Finding the Three Numbers Given Their Relationship

Let's denote the three numbers as \(a\) (the smallest number), \(b\) (the middle number), and \(c\) (the largest number). We can set up the following equations based on the information given: 1. The sum of the three numbers is 15: \[ a + b + c = 15 \] 2. The largest number is 4 times the smallest: \[ c = 4a \] 3. The sum of the smallest number and twice the largest number is 18: \[ a + 2c = 18 \] We now have a system of three equations: 1. \( a + b + c = 15 \) 2. \( c = 4a \) 3. \( a + 2c = 18 \) First, let's substitute the value of \(c\) from the second equation into the other equations. From equation 2, \( c = 4a \): Substitute \( c = 4a \) into equation 3: \[ a + 2(4a) = 18 \] \[ a + 8a = 18 \] \[ 9a = 18 \] \[ a = 2 \] Now, using \( a = 2 \), substitute into equation 2 to find \( c \): \[ c = 4a = 4(2) = 8 \] We now know \( a = 2 \) and \( c = 8 \). Substitute these values into equation 1 to find \( b \): \[ a + b + c = 15 \] \[ 2 + b + 8 = 15 \] \[ b + 10 = 15 \] \[ b = 5 \] So, the three numbers are: \[ a = 2, \, b = 5, \, c = 8 \]