# Trying to get to the "root" of things. Sorry.

April 5, 2009 12:16 PM Subscribe

Totally naive math-filter. What is the relationship between percentage difference between two real numbers and the percentage difference between their squares?

If I add 10% to x, then (x+10%)^2 will always be 21% larger than x^2. If I add 50% to x, then the squared difference will always be 225% Why? What's the relationship?

This is not a homework question. I'm actually studying for a standardized test which asks many questions along the lines of "if you increase the radius of a circle by 10% [or some such number], by what percentage will the total area increase?" Sometimes the value is given, sometimes it isn't, but the question runs more or less along the same lines.

One way of doing it is by plugging in the numbers and calculating it by hand: eg,

(1) let x=10.

(2) 10 + 10% = 11

(3) 10^2 = 100; 11^2=121

(4) 121 is 21% more than 10.

So, increasing the original number by 10% yields 21% on the other end.

In this context, it's easiest just to memorize the squares up to 20, but I'd like to know the "real" reason why this is. Bonus points if there's an elegant function that yields the difference when it's not so easily calculated in your head.

If I add 10% to x, then (x+10%)^2 will always be 21% larger than x^2. If I add 50% to x, then the squared difference will always be 225% Why? What's the relationship?

This is not a homework question. I'm actually studying for a standardized test which asks many questions along the lines of "if you increase the radius of a circle by 10% [or some such number], by what percentage will the total area increase?" Sometimes the value is given, sometimes it isn't, but the question runs more or less along the same lines.

One way of doing it is by plugging in the numbers and calculating it by hand: eg,

(1) let x=10.

(2) 10 + 10% = 11

(3) 10^2 = 100; 11^2=121

(4) 121 is 21% more than 10.

So, increasing the original number by 10% yields 21% on the other end.

In this context, it's easiest just to memorize the squares up to 20, but I'd like to know the "real" reason why this is. Bonus points if there's an elegant function that yields the difference when it's not so easily calculated in your head.

Just off the top of my head:

(x + .1x)^2 = x^2 + .2x^2 + .01x^2 = 1.21x^2

So to generalize, I'd say add up the square of the percentage, the double of the percentage, and 1.

posted by mad bomber what bombs at midnight at 12:27 PM on April 5, 2009

(x + .1x)^2 = x^2 + .2x^2 + .01x^2 = 1.21x^2

So to generalize, I'd say add up the square of the percentage, the double of the percentage, and 1.

posted by mad bomber what bombs at midnight at 12:27 PM on April 5, 2009

Oh, and to test with your other example:

.5^2 = .25

2*.5 = 1

And 1 + 1 + .25 = 2.25 = 225%

posted by mad bomber what bombs at midnight at 12:29 PM on April 5, 2009

.5^2 = .25

2*.5 = 1

And 1 + 1 + .25 = 2.25 = 225%

posted by mad bomber what bombs at midnight at 12:29 PM on April 5, 2009

This is really helpful. To extrapolate, you can obtain the percentage increase by rewriting the percentage as a value of x, and squaring that?

[(x + 22%)^2-x^2] = 1.22^2 + (x^2) = 1.4884; so, the answer is 48.84%

Am I thinking along the right lines?

posted by awenner at 12:36 PM on April 5, 2009

[(x + 22%)^2-x^2] = 1.22^2 + (x^2) = 1.4884; so, the answer is 48.84%

Am I thinking along the right lines?

posted by awenner at 12:36 PM on April 5, 2009

Yes. The general formula for percentage increase is given by the difference between the increased value and the original value, divided by the original value, times 100.

So if, eg, the radius of your original circle is r, and you increase the radius by 10%, that means the new radius is s = 1.1r.

Then according to the formula above, the percentage increase is

(pi * s^2 - pi * r^2) / (pi * r^2)

= (pi * (1.1^2 - 1) * r^2) / (pi * r^2)

The pi and r^2 on the top cancel with the pi and r^2 on the bottom, leaving you with the number (1.1^2 - 1) = 0.21. Then multiply by 100 to get 21%.

