eli5 The percentage rule

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Can someone explain how/why X% of Y is equal to Y% of X? I know it’s true but I can’t figure out why. Thanks!

In: 10

Think of it this way. Let’s take 25 percent of 3. We’ll you can write this as 0.25 x 3
Now flip it. 3 percent of 25. Well that’s just 0.03 X 25. They’re the same thing! Just a factor of 100 different both sides.

Lets say there are two numbers

X and Y

To get X% of Y

we’ll have,

X * (Y/100)
=(X*Y)/100

let this be equation 1.

Now lets say for getting Y% of X,

we’ll do Y * (X/100)
= (Y*X)/100

this our equation 2.

You see, equation 1 and equation 2 are same.

just X and Y are swapped.

so when you say 5% of 10.

you do 5*(10/100)

but when you say 10% of 5.

you do 10*(5/100).

but both will give same answer

50/100

Percent (the sign of percent is %) means the number of something per 100, so 5 percent is 5 per 100, 1 percent is 1 per 100, 100 percent is 100 per 100.

Per is the same as splitting something up:
100 percent = 100 cakes split up between 100 people = 1 cake each
50 percent = 50 cakes split up between 100 = half a cake each
10 percent = 10 cakes split up between 100 = a 10th of a cake each

Remember that percent means 1 per 100.
So 20 percent (20 times 1 cake split up between 100) becomes “20 times 1 per 100” and 40 percent becomes “40 times 1 per 100”

We write it like this:
20 percent of 40 cakes = 20 * 1/100 * 40 cakes = 8 cakes
40 percent of 20 cakes = 40 * 1/100 * 20 cakes = 8 cakes
You can see that they are the same, well remember 2 + 1 is the same as 1 + 2 & 1 * 2 is the same as 2 * 1; you can move these numbers around a bit.
So 40 * 20 * 1/100 = 8
and 1/100 * 20 * 40 = 8

Lets stop talking about cake.
20 * 1/100 * 40 = 8
40 * 1/100 * 20 = 8
And turn 1/100 back into the percent sign % from the start
20% of 40 = 8
40% of 20 = 8

X% of Y = Y% of X

It becomes clearer when you translate the words “per” and “of” into their mathematical operations. “per” means division and “of” means multiplication. Then it becomes straight-up algebra:

X% of Y

(X per 100) of Y

X
—- * Y
100

X * Y
——–
100

You get the same exact thing if you start the other way around:

Y% of X

Y
—- * X
100

Y * X
——–
100

Another way is with simple multiplication.

If you translate “x%” as (x • 0.01), then

x% of y = x • 0.01 • y

y% of x = y • 0.01 • x

And the commutative property says those two expressions are equal.

A “per cent” is just “divided by 100”.

50 percent is 50 / 100.

50 percent of a price is fifty one-hundredths of the original cost.

It’s just a fraction.

so X% of something is just (X / 100) times by Y.

Which is mathematically the same as (Y / 100) times by X.

x*y/100 is the same as y*x/100.

A “percent” means “out of 100”. So X% is is the same as X * 0.01
In math you learn something called the “commutative property of multiplication” which tells you that if you multiply a bunch of numbers together that the order you do the multiplication in doesn’t change the answer.
So X * 0.01 * Y = X * Y * 0.01

I saw a lot of math, but I find it helpful to visualize it as objects instead of abstractions.

First, percentages are just factions. If we have 25%, that is the same as 25/100, 0.25, or 1/4. 100% is the same as 1.00. 3% is 0.03.

If I have 1 pies, and you get 1/4 (or 25%) of it, then you have 25% of that pie (1.00 x 25%). If you reverse it, then I have 1/4 of a pie, and you get 100% (or “1”) of what I have (¼ x 100%).

Or, if I have 4 kilograms of gold, and you get a 10% share of it, you get 0.4 kg (4.00 x 10%). If shares of gold are 0.1 each (0.1 = 10%), and you get 4 shares, you still get 0.4 kg (0.1 x 400%).

Hope this helps.

So many example here. Good, but I’ll try the armchair mathematician explanation as to the why since it seems that’s what you were really asking. The idea of a percent of something would indicate a value between 0 and 100 percent. We have a number 100, which is easy to factor in two directions while we use a base 10 number system. That’s why doing basic math with 10 and 100 is easy in either direction (multiply or divide to achieve a percent)

% just means “divided by 100”

so “x% of Y” is equivalent to `X / 100 * Y` Because multiplication and division are transitive, we can rearrange to `X * Y / 100` and since % means “divided by 100”, then we can say “Y% of X”.

25% of 80 is just 0.25×80=20

80% of 25 is just 25×0.8=20

All you’re doing is multiplying one of the numbers by 100 and dividing the other one by 100 when you switch X and Y

“Percent” means “per 100”. You can write 25% as 25 / 100 or 25 x 1/100.

If you write it as 25 x 1/100, its pretty obvious that you’re just multiplying two numbers. X percent is just X x 1/100.

When you multiple numbers, the order that you multiply them doesn’t matter; you get the same results (2 x 3) x 4 = 2 x (3 x 4) = (2 x 4) x 3 = 24. You can visualize that with pennies: 4 groups of pennies laid out in 2 rows and 3 columns, or 2 groups of pennies laid out in 3 rows and 4 columns, etc.

So now X% x Y = X x Y% because the ‘%’ symbol means ‘x 1/100’: X x 1/100 x Y = X x Y x 1/100 — because the order you multiple the 3 numbers doesn’t matter.

X% of Y = Y% of X. A percentage is a multiplication. Like 69% of 70 = 70×0.69. So:

Y × X/100 = X × Y/100 //×100

Y × X = X × Y

True because multiplication is commutative.

