Marveling At The Historical

Math Oldies But Goodies

  • About This Blog

    This blog is mostly about math procedures in textbooks dated from about 1825-1900. I’m writing about them because some of the procedures are exquisite and much more powerful, and simpler, than some of the procedures in current text books. Really!

    I update this blog as frequently as possible ... every 2-3 days. And, if you are a lover of old texts and unique procedures, you might want to talk to me about them, at I’m not an antiquarian; the books I have are dusty, musty, brown-paged scribbled-in texts written by authors with insights into how math works. Unfortunately, most of their procedures have vanished. They’ve been overcome by more traditional perspectives, but you have to realize that at that time, they were teaching the traditional methods.

Percent Proportion

Posted by mark schwartz on March 24, 2016


It seems that the proportion idea fits the idea of percent well. In an 1850 text (sorry, no reference ̶ I misplaced this volume) provided a proportion approach for solving percent problems. The author preferred the proportion method over the standard percent equation method. Also in that text, many unusual problems such as “.006 is .04% of what number?” dramatically illustrated the value of the proportion method for solving percent problems. As it turns out, students in my developmental math responded quite favorably to the concept, verbalizing and emphasizing the importance of having a visual reference point.

The Story.

Most texts present the percent equations first and then perhaps note the percent proportion idea. I use the plural because typically 3 “percent equations” are presented, although it’s the same equation with each case having a different unknown. The basic equation amount = percent x base can have any of the 3 elements as the unknown. The usual solution is to state the percent as a decimal, if given the percent, and either find the base or the amount. If the percent is the unknown, then the amount divided by the base gives a number, usually a decimal, which is then restated as a percent (move the decimal point two places to the right and affix the percent sign). This latter “restating” is sometimes a catch-point for students, noticeably if the calculated percent is greater than 100 or a relatively small decimal, like .0005. This is so because, although the rationale for “moving the decimal point” is typically taught, it also typically fades into non-existence, so the “why” doesn’t get affixed to the operation.

This is what occurs with the percent equation concept, which is accurate as long as the decimal movements are performed before and/or after any calculation. Also, sometimes identifying which is the base and which is the amount is problematic, particularly with more sophisticated problems.

However, the equation form of amount/base = percent, can be rephrased if the percent value is placed over 100 (which is really what percent means). Allowing the percent expressed not as a decimal but rather in the fraction form as some number over 100, the result is the proportion form of the percent relationship, or

In this form, if the percent is given in the problem, that number without the percent sign is placed over the 100. No decimal point moving occurs! Simply, if the problem notes a .0005% rate, the student need not be concerned with “where does the decimal point go”? Here the .0005 is simply placed over 100, since .0005% is the same as .

Another advantage of the proportion form can be seen in a few examples. These examples show several things. First, the “1” and the “50” in the first proportion and the “3” and the “60” in the second proportion are equivalent values relative to their respective denominators; in essence, 2 equivalent fractions. The point is that if these are written as proportions then the “1” and the “50” in the first proportion and the “3” and the “60” in the second proportion will be at the same point on their respective ratio. Visually, these proportions can be seen as:

1          =            50              and                3         =         60
2                        100                                  5                     100

Other Stuff      Percent                   Other Stuff      Percent

So, the “1” in the “2” ratio IS at the same point as the “50” on the “100” ratio, and similarly for the other example. The word “IS” is used to indicate the values being at the same “height” on their respective scales. Using the 1/2 = 50/100, the 1/2 ratio labeled the “other stuff”, since the values can represent apples, giraffes, eggs, money, or some amount of something, and the 50/100 is labeled the percent ratio.

The second thing these examples show is that the percent is represented as a number over 100, so if the percent is given, the number is placed over the 100, or if the percent is what is being looked for, the number found is the percent, no decimal-point moving need occur.

Typically, proportions have been studied before looking at percent, so students know how to solve proportions. The proportion can now be solved by “cross-multiply and divide”, where cross-multiply is done with the 2 values which are in a position to be cross-multiplied. Note that in using the percent proportion form, the “100” is a fixed value.