I would say just be careful with the way you write things down in order not to confuse yourself. In itself, x + 22% doesn't really mean anything, since 22% really means 'multiply by 22/100'. What you mean to write is that increasing x by 22% gives you the new value of

x + 22/100 * x = 1.22x.

posted by number9dream at 12:57 PM on April 5, 2009

So if, eg, the radius of your original circle is r, and you increase the radius by 10%, that means the new radius is s = 1.1r.

Then according to the formula above, the percentage increase is

(pi * s^2 - pi * r^2) / (pi * r^2)

= (pi * (1.1^2 - 1) * r^2) / (pi * r^2)

The pi and r^2 on the top cancel with the pi and r^2 on the bottom, leaving you with the number (1.1^2 - 1) = 0.21. Then multiply by 100 to get 21%.

I would say just be careful with the way you write things down in order not to confuse yourself. In itself, x + 22% doesn't really mean anything, since 22% really means 'multiply by 22/100'. What you mean to write is that increasing x by 22% gives you the new value of

x + 22/100 * x = 1.22x.

posted by number9dream at 12:57 PM on April 5, 2009

let's say I add y% to x.

I can rewrite "y% of x" as x*(y/100). for example, 30% of x is x*(30/100) = x*0.3.

So if I'm starting with x and adding y% of x, the result is x(1 + (y/100)). e.g. for y=30%, result is 1.3x.

squaring:

[x(1+(y/100))]^2

= x^2 [ (1 + y/100)^2]

= x^2 [ (1 + y/100)(1 + y/100)]

= x^2 [ (1 + 2y/100 + y^2/10000) ]

let's try to make that look like a percentage of x^2 (1 + something/100)

= x^2 [ (1 + 2y/100 + 0.01y^2/100) ]

= x^2 [ (1 + (2y+0.01y^2)/100) ]

So the result is 2y+0.01y^2 percent larger than x^2.

Checking:

y=10%

2y+0.01y^2 = 20 +0.01(100) = 21%

y=50%

2y+0.01y^2 = 100 + 0.01(2500) = 125%

(note that 2.25X means 125% larger; 1.25X means 25% larger.)

y=30%

2y + 0.01y^2 = 60 + 0.01(900) = 69%

posted by PercussivePaul at 1:16 PM on April 5, 2009

I can rewrite "y% of x" as x*(y/100). for example, 30% of x is x*(30/100) = x*0.3.

So if I'm starting with x and adding y% of x, the result is x(1 + (y/100)). e.g. for y=30%, result is 1.3x.

squaring:

[x(1+(y/100))]^2

= x^2 [ (1 + y/100)^2]

= x^2 [ (1 + y/100)(1 + y/100)]

= x^2 [ (1 + 2y/100 + y^2/10000) ]

let's try to make that look like a percentage of x^2 (1 + something/100)

= x^2 [ (1 + 2y/100 + 0.01y^2/100) ]

= x^2 [ (1 + (2y+0.01y^2)/100) ]

So the result is 2y+0.01y^2 percent larger than x^2.

Checking:

y=10%

2y+0.01y^2 = 20 +0.01(100) = 21%

y=50%

2y+0.01y^2 = 100 + 0.01(2500) = 125%

(note that 2.25X means 125% larger; 1.25X means 25% larger.)

y=30%

2y + 0.01y^2 = 60 + 0.01(900) = 69%

posted by PercussivePaul at 1:16 PM on April 5, 2009

Though mad bomber's method is more intuitive and easier. You just have to think of percentages as decimals; 30% as 0.3 etc. Then you can do all that math with (1+y) instead of (1+y/100). Like he said, square the percentage and add twice the percentage. 2y + y^2.

posted by PercussivePaul at 1:50 PM on April 5, 2009

posted by PercussivePaul at 1:50 PM on April 5, 2009

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(1.1x)^2 = (1.1)^2 * x^2 = 1.21 * (x^2)

1.21 is 21% larger than 1 (the original multiplier of x).

Basically, you have to look at what happens to the multiplier of x (or whatever your independent variable is).

Likewise, x + 50% = 1.5x; (1.5x)^2 = 1.5^2 * x^2 = 2.25 * (x^2); Thus, 225% larger.

The mathematical property that governs this situation is the Distributive Property of Exponentiation.

I hope this gets at what you are asking...

posted by ThyroidBob at 12:24 PM on April 5, 2009