Question

The sum of the digits of a number consistingof three digits is 12, The middle digit is equalto half the sum of the other two. If the orderof the digits is reversed, the number isdiminished by 198. Find the number.​

Answer 1

Answer

The original number is 543

Given

• The sum of the digits of a 3digit number is 12
• The middle digit is equal to half the sum of other two
• When the order of the digit is reversed , the number is diminished by 198

Solution

Let us consider the digits be x , y and z respectively

According to question

$\sf\implies x + y + z = 12 \longrightarrow (1)$

$\sf\implies y = \dfrac{1}{2}(x + z) \\\\ \sf\implies 2y = x + z \\\\ \sf\implies x + z = 2y \longrightarrow(2)$

$\sf\implies 100z + 10y + x = 100x +10y +z - 198 \\\\ \sf\implies 99z - 99x = -198 \\\\ \sf\implies 99 (z - x) = -198 \\\\ \sf\implies z - x = -2 \\\\ \sf\implies x - z = 2 \longrightarrow (3)$

Now using the value of (2) in (1)

$\sf \implies 2y + y = 12 \\\\ \sf\implies 3y = 12 \\\\ \sf\implies y = 4$

Again using the value of 'y' in (2)

$\sf\implies x + z = 2\times 4 \\\\ \sf\implies x + z = 8 \longrightarrow(4)$

Adding (3) and (4) we have ,

$\sf\implies x + z + x - z = 8+2 \\\\ \sf\implies 2x = 10 \\\\ \sf\implies x = 5$

And using the value of 'x' in (4)

$\sf\implies 5 + z = 8 \\\\ \sf\implies z = 8-5 \\\\ \sf\implies z =3$

Therefore, the number is

➡ 100×5 + 10×4 + 3

➡ 543

Verification

• 5 + 4 + 3 = 12
• 4 = 1/2×(5+3)
• 543 - 345 = 198