The sum of the first two digits of a three digit number is equal to the third digit. If the digits are reversed, the number so obtained is 198 more than the original number. What are the possible original numbers?

প্রশ্নে বলা হচ্ছে, তিন সংখ্যা বিশিষ্ট একটি সংখ্যার তিনটির যোগফল 10. মাঝের অঙ্কটি অন্য দুটি সংখ্যার সমষ্টির সমান এবং ঐ দুটি সংখ্যা স্থান বিনিময় করলে যে সংখ্যাটি পাওয়া যায় তা মূল সংখ্যার চেয়ে 99 বেশি হলে সংখ্যাটি কত?

$$\begin{gathered} Let, the Hundredth digit is x, \\ Tenth digits is y \\ And Unit digits is z \\ \therefore The number is (100x+10y+z) \\ According to question, \\ x+y+z=10................ \left(i\right) \\ y=x+z............... \left(ii\right) \\ 100x+10y+z+99 =100z+10y+x................ \left(iii\right) \end{gathered}$$

From equation (iii) 

$$\begin{gathered} 100x+10y+z+99=100z+10y+x \\ \Rightarrow 100x+z+99+10y-10y=100z+x \\ \Rightarrow 100x+z+99=100z+x \\ \Rightarrow 100z-z+x-100z=99 \\ \Rightarrow 99z-99x=99 \\ \Rightarrow 99\left(z-x\right)=99 \\ \Rightarrow \left(z-x\right)=\frac{99}{99} \\ \Rightarrow z-x=1 \\ \therefore z=x+1........................\left(iv\right) \end{gathered}$$

putting the value of z in equation (ii) 

$$\begin{gathered} y=z+x \\ \Rightarrow y=x+x+1 \\ \Rightarrow y=2x+1................\left(v\right) \end{gathered}$$

putting the value of y & z in equation (i) 

$$\begin{gathered} x+y+z=20 \\ \Rightarrow x+2x+1+x+1=10 \\ \Rightarrow 4x+2=10 \\ \Rightarrow 4x=10-2 \\ \Rightarrow 4x=8 \\ \Rightarrow x=\frac{8}{4} \\ \therefore x=2 \end{gathered}$$

putting the value of x in equation (v) 

$$\begin{gathered} y=2x+1 \\ \Rightarrow y=2\times 2+1 \\ \Rightarrow y=4+1 \\ \therefore y=5 \end{gathered}$$

putting the value of x in equation (iv) 

$$\begin{gathered} z=x+1 \\ \Rightarrow z=2+1 \\ \therefore z=3 \end{gathered}$$

The number will be $$\begin{gathered} = 100x+10y+z \\ =\left(100\times 2\right)+\left(10\times 5\right)+\left(3\times 1\right) \\ =200+50+3 \\ =253 \end{gathered}$$

প্রশ্নে বলা হচ্ছে, দুই অঙ্ক বিশিষ্ট কোন সংখ্যা তার অঙ্ক দুটির যোগফলের 4 গুণ। অঙ্ক দুটিকে স্থান পরিবর্তন করলে যে সংখ্যা পাওয়া যায় তা মূল সংখ্যার চেয়ে 18 বেশি। মূল সংখ্যাটি কত?

Let, Tenth digit is x and unit digit is y

∴ The number = (10x + y)

According to first condition, $4(x + y) = 10x + y . . . . . . . . . . . . \left(i\right)$

And second condition, $(10x + y) + 18 = 10y + x . . . . . . . . . . . . . \left(ii\right)$

From equation (ii), $10x + y + 18 = 10y + x$

$$\begin{gathered} \Rightarrow 9x-9y=-18 \\ \Rightarrow 9y-9x=18 [Multiplying by - 1] \\ \Rightarrow 9(y-x) = 18 \\ \Rightarrow (y-x) = \frac{18}{9} = 2 \\ \Rightarrow y-x = 2 \\ \therefore y = x+2 . . . . . . . . . \left(iii\right) \end{gathered}$$

From equation (i), we get $4(x + y) = 10x + y$

$$\begin{gathered} \Rightarrow 4x+4y = 10x + y \\ \Rightarrow 3y = 6x \\ \Rightarrow y = 2x \\ \Rightarrow x+2 = 2x [Putting the value of y] \\ \therefore x = 2 \end{gathered}$$

Now, putting the value of x in equation (iii) we get

$$\begin{gathered} y= x+2 \\ \Rightarrow y = 2+2 = 4 \\ \therefore y = 4 \\ The number is (10x + y) = (10 \times 2)+4=20+4=24. ans. \end{gathered}$$

Let, unit digit is y and tenth digit is x

$\therefore$The number is (10x + y)

From 1st condition, 10x+y= 6(x + y).......(i)

And from 2nd condition, (10x + y) - (10y + x) = 9 ............... (ii)

Now, from equation (i)

10x + y = 6x + 6y

=4x = 5y

$\therefore$x = $\frac{5y}{4}$…………………(iii)

Now, putting the value of x in equation (ii)

10x + y - (10y + x) = 9

= 10x + y - 10y - x = 9

=9x - 9y = 9

=(x - y) $\times$ 9 = 9

=x - y = 1

=$\frac{5y}{4}$ - y = 1

= $\frac{5y-4y}{4}$ = 1

$\therefore$y = 4

Putting y = 4 in equation

(iii), we get

$\therefore$x$\frac{5\times 4}{4}=5$

$\therefore$The original number is 10x + y = (10 $\times$ 5) + 4 = 50 + 4 = 54 (answer)

Let the unit digit is b and tenth dight is a So the number = 10a + b

According to first condition, a + b = 16 ........(i)

According to second condition, 

10a + b + 18 = 10b + a

= 10b + a - 10a - b = 18

= 9b - 9a = 18

= b - a = 2 ........ (ii)

By adding equation (i) and (ii),

a + b + b - a = 16 + 2

= 2b = 18

= b = 9

From equation (i),

a + b = 16

= a + 9 = 16

= a = 7

The original number 

= 10a + b

= $10 \times 7$ + 9

= 79 (Answer)