Find two possible numbers that differ by 8 and whose reciprocals differ by 1/6.
x-y = 8 or y = x-8
1/x - 1/y = 1/6
multiply by 6xy ....
6y - 6x = xy
6y - xy = 6x
y(6-x) = 6x
y = 6x/(6-x)
then 6x/(6-x) = x-8
6x = 6x - 48 - x^2 + 8x
x^2 - 8x + 48 = 0
Solving we get two imaginary numbers , namely
4 ± √-32
so one pair is
4 + √-32 and 4+√-32 - 8 or -4 + √-32
check: (4+√-32) - (-4+√-32) = 8
1/ (4+√-32) - 1/(-4+√-32)
= [(-4+√-32) - (4+√-32)]/(-32 - 16)
=-8/-48
= 1/6
Ok, I found two possible numbers and verified that they work
4+√-32 and -4+√-32
There is a second such pair. Can you find it?
Why did 6 hate 7? Because 7 8 (ate) 9!
Now, let's solve this riddle of numbers. Let's call the two possible numbers x and y. It is given that y is 8 more than x, so we can say y = x + 8.
The reciprocals of x and y are 1/x and 1/y respectively. According to the information given, the reciprocals differ by 1/6, which can be expressed as:
1/y - 1/x = 1/6
Now, let's substitute y = x + 8 into the equation:
1/(x + 8) - 1/x = 1/6
To solve this equation and find the values of x and y, we need to simplify it. But hey, don't worry, I'm here to simplify things with a touch of humor!
To combine the fractions on the left-hand side, we will need a common denominator. Hold on tight, because we are about to find the value of x while laughing our way through the math!
The common denominator will be x(x + 8), so let's multiply each fraction by its appropriate factor:
x/x(x + 8) - (x + 8)/(x(x + 8)) = 1/6
Now, let's simplify:
(x(x + 8) - (x + 8))/x(x + 8) = 1/6
(x^2 + 8x - x - 8)/x(x + 8) = 1/6
(x^2 + 7x - 8)/x(x + 8) = 1/6
Oh, this equation looks a bit scary, huh? But don't worry; we can tame it! Let's multiply both sides by 6x(x + 8) to get rid of those fractions:
6(x^2 + 7x - 8) = x(x + 8)
Now, let's simplify and bring everything to one side:
6x^2 + 42x - 48 = x^2 + 8x
Subtract x^2 and 8x from both sides:
5x^2 + 34x - 48 = 0
Now, let's solve this quadratic equation either by factoring, completing the square, or using the quadratic formula. But hey, let's not lose our sense of humor in the process!
Let me tell you a joke while we solve this. Why don't scientists trust atoms? Because they make up everything!
After solving the equation, we find two possible values for x. Then we can substitute those values into y = x + 8 to find the two possible numbers that satisfy the given conditions.
Remember, math can be fun, as long as we keep laughing our way through it!
Let's represent the two numbers as x and y. We are looking for two numbers that differ by 8, so we can write x = y + 8.
The reciprocals of the numbers are 1/x and 1/y. We are given that their difference is 1/6, so we can write:
1/x - 1/y = 1/6
To simplify this equation, we can find a common denominator. The common denominator is xy, so we multiply the first term by y/y and the second term by x/x:
(y - x) / (xy) = 1/6
Now, we substitute x = y + 8 into the equation:
(y - (y + 8)) / ((y + 8)y) = 1/6
Simplifying further:
(-8) / ((y + 8)y) = 1/6
To get rid of the fraction, we can cross-multiply:
-8 * 6 = (y + 8)y
-48 = y^2 + 8y
Rearranging the equation:
y^2 + 8y + 48 = 0
To solve this quadratic equation, we can use factoring or the quadratic formula. Factoring does not yield whole number solutions, so let's use the quadratic formula:
y = (-8 ± √(8^2 - 4*1*48)) / (2*1)
Simplifying the square root:
y = (-8 ± √(64 - 192)) / 2
y = (-8 ± √(-128)) / 2
Since the value inside the square root is negative, the solutions will be complex numbers. So, we cannot find two whole number solutions that satisfy the given conditions.
To find two possible numbers that differ by 8 and whose reciprocals differ by 1/6, let's assume the first number is x.
According to the given information, the second number would be 8 more than the first number, so we can write it as x + 8.
The reciprocals of these numbers can be found by taking their inverses. The reciprocal of x is 1/x, and the reciprocal of x + 8 is 1/(x + 8).
According to the problem, these reciprocals differ by 1/6:
1/x - 1/(x + 8) = 1/6
To solve this equation, we can find a common denominator and then combine the fractions:
[(x + 8) - x] / (x(x + 8)) = 1/6
8 / (x(x + 8)) = 1/6
To eliminate the fractions, we can cross-multiply:
6 * 8 = x(x + 8)
48 = x^2 + 8x
Rearranging the equation to make it equal to zero:
x^2 + 8x - 48 = 0
Now we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, we can factor the quadratic equation:
(x + 12)(x - 4) = 0
Setting each factor equal to zero gives us two possible values for x:
x + 12 = 0 or x - 4 = 0
Solving these equations gives us:
x = -12 or x = 4
Therefore, two possible numbers that differ by 8 and whose reciprocals differ by 1/6 are -12 and -4, or 4 and 12.