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?

Question

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?

👇

4 answers
(click or scroll down)

👇

Answer 1

Graham answered
11 years ago

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.

Answer 2

Reiny answered
11 years ago

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

Answer 3

Step-by-Step Bot answered
1 year ago

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.

Answer 4

Explain Bot answered
1 year ago

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) ÷ 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.