2. The sum of the reciprocals of two consecutive positive integers is 17/72. Write an equation that can be used to find the two integers. What are the integers?
Steve helped me yesterday and gave me the hint 8+9=17. Then I thought about it and saw 8*9=72, but I am confused about how to put it into an equation.
x = first number
x + 1 = second number
1 / x + 1 / ( x + 1 ) = 17 / 72
[ 1 * ( x + 1 ) + 1 * x ] / [ x * ( x + 1 ) ] = 17 / 72
[ ( x + 1 ) + x ] / [ x * x + x * 1 ) ] = 17 / 72
( 2 x + 1 ) / ( x ^ 2 + x ) = 17 / 72 Multiply both sides by 72
72 * ( 2 x + 1 ) / ( x ^ 2 + x ) = 17
( 72 * 2 x + 72 * 1 ) / ( x ^ 2 + x ) = 17
( 144 x + 72 ) / ( x ^ 2 + x ) = 17 Multiply both sides by x ^ 2 + x
144 x + 72 = 17 * ( x ^ 2 + x )
144 x + 72 = 17 * x ^ 2 + 17 * x
144 x + 72 = 17 x ^ 2 + 17 x Subtract 144 x to both sides
144 x + 72 - 144 x = 17 x ^ 2 + 17 x - 144 x
72 = 17 x ^ 2 - 127 x Subtract 72 to both sides
72 - 72 = 17 x ^ 2 - 127 x - 72
0 = 17 x ^ 2 - 127 x - 72
17 x ^ 2 - 127 x - 72 = 0
The solutions are:
x = - 9 / 17 and x = 8
- 9 / 17 isn't positive integers so x = 8
first number = 8
second number = 8 + 1 = 9
Proof:
1 / x + 1 / ( x + 1 ) =
1 / 8 + 1 / 9 =
( 1 * 9 + 1 * 8 ) / ( 8 * 9 ) =
( 9 + 8 ) / 72 = 17 / 72
cope
bruh why is that so long
Ah, math can be a tricky circus sometimes! But fear not, I'm here to help you put this into an equation. Let's call the two consecutive positive integers "x" and "x+1". So, the equation to represent the sum of their reciprocals is:
1/x + 1/(x+1) = 17/72
Now, let's put on our clown shoes and solve this equation. To make it a little less intimidating, let's clear the fractions, shall we?
Multiplying every term by the common denominator, which is (x)(x+1), we get:
(x+1) + x = 17(x)(x+1)/72
Simplifying further:
2x + 1 = 17(x^2 + x)/72
Are you still following me? Great! Let's keep going.
Now, cross-multiply to get rid of the fractions:
72(2x + 1) = 17(x^2 + x)
Expanding both sides of the equation:
144x + 72 = 17x^2 + 17x
Rearranging everything and forming a quadratic equation:
17x^2 + 17x - 144x - 72 = 0
Now, simplify and combine the like terms:
17x^2 - 127x - 72 = 0
Now, we can solve this quadratic equation using either factoring, completing the square, or the quadratic formula. Let's use the quadratic formula just to keep things clownishly fun:
x = (-b ± √(b^2 - 4ac)) / (2a)
Plugging in the values:
x = (-(-127) ± √((-127)^2 - 4(17)(-72))) / (2(17))
After a bit of calculation, we find that x can be approximately 7.76 or -6.14. But since we're dealing with positive consecutive integers, we can discard the negative solution.
So, the first positive integer is approximately 8. (Well done, Steve, for that hint!)
And the second positive integer will be x+1, which is approximately 9.
So, the two integers are approximately 8 and 9. Voila!
To solve this problem, we can start by assigning variables to the two consecutive positive integers. Let's say the first integer is x.
The second consecutive positive integer will then be x + 1, since they are consecutive.
Next, we need to set up an equation using the given information.
The sum of the reciprocals of the two consecutive positive integers is 17/72.
The reciprocal of a number x is equal to 1/x. So, the sum of the reciprocals can be written as:
1/x + 1/(x + 1) = 17/72
Now we have an equation that can be used to find the two integers.
To solve this equation, we can start by getting rid of the denominators by multiplying both sides of the equation by 72:
72 * (1/x + 1/(x + 1)) = 72 * (17/72)
This simplifies to:
72/(x) + 72/(x + 1) = 17
Now, we need to simplify further by multiplying both sides of the equation by x(x + 1) to get rid of the fractions:
[x(x + 1)] * [72/(x) + 72/(x + 1)] = [x(x + 1)] * 17
Simplifying the left side of the equation:
72(x + 1) + 72(x) = 17x(x + 1)
Distributing and simplifying:
72x + 72 + 72x = 17x^2 + 17x
144x + 72 = 17x^2 + 17x
Bringing all the terms to one side of the equation:
17x^2 + 17x - 144x - 72 = 0
Simplifying:
17x^2 - 127x - 72 = 0
Now we have a quadratic equation which can be solved to find the values of x.
Using factoring, quadratic formula, or any other method, we find:
x = -4 or x = 9
Since we need to find positive integers, we discard the negative solution.
Therefore, the first integer is x = 9, and the second integer is x + 1 = 9 + 1 = 10.
So, the two consecutive positive integers that satisfy the given condition are 9 and 10.