The reciprocal of a number minus the reciprocal of two less than the number is -2/15. Find the number

Question

I have no idea how to solve this

Answer 1

When you have a word problem like this, it's best to set up an equation in terms of a variable to solve. The reciprocal of a number means 1 divided by that number. So the reciprocal of 4 is 1/4. Therefore, the equation we can set up will be:

1/x - 1/(x-2) = -2/15

Through some algebra (which I'll leave you to try to work through yourself), you should get x=5. You can test this by confirming that 1/5 - 1/3 = -2/15 via a common denominator. Hope that helps, let me know if you have any questions!

Answer 2

The reciprocal of a number can be represented as 1/x

The reciprocal of 2 less than the number can be represented as: 1/(x - 2)

Put them together (1/x) - (1/(x - 2)) = (-2/15)

Find a common denominator for all three terms by multiplying them all: x*(x-2)*15 and what you do to the bottom you must do to the top:

Thus: (1/x) multiply by (x-2) and 15 on the top and bottom to get [15(x-2)]/[x(x-2)(15)]

(1/(x-2)) multiply by x and 15 to get [15x/[x(x-2]15)

-2/15 multiply by x-2 and x to get [-2(x-2)(x)/(x(x-2)15]

All the denominators are now the same so they cancel out and you're left with the numerators

15(x-2) - 15x = -2(x-2)(x)

Distribute the values to get: 15x - 30 - 15x = -2x^2 +4x

Rearrange the values to get the problem equal to 0

-30 = -2x^2 +4x

2x^2 - 4x - 30 = 0

You can simplify by dividing all the terms by a common factor: 2

x^2 - 2x - 15 = 0

Then you can solve for x by factoring: find two numbers that add up to -2 and multiply to -15

Make a list and the two numbers end up being -5 and +3( see - 5 + 3 = -2 AND -5*3 = -15)

(x - 5)(x+3) = 0

T- off and set each equation in bubbles to 0 and solve for x

x - 5 = 0, so x = 5 AND x+ 3 = 0, so x = -3

Answer 3

The reciprocal of any number is 1 over that number.

Lets call our mystery number x. So the reciprocal is 1/x.

The reciprocal of 2 less than that number is 1/x-2 because it's the mystery number minus 2.

It says if we subtract them, we get -2/15.

This looks like 1/x - 1/x-2 = -2/15

Multiply each part by x to get rid of that x in the first fraction. This results in:

1 - x/x-2 = -2x/15

We will then multiply by (x-2) to get rid of the other denominator.

This will produce:

x - 2 - x = (-2x² + 4x) /15

The x's will cancel out since we have a positive and negative on the left side to the equal sign.

Let's now multiply by the last denominator of 15. This gives us

-30 = -2x² + 4x

We will set this equal to 0 and solve by factoring.

-2x² + 4x +30 = 0

Divide by the common factor of -2 gives us

x² -2x -15 = 0

So x=5 or x=-3.