The sum of three numbers is 113 . The second number is 3 times the third. The first number is 7 less than the third . What are the numbers?
a = first number
b = second number
c = thirdt number
b = 3 c
a = c - 7
a + b + c = 113
c - 7 + 3 c + c = 113
5 c = 113 + 7
5 c = 120
c = 120 / 5
c = 24
a = c - 7 = 24 - 7 = 17
b = 3 c = 3 * 24 = 72
first number = 17
b = 72
c = 24
17 + 72 + 24 = 113
Let's assume the third number to be x.
According to the given conditions:
The second number is 3 times the third, so the second number would be 3x.
The first number is 7 less than the third, so the first number would be x - 7.
The sum of the three numbers is given as 113, so we can write the equation: (x - 7) + 3x + x = 113.
Combining like terms, we can simplify the equation: 5x - 7 = 113.
Adding 7 to both sides of the equation: 5x = 120.
Dividing both sides of the equation by 5: x = 24.
Therefore, the third number is 24.
The second number is 3 times the third, so the second number is 3 * 24 = 72.
The first number is 7 less than the third, so the first number is 24 - 7 = 17.
Hence, the three numbers are 17, 72, and 24.
To solve this problem, let's first assign variables to the unknown numbers. Let's call the first number "x," the second number "y," and the third number "z."
Based on the information given, we can translate the given statements into equations:
1) The sum of the three numbers is 113:
x + y + z = 113
2) The second number is three times the third:
y = 3z
3) The first number is seven less than the third:
x = z - 7
Now, we have a system of three equations:
x + y + z = 113 (Equation 1)
y = 3z (Equation 2)
x = z - 7 (Equation 3)
We can solve this system of equations using substitution or elimination.
Let's use substitution to solve this system:
1) Substitute Equation 2 into Equations 1 and 3:
x + 3z + z = 113 (Substitute Equation 2 into Equation 1)
x = z - 7 (Equation 3)
Simplify Equation 1:
x + 4z = 113 (Simplified Equation 1)
2) Substitute Equation 3 into Equation 1:
z - 7 + 4z = 113 (Substitute Equation 3 into Equation 1)
Combine like terms:
5z - 7 = 113
3) Solve for z:
5z = 113 + 7
5z = 120
z = 24
So, we have found that the third number (z) is 24.
4) Substitute the value of z into Equation 3:
x = 24 - 7
x = 17
So, we have found that the first number (x) is 17.
5) Substitute the value of z into Equation 2:
y = 3(24)
y = 72
So, we have found that the second number (y) is 72.
Therefore, the three numbers are 17, 72, and 24.