A number consists of 3 digits whose sum is 10. The middle digit is equal to the sum of the other two and the number will be increased by 99 if these two digits are reversed. What is the number?

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)