The sum of two numbers is 4 and the reciprocal of one exceeds the reciprocal of other by twice the product of their reciprocals. What is the product of the reciprocals of two numbers?

Please help me solving.

Why did the number cross the road? Because it couldn't solve this equation! But don't worry, I'm here to help. Let's call the two numbers x and y. We know that x + y = 4. We're also told that the reciprocal of one number exceeds the reciprocal of the other by twice the product of their reciprocals. In other words, 1/x - 1/y = 2(1/x * 1/y).

Now, let's simplify this equation. Multiplying both sides by xy, we get y - x = 2. Rearranging the first equation, we have y = 4 - x. Substituting this into the second equation, we get 4 - x - x = 2. Simplifying further, we find 4 - 2x = 2. Subtracting 4 from both sides, we get -2x = -2. Dividing both sides by -2, we find x = 1.

So one of the numbers is 1. Substituting this value back into the first equation, we find y = 4 - 1 = 3. Therefore, the two numbers are 1 and 3. To find the product of the reciprocals, we simply take 1/1 * 1/3 = 1/3. So the product of the reciprocals of the two numbers is 1/3.

Hope that helps, and remember, math might be challenging, but it can always use a little silliness to lighten things up!

Let's assume the two numbers are x and y.

From the given information, we can form two equations:

1) x + y = 4 (the sum of the two numbers is 4)
2) 1/x - 1/y = 2(1/x * 1/y) (the reciprocal of one exceeds the reciprocal of the other by twice the product of their reciprocals)

To solve this system of equations, we can start by multiplying equation 2 by xy to eliminate the denominators:

xy(1/x - 1/y) = xy(2(1/x * 1/y))
y - x = 2y/x

Next, let's rearrange the equation to isolate y:

y - 2y/x = x
(1 - 2/x)y = x
y = x / (1 - 2/x)

Now we can substitute this value of y into equation 1:

x + x / (1 - 2/x) = 4
Multiplying through by (1 - 2/x), we get:
x(1 - 2/x) + x = 4(1 - 2/x)
x - 2 + x = 4 - 8/x
2x - 2 = 4 - 8/x
Multiplying through by x to eliminate the fraction, we get:
2x^2 - 2x = 4x - 8
2x^2 - 6x + 8 = 0

Now we can solve this quadratic equation. However, upon solving it, we find that it does not have real solutions. Therefore, there are no real numbers that satisfy the given conditions.

Hence, we cannot determine the product of the reciprocals of the two numbers.

To solve this problem, we can start by assigning variables to the two unknown numbers. Let's call the first number x and the second number y.

According to the problem, the sum of the two numbers is 4, so we can write the equation:
x + y = 4

The reciprocal of one number exceeds the reciprocal of the other by twice the product of their reciprocals. This can be written as an equation as well:
1/x - 1/y = 2(1/x * 1/y)

To make the equation easier to solve, we can eliminate the fractions by multiplying both sides of the equation by xy:
y - x = 2

Now we have a system of two equations:
x + y = 4
y - x = 2

We can solve this system of equations using the method of substitution or elimination. Let's solve it using substitution:

From the second equation, we can solve for y:
y = 2 + x

Substitute this expression for y in the first equation:
x + (2 + x) = 4
2x + 2 = 4
2x = 2
x = 1

Now substitute the value of x back into one of the original equations to solve for y. Let's use the second equation:
y - 1 = 2
y = 3

So, the two numbers are x = 1 and y = 3.

The product of reciprocals of two numbers can be found by multiplying their individual reciprocals:
1/x * 1/y = 1/1 * 1/3 = 1/3

Therefore, the product of the reciprocals of the two numbers is 1/3.

M+N=4

1/M-1/N=2/MN

using the second equation, then
N-M=2

add that to the first equation
2N=6
N=3, M=1