. Real cost of the goods is 100,000 TK.

LCM of 2 and 3 is 6

Work done in first 6 days = $\left(\frac{6}{30}+\frac{3}{15}+\frac{2}{10}\right)=\frac{6+6+6}{30}=\frac{18}{30}=\frac{4}{15}$

Remaining work = $\left(1-\frac{3}{5}\right)=\left(\frac{5-3}{5}\right)=\frac{2}{5}$

Next 3 days work done by them= $\left(\frac{3}{30}+\frac{1}{15}+\frac{1}{10}\right)=\frac{3+2+3}{30}=\frac{8}{30}=\frac{4}{15}$

Remaining work after 9 days = $\left(\frac{2}{5}-\frac{4}{15}\right)=\left(\frac{6-4}{15}\right)=\frac{2}{15}$

In 10^th days, B will help A, thus work done by them after 9 days= $\left(\frac{1}{30}+\frac{1}{15}\right)=\frac{1+2}{30}=\frac{3}{30}=\frac{1}{10}$

Remaining work= $\left(\frac{2}{15}-\frac{1}{10}\right)=\left(\frac{4-3}{30}\right)=\frac{1}{30}$

In 11^th days, A alone can do $\frac{1}{3}$ part of work which is the amount of remaining work.

Total 11 days will be required to complete the whole work. ans.

In 1 hour, one train goes (21-16)= 5 km more than another.

Now, In 1 hour total distance covered by them = (16+21) km = 37 km

If distance difference is 5 km then total distance = 37 km

If distance difference is 60 km then total distance = $\left(\frac{37\times 60}{5}\right)=444$ km 

Distance between Dhaka and Chittagong is 444 km. (ans.)

Let, Shakib borrowed Tk. and Lahib borrowed y Tk.

So, Shakib's principal + Interest of 3 years = Labib's principal + Interest of 2 years

$$\begin{gathered} \Rightarrow x+\frac{x\times 20\times 3}{100}=y+\frac{y\times 20\times 2}{} \\ \Rightarrow \frac{5x+3x}{5}=\frac{5y+2y}{5} \\ \Rightarrow \frac{8x}{5}=\frac{7y}{5} \\ \Rightarrow 8x=7y \\ \Rightarrow \frac{x}{y}=\frac{7}{8} \\ \Rightarrow x : y = 7 : 8 \end{gathered}$$

ans. Ratio of Shakib & Tabib's loan amount is 7. 8.

Let, width of the rectangle be x meter

Length of the rectangle be 3x meter

According to question, 

$$\begin{gathered} 3x\times x=1,200 \\ \Rightarrow 3x^2=1,200 \\ \Rightarrow x^2= \frac{1,200}{3} \\ {x}^2=400 \\ \therefore x= \sqrt{400} \\ x=\sqrt{{\left(20\right)}^2} \\ x=20 \end{gathered}$$

The perimeter of rectangle will be $2\left[\left(3\times 20\right)+20\right]=2\left(60+20\right)=2\times 80=160$ meter

Perimeter of the square, 4a=160 

One side of the square will be 40 meter

Area of the square will be = (40 x 40) = 1,600m² 

Now, length of 1 stone is 50cm = $\frac{50}{100}=0.5 m$ 

Area of one stone will be =$(0.5x0.5)=0.25 m²$

Number of stone required to cover the square = 1,600+0.25=6,400 pieces

Total amount of taka required at 50 tk per stone is $6,400 \times 50 = 3,20,000$ Tk. ans.