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
First digit = x
Second digit = y
x + y = 6
x*y = (1/3)*(10x + y)
How about 24? (x = 2; y = 4)
Smallest 2digistnumber
Let's assume the two-digit number is represented by AB, where A is the tens digit and B is the units digit.
Given that the sum of the digits is 6, we can write the equation A + B = 6. --(Equation 1)
Also, the product of the digits is 1/3 of the original number, which can be written as AB = (1/3)(10A + B). --(Equation 2)
To solve this system of equations, we can substitute Equation 1 into Equation 2:
A + B = 6 -> B = 6 - A
Plugging this into Equation 2:
A(6 - A) = (1/3)(10A + (6 - A))
Simplifying this equation:
6A - A^2 = (1/3)(9A + 6)
Multiplying both sides by 3:
18A - 3A^2 = 9A + 6
Rearranging and simplifying:
3A^2 - 9A - 6 = 0
Dividing by 3:
A^2 - 3A - 2 = 0
Factoring the equation:
(A - 2)(A + 1) = 0
Setting each factor to zero:
A - 2 = 0 or A + 1 = 0
Solving for A:
A = 2 or A = -1
Since A is a digit, we discard the negative value A = -1.
Now substitute A = 2 back into our original equation to find B:
B = 6 - A
B = 6 - 2
B = 4
Therefore, the two-digit number AB is 24, which satisfies the given conditions: the sum of its digits is 6, and the product of its digits is 1/3 of the original number.
To find the two-digit number, we need to solve the problem step by step. Let's break down the problem into smaller parts and find the solution.
Step 1: Represent the two-digit number
Let's assume the tens digit of the number is x and the units digit is y. Hence, the number can be represented as 10x + y.
Step 2: Sum of the digits
According to the problem, the sum of the digits is 6. So, we can write it as:
x + y = 6
Step 3: Product of the digits
The problem states that the product of the digits is 1/3 of the original number. Therefore, we can write it as:
xy = (1/3) * (10x + y)
Step 4: Simplify the equations
To make it easier, let's solve equations (1) and (2) step by step:
From equation (1), we can find x in terms of y:
x = 6 - y
Substituting this value of x into equation (2):
(6 - y) * y = (1/3) * (10 * (6 - y) + y)
Step 5: Solve the equation
Simplify the equation obtained in step 4 and solve it to find the value of y.
6y - y^2 = (20 - 2y)/3
Multiply the entire equation by 3 to eliminate the fraction:
18y - 3y^2 = 20 - 2y
Rearrange the equation:
3y^2 - 16y + 20 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula.
Step 6: Find the values of y
By factoring or quadratic formula, we can obtain two possible values for y. Let's assume the two possible values as y1 and y2.
Step 7: Find the corresponding values of x
Using equation (1), calculate the corresponding values of x for each value of y obtained in step 6.
Step 8: Find the two-digit numbers
Combine the values of x and y from step 7 to find the two-digit numbers in the form 10x + y.
Step 9: Check the condition
Verify if any of the two-digit numbers obtained satisfies the condition given in the problem.
Step 10: Find the final solution
If any of the two-digit numbers satisfy the condition, that will be the solution to the problem.