There is a two-digit number such that the sum of its digit is 6 while the product of the digits is 1/3 of the original number. Find this number. Explain your solution
The number is 24
the sum of the digits: 2 + 4= 6
the product of the digits is: 2X4= 8
1/3 of 24 is 8.
To find the two-digit number with the given conditions, we can start by letting the tens digit be "x" and the units digit be "y".
From the problem, we know that the sum of the digits is 6. So, we can write the equation: x + y = 6. This equation represents the fact that the sum of the tens digit (x) and the units digit (y) is 6.
We are also given that the product of the digits is 1/3 of the original number. Since the original number is a two-digit number, we can express it as 10x + y. The product of the digits is represented by x * y. So, we can write the equation: x * y = (1/3) * (10x + y).
Now, we have a system of two equations:
1) x + y = 6
2) x * y = (1/3) * (10x + y)
To solve this system of equations, we can use substitution or elimination. Let's use substitution.
From equation 1), we can rearrange it to express x in terms of y: x = 6 - y.
Substituting this expression for x into equation 2), we get:
(6 - y) * y = (1/3) * (10(6 - y) + y)
Simplifying this equation gives us:
6y - y^2 = (1/3)(60 - 9y)
Multiplying by 3 to eliminate fractions:
18y - 3y^2 = 60 - 9y
Rearranging the terms:
3y^2 + 27y - 60 = 0
We now have a quadratic equation in terms of y. To solve this equation, we can factor or use the quadratic formula. Factoring, we get:
(y - 2)(3y + 30) = 0
Setting each factor equal to zero:
y - 2 = 0 or 3y + 30 = 0
Solving these equations gives us:
y = 2 or y = -10
Since we are dealing with a two-digit number, y cannot be negative, so we discard y = -10 as a solution.
Now, substitute y = 2 back into x = 6 - y. We get:
x = 6 - 2 = 4
Therefore, the two-digit number that satisfies the given conditions is 42.