when the reciprocal of the larger of the two consecutive even integers is subtracted from 4 times the reciprocal of the smaller, the result is 5/6. find integers.
n , (n+2)
4/n - 1/(n+2) = 5/6
24/n - 6/(n+2) = 5
24(n+2) - 6 n = 5n(n+2)
24 n + 48 - 6 n = 5 n^2 + 10 n
5 n^2 - 8 n - 48 = 0
(5 n + 12)(n-4)
so n = 4
n+2 = 6
Let's start by assigning variables to the two consecutive even integers. Let's call the smaller integer "x" and the larger integer "x + 2".
According to the given information, the reciprocal of the larger integer is subtracted from 4 times the reciprocal of the smaller integer, resulting in 5/6:
4(1/x) - 1/(x + 2) = 5/6
To solve this equation, we need to find a common denominator for the fractions on the left side:
(4(x + 2))/(x(x + 2)) - (1/x) = 5/6
Multiplying the first fraction by (x/x) to get a common denominator:
(4(x + 2))/(x(x + 2)) - (1/x) = 5/6
Simplifying the equation further:
(4x + 8)/(x(x + 2)) - (1/x) = 5/6
Now we can combine the fractions:
(4x + 8 - (x(x + 2)))/(x(x + 2)) = 5/6
Expanding the numerator:
(4x + 8 - (x^2 + 2x))/(x(x + 2)) = 5/6
Simplifying:
(4x + 8 - x^2 - 2x)/(x(x + 2)) = 5/6
Combining like terms:
(2x^2 + 2x - 8)/(x(x + 2)) = 5/6
Now we can cross-multiply to get rid of fractions:
6(2x^2 + 2x - 8) = 5x(x + 2)
Expanding:
12x^2 + 12x - 48 = 5x^2 + 10x
Bringing all terms to one side:
7x^2 + 2x - 48 = 0
Now we have a quadratic equation. Let's solve it by factoring or using the quadratic formula. In this case, factoring does not work, so we will use the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac))/(2a)
In our equation, a = 7, b = 2, and c = -48:
x = (-2 ± sqrt(2^2 - 4(7)(-48)))/(2(7))
Simplifying:
x = (-2 ± sqrt(4 + 1344))/14
x = (-2 ± sqrt(1348))/14
Now we can calculate the two possible values for x:
x ≈ (-2 + sqrt(1348))/14 ≈ 3.87
x ≈ (-2 - sqrt(1348))/14 ≈ -1.37
Since we are dealing with even integers, we choose the nearest even integers to these values:
x ≈ 4
x + 2 ≈ 6
Therefore, the two consecutive even integers are 4 and 6.
To solve this problem, let's break it down step by step:
Step 1: Assign variables
Let's assume the smaller even integer as "x" and the larger even integer as "x + 2".
Step 2: Set up the equation
We know that the reciprocal of a number "n" is 1/n. Based on the given information, we can set up the equation:
4 * (1/x) - (1/(x + 2)) = 5/6
Step 3: Solve the equation
To remove the denominators, we can multiply the entire equation by 6x(x + 2):
(4 * 6x * (x + 2) * (1/x)) - (6x * (x + 2) * (1/(x + 2))) = 5 * 6x(x + 2)
Simplifying further:
(24(x + 2)) - (6x) = 30x(x + 2)
Multiplying:
24x + 48 - 6x = 30x^2 + 60x
Combine like terms and set the equation equal to zero:
30x^2 + 36x - 48 = 0
Step 4: Solve the quadratic equation
This equation can be simplified by dividing every term by 6:
5x^2 + 6x - 8 = 0
Now, we can either factorize or use the quadratic formula to solve for "x". Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
Plugging in the values:
x = (-6 ± √(6^2 - 4 * 5 * -8)) / (2 * 5)
Simplifying:
x = (-6 ± √(36 + 160)) / 10
x = (-6 ± √196) / 10
x = (-6 ± √(4 * 49)) / 10
x = (-6 ± 14) / 10
This gives us two possible solutions for "x":
1) x = (-6 + 14) / 10 = 8 / 10 = 0.8
2) x = (-6 - 14) / 10 = -20 / 10 = -2
Since we are looking for even integers, the second solution (-2) is valid. Thus, the smaller even integer is -2, and the larger even integer is -2 + 2 = 0.
Therefore, the two consecutive even integers are -2 and 0.