The sum of a two digit number is 6. The number with the digits interchanged is 3 times the tens digit of the original number. Find the original number.
t+u = 6
10u+t = 3t
Find t and u; the original number is 10t+u.
T =5 , u=1
Let's break down the given information step by step:
Step 1: Let's assume the tens digit of the original number is x.
Step 2: As per the given information, the units digit of the original number must be (6 - x) because the sum of the digits is 6.
Step 3: Now, let's consider the number with the digits interchanged, which is the units digit followed by the tens digit. The new number would be [(6 - x) * 10] + x.
Step 4: According to the given information, this new number is 3 times the tens digit of the original number, so we can write the equation as follows:
[(6 - x) * 10] + x = 3x
Step 5: Let's solve the equation to find the value of x:
60 - 10x + x = 3x
60 - 2x = 3x
60 = 5x
x = 12
Step 6: We have found that the tens digit of the original number is 12, so the units digit is (6 - 12) = -6.
Step 7: Since a number cannot have a negative digit, we can conclude that there is no two-digit number that satisfies the given conditions.
Therefore, there is no solution for the original number based on the given information.
To solve this problem, we can use algebraic equations.
Let's assume the tens digit of the original number is 'x' and the units digit is 'y'. According to the given information, we have two conditions:
1. The sum of the digits is 6:
Equation 1: x + y = 6
2. The number with the digits interchanged is 3 times the tens digit of the original number:
Equation 2: 10y + x = 3x
Now, let's solve these equations to find the values of x and y.
From Equation 1, we can express y in terms of x:
y = 6 - x
Substituting this value of y in Equation 2:
10(6 - x) + x = 3x
60 - 10x + x = 3x
60 - 7x = 3x
Now, solve for x:
60 = 10x
x = 6
Substituting the value of x in Equation 1:
6 + y = 6
y = 0
Therefore, the original number is 60.