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Can someone explain how/why X% of Y is equal to Y% of X? I know it’s true but I can’t figure out why. Thanks!

In: 10

Think of it this way. Let’s take 25 percent of 3. We’ll you can write this as 0.25 x 3
Now flip it. 3 percent of 25. Well that’s just 0.03 X 25. They’re the same thing! Just a factor of 100 different both sides.

Lets say there are two numbers

X and Y

To get X% of Y

we’ll have,

X * (Y/100)
=(X*Y)/100

let this be equation 1.

Now lets say for getting Y% of X,

we’ll do Y * (X/100)
= (Y*X)/100

this our equation 2.

You see, equation 1 and equation 2 are same.

just X and Y are swapped.

so when you say 5% of 10.

you do 5*(10/100)

but when you say 10% of 5.

you do 10*(5/100).

but both will give same answer

50/100

Percent (the sign of percent is %) means the number of something per 100, so 5 percent is 5 per 100, 1 percent is 1 per 100, 100 percent is 100 per 100.

Per is the same as splitting something up:
100 percent = 100 cakes split up between 100 people = 1 cake each
50 percent = 50 cakes split up between 100 = half a cake each
10 percent = 10 cakes split up between 100 = a 10th of a cake each

Remember that percent means 1 per 100.
So 20 percent (20 times 1 cake split up between 100) becomes “20 times 1 per 100” and 40 percent becomes “40 times 1 per 100”

We write it like this:
20 percent of 40 cakes = 20 * 1/100 * 40 cakes = 8 cakes
40 percent of 20 cakes = 40 * 1/100 * 20 cakes = 8 cakes
You can see that they are the same, well remember 2 + 1 is the same as 1 + 2 & 1 * 2 is the same as 2 * 1; you can move these numbers around a bit.
So 40 * 20 * 1/100 = 8
and 1/100 * 20 * 40 = 8

Lets stop talking about cake.
20 * 1/100 * 40 = 8
40 * 1/100 * 20 = 8
And turn 1/100 back into the percent sign % from the start
20% of 40 = 8
40% of 20 = 8

X% of Y = Y% of X

It becomes clearer when you translate the words “per” and “of” into their mathematical operations. “per” means division and “of” means multiplication. Then it becomes straight-up algebra:

X% of Y

(X per 100) of Y

X
—- * Y
100

X * Y
——–
100

You get the same exact thing if you start the other way around:

Y% of X

Y
—- * X
100

Y * X
——–
100

Another way is with simple multiplication.

If you translate “x%” as (x • 0.01), then

x% of y = x • 0.01 • y

y% of x = y • 0.01 • x

And the commutative property says those two expressions are equal.

A “per cent” is just “divided by 100”.

50 percent is 50 / 100.

50 percent of a price is fifty one-hundredths of the original cost.

It’s just a fraction.

so X% of something is just (X / 100) times by Y.

Which is mathematically the same as (Y / 100) times by X.

x*y/100 is the same as y*x/100.

A “percent” means “out of 100”. So X% is is the same as X * 0.01
In math you learn something called the “commutative property of multiplication” which tells you that if you multiply a bunch of numbers together that the order you do the multiplication in doesn’t change the answer.
So X * 0.01 * Y = X * Y * 0.01

I saw a lot of math, but I find it helpful to visualize it as objects instead of abstractions.

First, percentages are just factions. If we have 25%, that is the same as 25/100, 0.25, or 1/4. 100% is the same as 1.00. 3% is 0.03.

If I have 1 pies, and you get 1/4 (or 25%) of it, then you have 25% of that pie (1.00 x 25%). If you reverse it, then I have 1/4 of a pie, and you get 100% (or “1”) of what I have (¼ x 100%).

Or, if I have 4 kilograms of gold, and you get a 10% share of it, you get 0.4 kg (4.00 x 10%). If shares of gold are 0.1 each (0.1 = 10%), and you get 4 shares, you still get 0.4 kg (0.1 x 400%).

Hope this helps.

So many example here. Good, but I’ll try the armchair mathematician explanation as to the why since it seems that’s what you were really asking. The idea of a percent of something would indicate a value between 0 and 100 percent. We have a number 100, which is easy to factor in two directions while we use a base 10 number system. That’s why doing basic math with 10 and 100 is easy in either direction (multiply or divide to achieve a percent)

% just means “divided by 100”

so “x% of Y” is equivalent to `X / 100 * Y` Because multiplication and division are transitive, we can rearrange to `X * Y / 100` and since % means “divided by 100”, then we can say “Y% of X”.

25% of 80 is just 0.25×80=20

80% of 25 is just 25×0.8=20

All you’re doing is multiplying one of the numbers by 100 and dividing the other one by 100 when you switch X and Y

“Percent” means “per 100”. You can write 25% as 25 / 100 or 25 x 1/100.

If you write it as 25 x 1/100, its pretty obvious that you’re just multiplying two numbers. X percent is just X x 1/100.

When you multiple numbers, the order that you multiply them doesn’t matter; you get the same results (2 x 3) x 4 = 2 x (3 x 4) = (2 x 4) x 3 = 24. You can visualize that with pennies: 4 groups of pennies laid out in 2 rows and 3 columns, or 2 groups of pennies laid out in 3 rows and 4 columns, etc.

So now X% x Y = X x Y% because the ‘%’ symbol means ‘x 1/100’: X x 1/100 x Y = X x Y x 1/100 — because the order you multiple the 3 numbers doesn’t matter.

X% of Y = Y% of X. A percentage is a multiplication. Like 69% of 70 = 70×0.69. So:

Y × X/100 = X × Y/100 //×100

Y × X = X × Y

True because multiplication is commutative.