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First, to understand my question, checkout this one:

Calculating percentages for taxes.

Second, consider that I'm a layman in math.

So, after trying to understand the logic used to get the final result. I was wondering:

Why multiply $20,000 by 100 and then divide by 83? I know this is the rule of three, but, I can't understand the "internals" of this approach. It isn't intuitive as think in this way:

Say 100% of one value, is the same of divide this value by 100. In other words: I have 100 separeted parts of this integer.

It's intuitive think about the taxes like this:

$$X - 17\% = \$20.000$$

So:

$$\$20.000 = 83\%$$

For me, the most easy and compreensive way to solve this is:

$$\$20.000 / 83 = 240.96$$

It's the same as think, if 100% is 100 parts of one integer, 83% of that integer is the same of divide this integer by 83.

And finally to get the result:

$$\$20.000 + 17 * 240.96$$

My final question is:

How can I think intuitively like this using the Rule of Three? In other words, why multiply 20.000 by 100 and then divide by 83 is a shorcut to get the result?

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I would point out that writing $X-17\%=\$20,000$ is rather bad form, since you are mixing quantities (kind of like trying to add 1 mile and 2 gallons, not very mathematical). You need to replace that $17$% with $0.17X$ (seventeen percent of $X$) to get something that actually makes sense. –  Arturo Magidin Dec 17 '10 at 21:18

3 Answers 3

up vote 6 down vote accepted

The Rule of Three is just doing the same thing you are doing: if $\$20,000$ is $83$%, then $\frac{20000}{83}$ is $1$% of the original. Since $X$ is the original, then $\frac{X}{100}$ is also $1$% of the original. That means that the two quantities are equal, so we have: $$\frac{20,000}{83} = \frac{X}{100}.$$ From here it is easy to solve for $X$: Edited:

Since both sides are equal, if you do the same thing to both sides you will get equal quantities. Or, intuitively, since each of these quantities is 1% of what we want, multiplying them by 100 will give us 100% (which is what we want). So we have $$100\left(\frac{20,000}{83}\right) = 100\left(\frac{X}{100}\right).$$ Now, on the right hand side, the $100$ cancels with the $100$ on the denominator (which again, makes sense: remember that $X$ was the total, $\frac{X}{100}$ was 1% of $X$, so a hundred times 1% of $X$ gives us back $X$), so we are left with $$\frac{(100)(20,000)}{83} = X.$$

In general, the Rule of Three works because you have proportionality: if $a$ is to $b$ like $x$ is to $c$, (if $\$20,000$ is to $83$% like $X$ is to $100$%) $$\begin{array}{ccc} a & \longrightarrow &b\\\ x & \longrightarrow &c \end{array}$$ then that means that $$\frac{a}{b} = \frac{x}{c}.$$ which is solved by multiplying through by $c$, which cancels the $c$ in the denominator on the right hand side; it comes out to exactly the same as the Rule of Three says: multiply across the diagonal of known quantities (in this case, $a$ and $c$) and divide by the remaining one (in this case, $b$). The point of the Rule of Three is to do it by rote and not having to think about setting everything up carefully to get the answer.

It also works if your unknown were the percentage: if $\$20,000$ is $87$%, then what percentage of the original is $\$15,575$? (I just made that number up) Well, using the logic from above, since $\frac{20,000}{87}$ is $1$%, and $\frac{15,575}{p}$ is $1$% (where $p$ is the percentage we are looking for) then we have $$\frac{20,000}{87} = \frac{15,575}{p}$$ which solving for $p$ by first multiplying across by $p$, then multiplying across by $87$, and finally dividing through by $20,000$ (aka "cross-multiplying"): \begin{align} \frac{20,000}{87} &= \frac{15,575}{p}\\ \frac{20,000p}{87} &= \frac{15,575p}{p}\\ \frac{20,000p}{87} &= 15,575\\ \frac{20,000p(87)}{87} &= (15,575)(87)\\ 20,000p &= (15,575)(87)\\ \frac{20,000p}{20,000} &= \frac{(15,575)(87)}{20,000}\\ p &= \frac{(15,575)(87)}{20,000} \approx 67.75 \end{align} so the percentage here is approximately $67.75$%. Using the Rule of Three, you have that $20,000$ is to $87$ like $15,575$ is to $p$, so $$\begin{array}{ccc} 20,000 &\longrightarrow & 87\\\ 15,575 & \longrightarrow & p \end{array}$$ so to get $p$, you multiply across the known diagonal ($15,575$ times $87$) and divide by the remaining quantity ($20,000$), same operation as before. Added: Again, intuitively, what we are doing is noting that if $20,000$ is 87%, then $\frac{20,000}{87}$ is 1% of the total; we want to know the $p$ such that $15,575$ is $p$%; whatever it is, $\frac{15,575}{p}$ will also be $1$%, so the two quantities are equal. Multiplying both sides by $p$ makes both sides equal to $p$%. of the total. Multiplying both sides by 87 makes both sides $87p$&%. Since $20,000$ is $87$%, dividing both sides by $20,000$ will give $p$, the percentage we are looking for (since we have $87p$% divided by $87$%).

