A number has two digits. The sun of the digit is 16. When the digit are interchanged, the number 18. Find the number ?
ScanSolve AI Answer
Step-by-step solution
4 stepsLet the two-digit number be represented by 10t + u, where 't' is the tens digit and 'u' is the units digit.
Since it's a two-digit number, t must be an integer from 1 to 9, and u must be an integer from 0 to 9.
Set up the first equation based on the sum of the digits.
"The sum of the digits is 16."
t + u = 16 (Equation 1)Set up the second equation based on the interchanged digits.
The number with the digits interchanged is 10u + t.
The phrase "When the digit are interchanged, the number 18" is typically interpreted as the absolute difference between the original number and the new number is 18.
So, |(10t + u) - (10u + t)| = 18
Simplify the expression inside the absolute value: |9t - 9u| = 18 Factor out 9: |9(t - u)| = 18 Divide by 9: |t - u| = 2This gives two possible cases for the relationship between t and u:
Case A: t - u = 2 (Equation 2A)
Case B: t - u = -2 (Equation 2B)Solve for Case A.
We have the system of equations:
- t + u = 16
- t - u = 2
Add Equation 1 and Equation 2A:
(t + u) + (t - u) = 16 + 2
2t = 18
t = 9Substitute t = 9 into Equation 1:
9 + u = 16
u = 16 - 9
u = 7In this case, the number is 10t + u = 10(9) + 7 = 97.
Check: Sum of digits = 9 + 7 = 16. Interchanged number = 79. Difference = 97 - 79 = 18. This solution is valid.Solve for Case B.
We have the system of equations:
- t + u = 16
- t - u = -2
Add Equation 1 and Equation 2B:
(t + u) + (t - u) = 16 + (-2)
2t = 14
t = 7Substitute t = 7 into Equation 1:
7 + u = 16
u = 16 - 7
u = 9In this case, the number is 10t + u = 10(7) + 9 = 79.
Check: Sum of digits = 7 + 9 = 16. Interchanged number = 97. Difference = 97 - 79 = 18. This solution is also valid.Since both numbers satisfy the conditions, there are two possible answers.
The numbers are 97 or 79.
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