A two digit number is six times
the sun of its digits The Number
obtained by interchanging the
digits is less by 9 than the original
number. Find the original no.
ho
Answers
Solution :-
Let the ones and tens digits of a number be x and y respectively.
Case I : A two digit number is six times
the sum of its digits.
=> 6(x + y) = 10y + x
=> 6x + 6y = 10y + x
=> 6x - x = 10y - 6y
=> 5x = 4y
=> x = 4y/5 _____(i)
Case II : The Number obtained by interchanging the digits is less by 9 than the original number.
=> 10x + y + 9 = 10y + x
=> 10x - x + y - 10y = - 9
=> 9x - 9y = - 9
=> 9(x - y) = - 9
=> x - y = - 1
=> 4y/5 - y = - 1 [from equation (i)]
=> (4y - 5y)/5 = - 1
=> - y = - 5
=> y = 5
Substituting the value of y in equation (i),
x = (4 × 5)/5 = 4
Therefore,
x = 4 ; y = 5
10y + x = 10 × 5 + 4 = 54
Hence,
The original number is 54
• Let one's digit be M and ten's digit be N.
• Original number = 10N + M
» A two digit number is six times the sum of its digits i.e. (M + N)
A.T.Q.
→ 6(M + N) = 10N + M
→ 6M + 6N = 10N + M
→ 6M - M + 6N - 10N = 0
→ 5M - 4N = 0
→ 5M = 4N
→ M = 4N/5 __________ (eq 1)
_____________________________
» The Number obtained by interchanging the digits is less by 9 than the original number.
• Interchanged number = 10M + N
A.T.Q.
→ 10M + N = 10N + M - 9
→ 10M - M + N - 10N = - 9
→ 9M - 9N = - 9
→ M - N = - 1
→ (4N/5) - N = -1
→ (4N - 5N)/5 = - 1
→ -N/5 = - 1
→ - N = - 5
→ N = 5
Put value of N in (eq 1)
→ M = (4 × 5)/5
→ M = 4
_____________________________
From above calculations we have M = 4 and N = 5
Also, we have original number = 10N + M
→ 10(5) + 4
→ 50 + 4
→ 54
____________________________
54 is the original number.
________ [ ANSWER ]
____________________________
✡ VERIFICATION :
From above calculations we have M = 4 and N = 5
Also, we have equation : 10M + N = 10N + M - 9
Put value of M and N in above equation.
=> 10(4) + 5 = 10(5) + 4 - 9
=> 40 + 5 = 50 + 4 - 9
=> 45 = 45
__________________________