In using this proportion form, the first question to be asked when given a problem is: “are you given a percent or are you looking for a percent?” Sketch the proportion and fill in the “100”. If given a percent, place that value above the 100. If looking for a percent, put an “x’ over the 100. Two of the four places on in the proportion are now filled, and the problem will now have 1 value to put in one of the two remaining places on the “other stuff” scale. Please note that in more sophisticated percent problems, some work may need to be done before values can be placed (for example, in percent increase or decrease problems). But take the problem “16 is what percent of 40?” Since percent is not given, an “x” goes on the percent scale. Looking at the phrasing of the problem, it can be seen that “16 is what percent…” indicates that the 16 goes on the “other stuff” scale opposite the percent “x”. This is:

16  = x 
40   100

“Cross-multiply and divide” gives “1600 ÷ 40” and entering this expression in a calculator gives the answer “40”. Since no decimal point moving happens in this percent proportion form, the answer is 40%.

Another example illustrates further the value of the percent proportion form. Given the problem, “15% of 60 is what number”? First, given the 15% in the problem, the 15 goes over the 100. Now the question is “where does the 60 go on the “other stuff” scale?” Look back at the problem and realize that it can be rephrased. It can be stated as “what number is 15% of 60?” and parsing this phrase, the outcome is “what number is 15%…” which identifies the unknown “what number” on the “other stuff” ratio as opposite the “15” on the percent ratio. The proportion is:

x  =  15
60    100

“Cross-multiply and divide” gives “(15 x 60) ÷ 100 =” and the calculation gives the answer “9”.

Finally, “short-cuts” or short-forms for this percent proportion form can be found, but presenting percent as a proportion is a good way of understanding the relationships in percent problems, particularly more sophisticated problems, like … .0056 is .0004% of what number? … or … After a 25% reduction in price and an additional 10% reduction, the cost of an item is $81 dollars. What was the original price?

I took license with the 1850 model and pointed out to the students that labeling the values on the percent and “other stuff” ratios was a good reminder of the relationships between the numbers in the problem and what is being solved for. For example, Jane earns a 12% commission on every sale she makes. If she makes a $1575 sale, what commission did she earn?

Amount comm.          % comm.

     x      =       12
    1575           100

Writing a few simple words first of all slows things down; students have to think about the relationships. Basically, the label on the percent ratio will be “% something” and on the other stuff ratio “amount of something”, where “something” is the same label on both scales. Then, after the calculation is done, the answer is already labelled.

I found this labeled proportion method to be an effective method for several reasons. First, it links to something already known – how to solve a proportion. Second, it’s a visual and verbal representation of the number relationships. And third, there is no need to move decimal points. In the sample problem from the 1850 text this method handles quite easily all the decimal point concerns. Further, even if the given percent is greater than 100% or the percent found is greater than 100%, this conception still works well it’ you don’t have to mess with moving a decimal point. Two examples follow. First, using the 1850 problem “.006 is .04% of what number?” sets up as:

.006  =  .04
  x      100

This problem sets up quite readily based on the key word “IS” in the statement of the problem. The answer is 15. Before doing the problem, most students believed that the answer also had to be a decimal value. Again, no messing with moving decimal points.

The following problem indicates that thinking about and rephrasing the original problem sometimes clarifies where the values need to be placed on the scales. Take the problem “Mary always marks up items for sale in her store by 15%. If she sells an item for $82.80, what was the original price?” I proposed to the class that before they write anything, try to identify the relationships. I only offered questions as we progressed through this problem. First, what does the $82.80 represent? The class agreed that that was the cost after the 15% was added, so rephrased, it was noted that it could be stated as “cost + markup” (abbreviate this as c+m). Given that, the question was “what was the original percent?” This seemed puzzling at first but there was collective acceptance that the original percent was 100%. So, if the “cost + markup” is the “other stuff” element, what would be the percent element? Again, the class determined that it would have to be the original percent + the percent markup (abbreviate this as p+pm), or 115%. So, drawing the scales and labeling them, they derived:

c + m                   p + pm

82.80   =    115
  x          100

Several other problems like these gave these developmental math students the confidence to approach other kinds of percent problems and further made them aware that many math concepts are related – equivalent fractions, proportion and percent are “all the same” one student said.



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