Question

The tens digit of a two digit number is less by 3 than the units digit. If the product of the
two digits is subtracted from the number, the result is 15. Let us write the unit digit of the
number by calculation.​

Answer 1

$\LARGE{ \underline{ \bf{Required \: answer:}}}$

• $\large{ \boxed{ \boxed{ \rm{5 \: or \: 9}}}}$

Step-by-step explanation:

Let us assume that the unit digit be x. Then, the ten's digit will be x - 3 according to first condition given in Q.

Now,

According to 2nd condition, the product of the two digits of the number will be: x(x - 3). And the no. is 10(x - 3) + x.

And when the product of the two digits is subtracted from the number, the result is 15. Which means,

⇒ 10(x - 3) + x - x(x - 3) = 15

⇒ 10x - 30 + x - x² + 3x = 15

⇒ - x² + 10x + 3x + x - 30 - 15 = 0

⇒ - x² + 14x - 45 = 0

⇒ x² - 14x + 45 = 0

Finding the values of x by Middle term factorisation,

⇒ x² - 5x - 9x + 45 = 0

⇒ x(x - 5) - 9(x - 5) = 0

⇒ (x - 9)(x - 5) = 0

From here, we can say that x = 5 or 9

Let x = 5 ⇒ Then, x - 3 = 2

• Then the number is 25.

Let x = 9 ⇒ Then, x - 3 = 6

• Then the number is 69.

Quick check:

The number is 25. Product of digits = 10

• 25 - 10 = 15 (Condition satisfied✓)

The next possibility is 69. Product of digits = 54

• And, 69 - 54 = 15 (Condition satisfied✓)

Answer 2

$\large{\underbrace{\sf{\red{Required\:answer:}}}}$

$\huge{\boxed{\rm{\orange{5\:or\: 9}}}}$

$\large{\underbrace{\sf{\purple{Explanation:}}}}$

Let's assume that the unit digit be x.

Then, the ten's digit will be x - 3.

And original no. is x(x -3) and the interchanged or reserved no. will be 10(x - 3) + x.

It is given that the difference of the products is 15.

Therefore,

:$\implies{\sf{\big[10(x - 3) + x \big] - \big[x(x^{2}-3) \big] = 15}}$

:$\implies{\sf{10x- 30 + x - x^{2}+3x = 15}}$

:$\implies{\sf{-x^{2}+10x+3x+x-30 = 15}}$

:$\implies{\sf{-x^{2}+10x+3x+x-30-15 = 0}}$

:$\implies{\sf{x^{2}-14x+45= 0}}$

• Find the values of x by Split the middle term and factorise and simplify the equation.

:$\implies{\sf{x^{2}-5x-9x+45= 0}}$

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

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

Therefore, we can say that x = 5 or 9.

If x = 5 → Then, x - 3 = 2

Then the number is 25.

If x = 9 → Then, x - 3 = 6

Then the number is 69.

And we are done! :D

$\large{\underbrace{\sf{\pink{Justification:}}}}$

The number is 25. Product of digits = 10

25 - 10 = 15

Hence, verified!

The next possibility is 69. Product of digits = 54

And, 69 - 54 = 15

Hence, verified!

__________________________