Question

The sum of the digits of a number consisting
of three digits is 12, The middle digit is equal
to half the sum of the other two. If the order
of the digits is reversed, the number is
diminished by 198. Find the number.
​

Answer 1

Answer:

543

Step-by-step explanation:

Let the digits be x, y and z respectively.

So, the number would be 100x + 10y + z.

and the reverse would be 100z + 10y + x

Given sum of digits = x + y + z = 12

Also, Given that, middle digit is equal to half the sum of other two.

⇒ y = (x + z)/2

So, the number becomes

100x + 10y + z

⇒ 100x + 10(x + z)/2 + z

⇒ 100x + 5(x + z) + z

⇒ 100x + 5x + 5z + z

⇒ 105x + 6z

Now, the reverse of the number would be

100z + 10y + x

⇒ 100z + 10(x + z)/2 + x

⇒ 100z + 5x + 5z + x

⇒ 105z + 6x

Now, given that reverse is diminished by 198

⇒ 105x + 6z = 105z + 6x + 198

⇒ 105x - 6x + 6z - 105z = 198

⇒ 99x - 99z = 198

⇒ 99(x - z) = 198

⇒ x - z = 198/99

⇒ x - z = 2

Also, given that x + y + z = 12

but, y = (x + z)/2

⇒ x + (x + z)/2 + z = 12

⇒ 2x/2 + (x + z)/2 + 2z/2 = 12

⇒ (2x + x + z + 2z)/2 = 12

⇒ (3x + 3z)/2 = 12

⇒ 3x + 3z = 12 × 2

⇒  3(x + z) = 24

⇒ x + z = 24/3

⇒ x + z = 8

And, x - z = 2

so, adding the two equations, we get

2x = 10

⇒ x = 5

and, x + z = 8

⇒ 5 + z = 8

⇒ z = 8 - 5

⇒ z = 3

And, y = (x + z)/2

⇒ y = 8/2

⇒ y = 4

So, the number is 100x + 10y + z

⇒ 100(5) + 10(4) + 3

⇒ 500 + 40 + 3

⇒ 543

Answer 2

Let a, b, c = the three digits

then

100a+10b+c = "the number"

Write an equation for each statement simplify as much as possible:

the sum of the digits of a number consisting of three digit is 12.

a + b + c = 12

he middle digit is equal to half of the sum of the other two.

.b = .5(a+c)

if the order of the digit be reversed the number is diminished by 198 find be number

100c + 10b + a = 100a + 10b + c - 198

100c - c + 10b - 10b = 100a - a - 198

99c = 99a - 198

simplify, divide by 99

c = a - 2

In the 2nd equation, b = .5(a+c), replace c with (a-2)

b = .5(a + a - 2)

b = .5(2a-2)

b = a - 1

In the 1st equation replace b with (a-1); replace c with (a-2)

a + (a-1) + (a-2) = 12

3a - 3 = 12

3a = 12 + 3

a = 15/3

a = 5

use the above equations we know that:

b = 4

and

c = 3

543 is the number

Check that in the last statement

345 = 543 - 198

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