Question

the sum of the digits of a number consisting of 3 digits is 12 the middle digit is equal to half of the sum of the other two of the order of the digits is reversed the number is diminished by 198 find the number​

Answer 1

Answer:

The required three digit number is 543.

Step-by-step-explanation:

Let the digit at hundreds place be x.

The digit at tens place be y.

The digit at units place be z.

∴ The three digit number = xyz

⇒ Original number = 100x + 10y + z

From the first condition,

The sum of the digits of the three digit number is 12.

∴ x + y + z = 12 - - - ( 1 )

From the second condition,

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

∴ y = 1 / 2 * ( x + z )

⇒ y = ( x + z ) / 2 - - - ( 2 )

By substituting this in equation ( 1 ), we get,

x + y + z = 12 - - - ( 1 )

⇒ x + [ ( x + z ) / 2 ] + z = 12

⇒ ( 2x + x + z + 2z ) / 2 = 12

⇒ 3x + 3z = 12 * 2

⇒ x + z = 4 * 2 - - - [ Dividing each term by 3 ]

⇒ x + z = 8

⇒ x = 8 - z

⇒ x = - z + 8 - - - ( 3 )

Now,

By substituting this in equation ( 2 ), we get,

y = ( x + z ) / 2 - - - ( 2 )

⇒ y = ( - z + 8 + z ) / 2

⇒ y = 8 / 2

⇒ y = 4 - - - ( 4 )

Now,

If the order of digits is reversed, new number is less than the original by 198.

∴ New number = zyx

⇒ New number = 100z + 10y + x

Now, from the given condition,

Original number - New number = 198

⇒ 100x + 10y + z - ( 100z + 10y + x ) = 198

⇒ 100x + 10y + z - 100z - 10y - x = 198

⇒ 100x - x + 10y - 10y + z - 100z = 198

⇒ 99x + 0 - 99z = 198

⇒ 99x - 99z = 198

⇒ 99 ( x - z ) = 198

⇒ x - z = 198 ÷ 99

⇒ x - z = 2

⇒ - z + 8 - z = 2 - - - [ From ( 3 ) ]

⇒ - z - z = 2 - 8

⇒ - 2z = - 6

⇒ 2z = 6

⇒ z = 6 ÷ 2

⇒ z = 3

By substituting z = 3 in equation ( 3 ), we get,

x = - z + 8 - - - ( 3 )

⇒ x = - 3 + 8

⇒ x = 5

Now,

The three digit number = 100x + 10y + z

⇒ The three digit number = 100 * 5 + 10 * 4 + 3

⇒ The three digit number = 500 + 40 + 3

⇒ The three digit number = 540 + 3

⇒ The three digit number = 543

∴ The required three digit number is 543.

Answer 2

Answer:

The number is 543

Step-by-step explanation:

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

The 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

then using 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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