Percentage formula to return to the starting point

[Hendrixxxxxxxx]

Hi All,
Firstly, I hope I am posting this in the correct forum. My apologies if not.

I was wondering if there is a formula to calculate the percentage change to return to the starting point, given a particular known percentage change.

For example, starting at 100:
A 100% (y%) move up takes us to 200.
A 50% (x%) move down takes us back to 100.

A 50% (y%) move up takes us to 150.
A 33% (x%) move back down takes us back to 100.

A 25% (y%) move up takes us to 125.
A 20% (x%) move back down takes us back to 100.

A 2% (y%) move takes us to 102.
An x% move back down takes us back to 100.

Does anyone know if there is a formula to calculate x, given a known y please?

Thanks in anticipation
Hendrix

 

[Otis]

A 50% (y%) move up takes us to 150.
A 33% (x%) move back down takes us back to 100.

Hi Hendrix. The end result above is not quite correct because you've rounded $33.\overline{3}$% to $33$%. Setting x=33 brings you back down to 100.5, unless you're using a calculator.

Does anyone know if there is a formula to calculate x, given a known y please?

Yes, you could write and solve an equation, to find a formula for x in terms of y.

Is this part of a school assignment?



$\;$

 

[Hendrixxxxxxxx]

A 50% (y%) move up takes us to 150.
A 33% (x%) move back down takes us back to 100.

Hi Hendrix. The end result above is not quite correct because you've rounded $33.\overline{3}$% to $33$%. Setting x=33 brings you back down to 100.5, unless you're using a calculator.

Does anyone know if there is a formula to calculate x, given a known y please?

Yes, you could write and solve an equation, to find a formula for x in terms of y.

Is this part of a school assignment?



$\;$

Hi Otis,

Apologies. My error.
No, it’s a work “assignment” (I left school about 40 years ago ?).
Is there a formula to calculate x given y?

 

[JeffM]

Following up from otis, do you know how to use algebra?

 

[Hendrixxxxxxxx]

I only recall some simple algebra from many many many years ago.

 

[JeffM]

$$\text {starting point} = s;\\ \text {amount of increase} = a;\\ \text {amount of offsetting decrease} = a;\\ \text {percentage increase} = x = 100 * \dfrac{a}{s}; \text { and}\\ \text {percentage offsetting decrease} = y = 100 * \dfrac{a}{s + a}.$$
Any trouble to here?

Now we do a little algebraic manipulation.

$$x =100 * \dfrac{a}{s} \implies a = \dfrac{sx}{100}.\\ \therefore \ \dfrac{a}{s + a} = \dfrac{\dfrac{sx}{100}}{s + \dfrac{sx}{100}} = \dfrac{\dfrac{sx}{100}}{\dfrac{100s +sx}{100}} =\\ \dfrac{sx}{100} * \dfrac{100}{s(100 +x)} = \dfrac{x}{100 + x}.\\ \text {But } y = 100 * \dfrac{a}{s + a} = 100 * \dfrac{x}{100 + x}.\\ \therefore \ y = \dfrac{100x}{100 + x}.$$
Let's check.
Increase 30 to 45.
Percentage increase 50.
Percentage decrease to get back to 30 =
(100 * 50)/(100 + 50) = 5000 / 150 $\approx$ 33.333333333.

Now that is an approximate percentage because 150 does not divide evenly into 5000, but you can take it out as many decimal places as you want.

Let's try your first example
Increase 100 to 200.
Percentage increase 100%.
Percentage decrease to get back to 100 =
(100 * 100)/(100 + 100) = 10000/200 = 50.

Let's try your last example.
Increase 100 to 102.
Percentage increase 2.
Percentage decrease to get back to 100 =
(100 * 2)(100 + 2) = 200/102 $\approx$ 1.961.

Now if we calculate 1.961 * 102 /100, we get 2.00022. So that takes us to 99.9978, which is pretty close to 100. Again it is an approximation because 102 does not divide evenly into 200. If you need more accuracy, just compute y to more decimal places.

 

[Hendrixxxxxxxx]

$$\text {starting point} = s;\\ \text {amount of increase} = a;\\ \text {amount of offsetting decrease} = a;\\ \text {percentage increase} = x = 100 * \dfrac{a}{s}; \text { and}\\ \text {percentage offsetting decrease} = y = 100 * \dfrac{a}{s + a}.$$
Any trouble to here?

Now we do a little algebraic manipulation.

$$x =100 * \dfrac{a}{s} \implies a = \dfrac{sx}{100}.\\ \therefore \ \dfrac{a}{s + a} = \dfrac{\dfrac{sx}{100}}{s + \dfrac{sx}{100}} = \dfrac{\dfrac{sx}{100}}{\dfrac{100s +sx}{100}} =\\ \dfrac{sx}{100} * \dfrac{100}{s(100 +x)} = \dfrac{x}{100 + x}.\\ \text {But } y = 100 * \dfrac{a}{s + a} = 100 * \dfrac{x}{100 + x}.\\ \therefore \ y = \dfrac{100x}{100 + x}.$$
Let's check.
Increase 30 to 45.
Percentage increase 50.
Percentage decrease to get back to 30 =
(100 * 50)/(100 + 50) = 5000 / 150 $\approx$ 33.333333333.

Now that is an approximate percentage because 150 does not divide evenly into 5000, but you can take it out as many decimal places as you want.

Let's try your first example
Increase 100 to 200.
Percentage increase 100%.
Percentage decrease to get back to 100 =
(100 * 100)/(100 + 100) = 10000/200 = 50.

Let's try your last example.
Increase 100 to 102.
Percentage increase 2.
Percentage decrease to get back to 100 =
(100 * 2)(100 + 2) = 200/102 $\approx$ 1.961.

Now if we calculate 1.961 * 102 /100, we get 2.00022. So that takes us to 99.9978, which is pretty close to 100. Again it is an approximation because 102 does not divide evenly into 200. If you need more accuracy, just compute y to more decimal places.

That is awesome Jeff.
Exactly what I was looking for.
I appreciate your help. Thank you.