The sum of the reciprocals of two consecutive positive integers is 17/12. Write an equation that can be used to find the two integers. What are the integers?
PLEASE HELP ME
surely you know
(a) what a reciprocal is
(b) what a sum is
(c) what consecutive integers are
So, if the smaller integer is x, the equation is just
1/x + 1/(x+1) = 17/12
Unfortunately, I detect a typo. Fix it, and it will come out a lot better.
Let's assume the two consecutive positive integers are x and x+1.
The reciprocal of x is 1/x, and the reciprocal of x+1 is 1/(x+1).
According to the given information, the sum of the reciprocals is 17/12:
1/x + 1/(x+1) = 17/12
To solve this equation, we can find a common denominator, which in this case is 12x(x+1):
12(x+1) + 12x = 17x(x+1)
Now we can simplify and solve for x:
12x + 12 + 12x = 17x^2 + 17x
24x + 12 = 17x^2 + 17x
Rearranging the equation:
0 = 17x^2 + 17x - 24x - 12
0 = 17x^2 - 7x - 12
Now we can factor the quadratic equation:
0 = (17x + 9)(x - 4)
Setting each factor equal to zero:
17x + 9 = 0 or x - 4 = 0
Solving for x:
17x = -9 or x = 4
x = -9/17 or x = 4
Since we are looking for positive integers, the solution is x = 4.
Therefore, the two consecutive positive integers are 4 and 5.
To solve this problem, we can start by assigning variables to the two consecutive positive integers. Let's call the first integer "x" and the second integer "x + 1".
The reciprocal of a number is found by taking the inverse of the number. So, the reciprocal of "x" can be represented as 1/x, and the reciprocal of "x + 1" can be represented as 1/(x + 1).
The problem states that the sum of the reciprocals of these two integers is 17/12. So, we can write an equation to represent this:
1/x + 1/(x + 1) = 17/12
Now, we need to solve this equation to find the values of "x" and "x + 1" that satisfy this equation.
To solve the equation, we can start by eliminating the denominators. We can do this by multiplying both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 12, x, and x+1 is 12x(x + 1).
Multiplying both sides of the equation by 12x(x + 1) gives us:
12(x + 1) + 12x = 17x(x + 1)
Simplifying this equation further:
12x + 12 + 12x = 17x^2 + 17x
Combining like terms:
24x + 12 = 17x^2 + 17x
Moving all the terms to one side:
17x^2 - 7x - 12 = 0
This quadratic equation can be factored or solved using the quadratic formula to find the values of "x". Once we find the value of "x", we can find the consecutive positive integers by adding 1 to "x".
So the equation that can be used to find the two integers is:
17x^2 - 7x - 12 = 0
To find the values of "x" and the consecutive positive integers, you can solve this equation either by factoring or using the quadratic formula.