a two digit no is 4 times the sum of its digits and also 16 more than the product of digits find the no.
Answers
Given :-
◉ A two digit number is 4 times the sum of the digits and also 16 more than the product of digits.
To Find :-
◉ The number
Solution :-
We know, A two digit number is of the form 10m + n where,
- m = ten's digit
- n = one's digit
Let the ten's and one's digit of the number be x and y respectively.
It is given that,
⇒ 4 × Sum of digits = Number
⇒ 4(x + y) = 10x + y
⇒ 4x + 4y = 10x + y
⇒ 6x - 3y = 0
⇒ 2x = y ...(1)
Also,
⇒ 16 + Product of digits = Number
⇒ 16 + xy = 10x + y
Substituting y = 2x from (1), we have
⇒ 16 + x(2x) = 10x + 2x
⇒ 16 + 2x² = 12x
⇒ 2x² - 12x + 16 = 0
⇒ x² - 6x + 8 = 0
Splitting the middle term,
⇒ x² - 4x - 2x + 8 = 0
⇒ x(x - 4) - 2(x - 4) = 0
⇒ (x - 4)(x - 2) = 0
x = 4, 2
Case 1 : [ x = 4 ]
Substituting x = 4 in (1), we get
⇒ 2×4 = y
⇒ y = 8
Now, The number was 10x + y
Substituting values,
⇒ Number = 10×4 + 8
⇒ Number = 48
Case 2 : [ x = 2 ]
Substituting x = 2 in (1), we get
⇒ 2×2 = y
⇒ y = 4
Now, The number we assumed was 10x + y
Substituting values,
⇒ Number = 10×2 + 4
⇒ Number = 24
We got two numbers that satisfies the given conditions. Hence, The numbers can be 48 and 24.
Answer:
Let the required Number be (10x + y).
• According to First Statement :
⇢ Number = 4 times sum of Digits
⇢ 10x + y = 4(x + y)
⇢ 10x + y = 4x + 4y
⇢ 10x - 4x = 4y - y
⇢ 6x = 3y
- Dividing both by 3
⇢ 2x = y ⠀⠀⠀— eq. ( I )
___________________________
• According to Second Statement :
⇢ Number = 16 more than product
⇢ 10x + y = 16 + xy
- Substituting value of y from eq. ( I )
⇢ 10x + 2x = 16 + (x × 2x)
⇢ 12x = 16 + 2x²
⇢ 2x² - 12x + 16 = 0
⇢ x² - 6x + 8 = 0
⇢ x² - 4x - 2x + 8 = 0
⇢ x(x - 4) - 2(x - 4) = 0
⇢ (x - 4)(x - 2) = 0
⇢ x = 4 ⠀⠀or, ⠀⠀x = 2
━━━━━━━━━━━━━━━━━━
◗ When x = 4
⇒ 2x = y ⠀⇒ 2(4) = y ⠀⇒ y = 8
Number : (10x + y) = 48
◗ When x = 2
⇒ 2x = y ⠀⇒ 2(2) = y ⠀⇒ y = 4
Number : (10x + y) = 24
∴ Hence, Number can be 48 or 24.