(But it's important that the Rule of Three only works when you do have proportionality. It works for percentages, it works for linear aggregates, but it does not work for things like exponential growth or decay, because then you don't have proportionality between growth and time elapsed.)

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Thank you. You were very clear in the first part. But when you cross multiply you're missing that I can't see what is really happening behind the scenes. @bill-dubeque got the point about that and I'm trying to understand his answer. Could you try to clarify this part too? I would appreciate it. –  Telephone Dec 20 '10 at 0:31
    
@Keyne: I don't understand what you mean by "behind the scenes". To me, the cross multiplication is the "behind the scenes". Do you mean, why is it that "moving" the 100 form the denominator on the right to the numerator on the left gives the answer? Or why we multiply by 100? I've tried to address that, but I'm not clear on what you mean. –  Arturo Magidin Dec 20 '10 at 0:43
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@Keyne: I've tried to add some explanation of what the algebraic manipulations might "mean"; but the point of translating everything into mathematics is that we can perform the operations without having to give them "physical" or "real" interpretations. It's the power of abstraction that makes mathematics so powerful; trying to anchor the operations to the original interpretation kind of defeats the purpose of making things mathematical in the first place. –  Arturo Magidin Dec 20 '10 at 1:16
    
What I'm trying to understand is the relationship between $$20.000 + 17 * 240,96$$ and $$20.000 * 100 / 83$$. Now, after your edit, I see why we have to multiply by 100. Actually the question would be: Why multiply 1% of that integer by 17 and then add 20.000 if we already know what is that 1%. Then, we could just mutiply 1% of that value by 100 to get the final result. Am I right? (I hope so) Thank you very much! –  Telephone Dec 20 '10 at 1:56
    
@Keyne: In your method, 20,000 is 83% and 240.96 is 1%; multiplying the latter by 17 gives you 17%; so the addition is adding 83% to 17% (to get 100%). To connect it, notice that $240.96$ is also equal to $20,000/83$. Substituting that into the sum you have $20,000+17(240.96) = 20,000+17\left(\frac{20,000}{38}\right) = 20,000\left(1 + \frac{17}{83}\right) = 20,000\left(\frac{83}{83}+\frac{17}{83}\right) = 20,000\left(\frac{100}{83}\right)$, so that is how you connect the two expressions. And, yes: multiplying 1% by 100 gives you 100%; that is exactly why the answer is also 20,000*100/83. –  Arturo Magidin Dec 20 '10 at 2:49

I think that the form to see this more easy,is to see it as proportionally; is say:

$$\frac{X}{100}=\frac{20000}{83}$$ (Are equals as they have the same proportion)

Thus only, you must solve the equation for $X$.

Then $$X= \frac{(20000)(100)}{83}$$ Passing to multiply to 100, remember this is solved in cross multiplying .

Arturo had been clear in the other post.

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The "rule of three" is an ancient ad-hoc mindless rote rule of inference that is best ignored. Instead, you should strive to learn the general principles behind it - namely, the laws of fraction arithmetic. Let's consider the example at hand. You seek the number of dollars $\rm\:X\:$ such that when decremented by $\:17\%\:$ yields $\rm\:N\:$ dollars. First, recall that $\rm\ 17\%\:$ of $\rm\:X\:$ means $\rm\displaystyle \frac{17}{100}\ X\:.\ $ Thus your equation is:

$$\rm N\ =\ X - \frac{17}{100}\ X\ =\ \bigg(1 - \frac{17}{100}\bigg)\ X\ =\ \bigg(\frac{100}{100}-\frac{17}{100}\bigg)\ X\ =\ \frac{83}{100}\ X $$

Thus $\rm\displaystyle\ \frac{83}{100}\ X\ =\ N\ \ \Rightarrow\ \ X\ =\ \frac{100}{83}\ N\ $ follows by multiplying both sides by $\rm\displaystyle\ \frac{100}{83}$

Note that we applied no ad-hoc rules above - just the basic laws of the arithmetic of fractions. These are the laws that are worthy of mastering.

It's interesting to look at the decline of the use of the "rule of three" over the last two centuries as the knowledge of general (abstract) algebra evolved. This is very easy using the recently-released Google Books Ngram viewer - which searches for phrases over 5 million books back to 1500. Browsing one of the earliest textbooks in the Google corpus containing the rule of three I noticed that it is immediately followed by a section titled "method of making taxes". So it seems this was a big application in the old days. Also notice how "fraction arithmetic" really ramped up circa 1960 (perhaps due to "new math" programs?).

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Interesting graph... thanks for posting! –  J. M. Dec 20 '10 at 1:03

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