A two-digit number is obtained by either multiplying the sum of the digits by 8 andthen subtracting 5 or by multiplying the difference of the digits by 16 and thenadding 3. Find the number.
Answers
Let ten's digit number be M and one's digit be N.
A two-digit number is obtained by either multiplying the sum of the digits by 8.
Two digit number = 10M + N
Sum of digits = M + N
According to question,
⇒ 10M + N = 8(M + N) - 5
⇒ 10M + N = 8M + 8N - 5
⇒ 10M - 8M + N - 8N = - 5
⇒ 2M - 7N = - 5 ___ (eq 1)
The difference of the digits by 16 and then by adding 3.
⇒ 10M + N = 16(M - N) + 3
⇒ 10M + N = 16M - 16N + 3
⇒ 10M - 16M + N + 16N = 3
⇒ - 6M + 17N = 3 ___ (eq 2)
Now, Multiply (eq 1) with 3
⇒ 2M - 7N = - 5 (×3)
⇒ 6M - 21N = - 15 ___ (eq 3)
Add (eq 2) from (eq3)
⇒ - 6M + 17N + 6M - 21N = 3 + (-15)
⇒ - 4N = - 12
⇒ 4N = 12
⇒ N = 3
Substitute value of N in (eq 1)
⇒ 2M - 7(3) = - 5
⇒ 2M - 21 = - 5
⇒ 2M = 16
⇒ M = 8
So,
Number :- 10M + N
⇒ 10(8) + 3
⇒ 80 + 3
⇒ 83
Given question:-
A two-digit number is obtained by either multiplying the sum of the digits by 8 andthen subtracting 5 or by multiplying the difference of the digits by 16 and thenadding 3.
To Find:-
Find the number.
Hence..
Let x be the 10th number and let y be the 1st number.
•♦• 10x + y = 2 digit number.
•♦• x + y = sum of digit.
So,
= 10x + y = 8 ( x + y ) - 5
= 10x 3 y = 8x + 8y - 5
= 10x - 8x + y - 8y = -5
= 2x - 7y = -5 – Equation (1)
Then,
= 10x + y = 16 (x + y ) - 3
= 10x + y = 16x - 16y + 3
= 10x - 16x + y + 16y = 3
= -6x + 17y = 3 – Equation (2)
Now , we have to multiply Equation (1) with number 3.
= 2x - 7y = -5 multiply ( × 3)
= 6x - 21y = -15 – Equation (3)
Hence, now adding Equation (2) from Equation (3)
= - 6x + 17y + 6x - 21y = 3 + ( - 15)
= 4y = -12
= 4y = 12
= (y = 3)
Substituting values of y in Equation (1)
= 2x - 1 ( 3 ) = -5
= 2x - 21 = -5
= 2x = 16
Hence,
10x + y is the number.
So,
➟ 10 ( 8 ) + 3
➟ 80 + 3
➟ 83