To solve this problem, we need to break it down into smaller steps. Let's consider the given information and figure out how to find the three consecutive even integers.
Step 1: Define the variables
Let's represent the three consecutive even integers as x, (x + 2), and (x + 4). Here, x represents the smallest even integer.
Step 2: Formulate the equation
According to the given information, "the quotient obtained by dividing twice the largest integer by the smallest integer is three less than three-fifths of the second integer." We can write this as an equation:
2(x + 4) รท x = (3/5) * (x + 2) - 3
Step 3: Simplify the equation
Now, let's simplify the equation and solve for x.
2x + 8 = (3/5)(x + 2) - 3
Step 4: Solve the equation
To solve the equation, we can start by clearing the fractions. Multiply both sides of the equation by 5 to get rid of the fraction.
5(2x + 8) = 3(x + 2) - 15
10x + 40 = 3x + 6 - 15
Step 5: Continue to solve the equation
Combine like terms and solve for x.
10x + 40 = 3x - 9
10x - 3x = -9 - 40
7x = -49
x = -49/7
Therefore, x = -7.
Step 6: Find the consecutive even integers
Now that we have found the value of x, we can substitute it back into our initial representation of the three consecutive even integers:
x = -7
x + 2 = -7 + 2 = -5
x + 4 = -7 + 4 = -3
Hence, the three consecutive even integers are -7, -5, and -3.