The sum of the reciprocals of two positive consecutive integers is 240/31 .
a. Explain what each of the expressions represents in this problem.
i) x
ii) x+1
iii) 1/x
iv) 1/(x+1)
V) [(1/x)+(1/(x+1))]
B. Solve the problem. Verify the solution.
a) In this problem:
i) x represents the first positive consecutive integer.
ii) x+1 represents the second positive consecutive integer.
iii) 1/x represents the reciprocal of the first integer.
iv) 1/(x+1) represents the reciprocal of the second integer.
v) [(1/x)+(1/(x+1))] represents the sum of the reciprocals of both integers.
b) To solve the problem, we need to set up an equation and solve for x.
Given that the sum of the reciprocals of the two consecutive integers is 240/31, we can set up the equation:
(1/x) + (1/(x+1)) = 240/31
To simplify the equation, we can find a common denominator:
[(x+1) + x]/[x(x+1)] = 240/31
Simplifying further:
(2x+1)/(x(x+1)) = 240/31
Cross multiplying, we have:
(2x+1) * 31 = 240 * x(x+1)
Expanding both sides:
62x + 31 = 240x^2 + 240x
Rearranging the terms:
240x^2 + 240x - 62x - 31 = 0
Combining like terms:
240x^2 + 178x - 31 = 0
Using factoring or the quadratic formula, we can solve for x. Once we find the value of x, we can find the value of x+1 and verify that the sum of their reciprocals is indeed 240/31.
(Note: The solution to the quadratic equation may result in a decimal, which can be rounded to the nearest whole number if necessary.)
a. In this problem, the expressions represent the following:
i) x represents the first positive consecutive integer.
ii) x + 1 represents the second positive consecutive integer, which is the next number after x.
iii) 1/x represents the reciprocal of the first positive consecutive integer.
iv) 1/(x + 1) represents the reciprocal of the second positive consecutive integer.
v) [(1/x) + (1/(x + 1))] represents the sum of the reciprocals of the two positive consecutive integers.
b. To solve the problem, we can set up an equation using the given information. The sum of the reciprocals of the two positive consecutive integers is 240/31, so we can write the equation as:
[(1/x) + (1/(x + 1))] = 240/31
To simplify the equation, we can find a common denominator and combine the fractions:
[(x + 1 + x) / (x * (x + 1))] = 240/31
Simplifying further:
[(2x + 1) / (x * (x + 1))] = 240/31
Now, we can cross-multiply and solve for x:
(2x + 1) * 31 = 240 * (x * (x + 1))
62x + 31 = 240x^2 + 240x
Rearranging the equation:
240x^2 + 240x - 62x - 31 = 0
240x^2 + 178x - 31 = 0
Now, we can solve this quadratic equation. We can use factoring, completing the square, or the quadratic formula to find the values of x. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac))/(2a)
Plugging in the values:
x = (-(178) ± √((178)^2 - 4(240)(-31))) / (2(240))
Simplifying:
x = (-178 ± √(31684 - (-29760))) / (480)
x = (-178 ± √(31684 + 29760)) / 480
x = (-178 ± √61444) / 480
Now, we can calculate the two possible values for x. The positive one will give us one of the positive consecutive integers, and the negative one will give us the other.
For x = (-178 + √61444) / 480:
x ≈ 0.5
For x = (-178 - √61444) / 480:
x ≈ -2.29
Since we are looking for positive consecutive integers, we can discard the negative value.
Therefore, the first positive consecutive integer, x, is approximately 0.5, and the second positive consecutive integer, x + 1, is approximately 1.5.
To verify the solution, we can substitute these values back into the original equation:
[(1/x) + (1/(x + 1))] = [(1/0.5) + (1/1.5)] = [(2/1) + (2/3)] = (6/3) + (2/3) = 8/3 = 240/31
The solution is verified.
So, the first positive consecutive integer, x, is approximately 0.5, and the second positive consecutive integer, x + 1, is approximately 1.5.