There are 3 consecutive even integers such that the quotient obtained by dividing twice the largest integer by the smallest integer is three less than three-fifths of the second integer. What are the integers?
Three consecutive even integers are:
2n, 2n+2, 2n+4, where n is an integer.
The rule is thus:
2(2n+4)/2n = (3/5)(2n+2) - 3
Rearranging gives:
0 = 6 n2 - 19n - 20
Find the integer root(s) of this quadratic. Solve for integer n.
Use n to find the three consecutive even numbers.
I would let my 3 consecutive even integers be
x-2, x, and x+2
2(x+2)/(x-2) =3x/5 - 3
expanding and collecting like terms gives us
3x^2 - 31x + 10 = 0
(x-10)(3x - 1) = 0
x = 10 or x = 1/3, but x is an integer, so
x = 10
the integers are 8, 10 , and 12
Let's begin by assigning variables to the three consecutive even integers. Let's call the smallest even integer x, the second one (which is one greater than the first) x + 2, and the largest one (which is one greater than the second) x + 4.
According to the given information, we can write an equation:
2(x + 4) = (3/5)(x + 2) - 3
Now let's solve this equation step by step:
Step 1: Distribute on the right side:
2x + 8 = (3/5)x + (6/5) - 3
Step 2: Combine like terms on the right side:
2x + 8 = (3/5)x + (6/5) - 15/5
Step 3: Simplify the right side:
2x + 8 = (3/5)x - 9/5
Step 4: Move all the terms with x to one side:
2x - (3/5)x = -9/5 - 8
Step 5: Simplify the left side and the right side:
(10/5)x - (3/5)x = -9/5 - 40/5
(7/5)x = -49/5
Step 6: Multiply both sides by the reciprocal of (7/5), which is (5/7):
x = (-49/5)(5/7)
Step 7: Simplify:
x = -49/7
x = -7
Now we can find the other two consecutive even integers:
First even integer = x = -7
Second even integer = x + 2 = -7 + 2 = -5
Third even integer = x + 4 = -7 + 4 = -3
Therefore, the three consecutive even integers are -7, -5, and -3.
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.