Question

A certain number between 10 and 100 is 8 times the sum of its digits, and if 45 be subtracted for
will be reversed. Find the number.
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Answer 1

Given,

• A certain number between 10 and 100 is 8 times the sum of its digits .
• If 45 be subtracted for will be reversed.

To Find,

• Two Digit Number

Solution :

$\implies$ Suppose the digit at the ten's place be x

And, Suppose the digit at the one's place be y

Therefore,

• Sum of its digit = x + y
• Two Digit Number = 10x + y
• Reversed Number = 10y + x

$\mapsto \underline{\sf{\pink{According \ to \ the \ First \ Condition :}}}$

• A certain number between 10 and 100 is 8 times the sum of its digits.

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

$\implies \sf{10x + y = 8x + 8y}$

$\implies \sf{10x - 8x = 8y - y}$

$\implies \boxed{\sf{2x = 7y}}$      1) Equation

$\mapsto \underline{\sf{\pink{According \ to \ the \ Second \ Condition:}}}$

• If 45 be subtracted from Two digit number then its will be Reversed .

$\implies \sf{10x + y - 45 = 10y + x}$

$\implies \sf{10x - x - 45 = 10y - y}$

$\implies \sf{9x - 45 = 9y}$

$\implies \sf{9x - 9y = 45}$

$\implies \sf{9(x - y ) = 45}$

$\implies \sf{x - y = \dfrac{45}{9}}$

$\implies \sf{x - y = 5}$

$\implies \boxed{\sf{x = 5 + y}}$      2) Equation

Now Put the Value of x in First Equation :

$\implies \sf{2x = 7y}$

$\implies \sf{2(5 + y) = 7y}$

$\implies \sf{10 + 2y = 7y}$

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

$\implies \sf{5y = 10}$

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

$\implies \boxed{\sf{y = 2}}$

Now Put the Value of y in Second Equation :-

$\implies \sf{x = 5 + y}$

$\implies \sf{x = 5 + 2}$

$\implies \boxed{\sf{x = 7}}$

Therefore,

$\boxed{\bold{\red{Two \ Digit \ Number = 10x + y = 10(7) + 2 = 72 }}}$

$\rule{200}2$

Answer 2

Given

Number b/w 10 - 100 is 8 times sum of digits.

If 45 substracted from original number, digits will be reversed.

To find

Original number

Solution

Here, number b/w 10 - 100 refers to such a number which consists of 2 digits.

Let the digit at unit's place be y & digit at ten's place be z.

⇾ Original number = y + 10z

⇾ Reversed number = z + 10y

According to 1st case :

⇒ y + 10z = 8(y + z)

⇒ y + 10z = 8y + 8z

⇒ y + 10z - 8y - 8z = 0

⇒ 2z - 7y = 0

⇒ 2z = 7y

⇒ z = 7y/2 ..(1)

According to 2nd Case :

⇾ y + 10z - 45 = z + 10y

⇾ y + 10z - z - 10y = 45

⇾ 9z - 9y = 45

⇾ 9(z - y) = 45

⇾ z - y = 45/9

⇾ z - y = 5

• Putting value from (1)

⇾ 7y/2 - y = 5

⇾ (7y - 2y)/2 = 5

⇾ 5y = 10

⇾ y = 10/5

⇾ y = 2

Now putting this value in (1)

⟼ z = 7(2)/2

⟼ z = 14/2

⟼ z = 7

Now finding the original number :

➺ Number = 2 + 10(7)

➺ Number = 2 + 70

➺ Number = 72

Therefore,

Original number = 72