Question

A certain number between 10 and 100 is 8 Times the sum of its digits,and if 45 be subtracted from it the digits will be reversed.find the number.

Answer 1

Answer:

Given :-

• A certain number between 10 and 100 is 8 times the sum of its digits, and if 45 be subtracted from it, the digits will be reversed.

To Find :-

• What is the number.

Solution :-

Let,

$\mapsto$ Digits at unit's place = x

$\mapsto$ Digits at ten's place = y

Then, the original number will be :

$\leadsto \sf\bold{\pink{10x + y}}$

According to the question,

$\implies \sf 10x + y =\: 8(x + y)$

$\implies \sf 10x + y =\: 8x + 8y$

$\implies \sf 10x - 8x =\: 8y - y$

$\implies \sf 2x =\: 7y$

$\implies \sf\bold{\purple{x =\: \dfrac{7y}{2}\: ------\: (Equation\: No\: 1)}}$

Again,

$\implies \sf 10x + y - 45 =\: 10y + x$

$\implies \sf 10x + y - 45 - 10y - x =\: 0$

$\implies \sf 10x - x + y - 10y - 45 =\: 0$

$\implies \sf 9x - 9y - 45 =\: 0$

$\implies \sf 9(x - y - 5) =\: 0$

$\implies \sf (x - y - 5) =\: 0 \times 9$

$\implies \sf\bold{\purple{x - y - 5 =\: 0\: ------\: (Equation\: No\: 2)}}$

Now, by putting the value of x in the equation no 2 we get,

$\implies \sf \dfrac{7y}{2} - y - 5 =\: 0$

$\implies \sf \dfrac{7y - 2y - 10}{2} =\: 0$

By doing cross multiplication we get,

$\implies \sf 7y - 2y - 10 =\: 2(0)$

$\implies \sf 7y - 2y - 10 =\: 0$

$\implies \sf 7y - 2y =\: 10$

$\implies \sf 5y =\: 10$

$\implies \sf y =\: \dfrac{\cancel{10}}{\cancel{5}}$

$\implies \sf \bold{\green{y =\: 2}}$

Now, by putting the value of y in the equation no 2 we get,

$\implies \sf x - y - 5 =\: 0$

$\implies \sf x - 2 - 5 =\: 0$

$\implies \sf x - 7 =\: 0$

$\implies \sf\bold{\green{x =\: 7}}$

Hence, the required original number is :

$\longrightarrow \sf Original\: number =\: 10x + y$

$\longrightarrow \sf Original\: number =\: 10(7) + 2$

$\longrightarrow \sf Original\: number =\: 70 + 2$

$\longrightarrow \sf\bold{\red{Original\: number =\: 72}}$

$\therefore$ The original number is 72.

Answer 2

Solution -

• A certain number between 10 and 100 is chosen .

• It is 8 times the sum of its digits.

• If 45 is subtracted from the digits, the digits get reversed .

We have to find the number .

It is clear that this is a two digit number. Let this number be ab , 9 ≥ a≥ 1 and 9 ≥ b ≥ 0

It is 8 times the sum of its digits .

> 10a + b = 8(a+b)

> 10a + b = 8a + 8b

> 2a = 7b

Now , ab - 45 = ba

> ab - ba = 45

> 10a + b - 10b - a = 45

> 9a - 9b = 45

> a - b = 5

> a = b + 5

> 2a = 2b + 10

But , 2a = 7b for the first eqn

> 7b = 2b + 10

> 5b = 10

> b = 2

> a = b + 5 = 2+5 = 7

Answer - The required number is 72.

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