Finding the Three Numbers in a Sum Equation

Let's denote the three numbers as \( a \) (the smallest number), \( b \) (the middle number), and \( c \) (the largest number). We are given the following information: 1. The sum of the three numbers is 15. 2. The largest number is 4 times the smallest number. 3. The sum of the smallest number and twice the largest number is 18. From this information, we can set up the following equations: 1. \( a + b + c = 15 \) 2. \( c = 4a \) 3. \( a + 2c = 18 \) First, we can use the second equation (\( c = 4a \)) and substitute \( c \) in the first and third equations. Substituting \( c = 4a \) into the first equation: \[ a + b + 4a = 15 \] \[ 5a + b = 15 \] \(\Rightarrow (4a)\) Substituting \( c = 4a \) into the third equation: \[ a + 2(4a) = 18 \] \[ a + 8a = 18 \] \[ 9a = 18 \] \[ a = 2 \] Now we can use \( a = 2 \) to find \( c \) and \( b \): From \( c = 4a \): \[ c = 4 \times 2 = 8 \] Now we can find \( b \) using \( 5a + b = 15 \): \[ 5(2) + b = 15 \] \[ 10 + b = 15 \] \[ b = 5 \] So, the three numbers are: \[ a = 2, \, b = 5, \, c = 8 \] Let's check these values: 1. The sum of the numbers: \( 2 + 5 + 8 = 15 \) (correct) 2. The largest number is 4 times the smallest: \( 8 = 4 \times 2 \) (correct) 3. The sum of the smallest number and twice the largest number: \( 2 + 2 \times 8 = 2 + 16 = 18 \) (correct) Therefore, the three numbers are \( 2 \), \( 5 \), and \( 8 \).