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.

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Posts Tagged ‘Proportions’

In 1877, Mr. Ray Reasons with Fractions

Posted by mark schwartz on September 8, 2016


In Mr. Ray’s 1877 Ray’s New Intellectual Arithmetic, an elementary school text, he presents some of the problems with their solution. A sample of these are worth looking at because in every case he shows a solution method which is based on fractions and knowing how to handle a sequence of fractions. But it’s not only the sequence of fraction operations but also the logic of these fraction operations that elementary school children had to follow. This required them to think about the relationships in the problem. I’d like to further note that this method of solution for all 7 problems presented here is seen in many of the texts of that era. It really required students to understand fractions! I’m not proposing that we use this “fractional” method in lieu of solving them by either proportions – the first 4 problems – or simple equations, the last 3 problems.

The Story

All these problems are from his text. Read the solutions slowly to really enjoy the subtlety of the method.

  1. A yard of cloth costs $6, what would 2/3 of a yard cost?  (Pg. 48, # 3)

Solution: 1/3 of a yard would cost 1/3 of $6, which is $2; then, 2/3 of a yard would cost 2 times $2, which are $4.

  1. If 3 oranges are worth 15 cents, what are 2 oranges worth?  (Pg. 49, #19)

Solution: 1 orange is worth 1/3 of 15, or 5 cents; then 2 oranges are worth 2 times 5 cents, which are 10 cents.

  1. At $2/3 a yard, how much cloth can be purchased for $3/4?  (Pg. 75, # 5)

Solution: For $1/3, 1/2 a yard can be purchased, and for $1, 3/2 of a yard; then, for $1/4, 1/4 of 3/2, or 5/8 of a yard can be purchased, and for $3/4, 9/8 = 1 and 1/8.

  1. If 2/3 of a yard o cloth costs $5, what will 3/4 of a yard cost?  (Pg. 101, # 2)

Solution: The cost of 1/3 of a yard will be 1/2 of $5 = $5/2; and a yard will cost 3 times $5/2 = $15/2; then, 1/4 of a yard will cost 1/4 of $15/2 = $15/8; and 3/4 of a yard will cost 3 times $15/8 = $5 and 5/8.

Note that these 4 problems lend themselves well to being solved using proportions. What follows now are 3 more problems, which if presented in today’s texts would likely be solved with simple equations, but again Mr. Ray’s solutions are a sequence of fraction operations.

  1. If you have 8 cents and 3/4 of your money equals 2/3 of mine, how many cents have I? (Pg. 52, #17)

Solution: ¾ of 8 cents = 6 cents; then 2/3 of my money = 6 cents, 1/3 of my money is 1/2 of 6 cents = 3 cents, and all my money is 3 times 3 cents = 9 cents.

  1. Divide 15 into two parts, so that the less part may be 2/3 of the greater.  (Pg. 106, #1)

Solution: 3/3 + 2/3 = 5/3; 5/3 of the greater part = 15; then, 1/3 of the greater part is 1/5 of 15 = 3, and the greater part is 3 times 3 = 9; the less part is 15 ̶ 9 = 6.

  1. A and B mow a field in 4 days; B can mow it alone in 12 days: in what time can A mow it?  (Pg. 110, #14)

Solution: A can mow 1/4 ̶ 1/12 = 1/6 of the field in 1 day; then he can mow the whole field in 6 days.

I hope you appreciate what elementary school students had to do at that time. Since it was elementary school, they weren’t taught proportions and simple equations but they were “exercised” with fractions in a way that I believe could benefit today’s students understanding of fractions.


Posted in algebra, basic math operations, fractions, Historical Math, math instruction, mathematics, proportion, Proportions, remedial/developmental math | Tagged: , , , , , , | Leave a Comment »

The Importance of a Clearly Stated Algorithm

Posted by mark schwartz on August 22, 2016


I posted a piece earlier in this blog titled Sheldon’s Compound Proportions. It describes what Sheldon labels the “cause and effect” method for solving compound proportions, which as far as I can tell, aren’t in todays’ texts. His work was in 1886. You might want to take a look at his idea because this posting talks about other compound proportion procedures at that time and I did it to emphasize the importance of a clearly stated procedure for doing an operation.

The Story

I strolled through my collection of old texts and in quite a few of them found the same prescription for solving compound proportions not using cause and effect. I picked 5 which cover about a 20 year span from 1864 to 1883. They all have the same procedure and what I suspect is that it was the established and accepted solution method at that time. As in todays’ texts, it was just a simple matter of “borrowing” a basic algorithm from someone else’s work. There are other texts of that era which reference Sheldon’s cause and effect method and a few of them introduce it along with the procedure I’ll cite below.

The point is that his method is a much clearer statement of how to handle the information in a compound proportion problem. Further, what I’m suggesting is that we should carefully examine some of our current traditional algorithms to see if the reason students have trouble with them is because of the way they are worded and presented. For example, finding the lowest common denominator (LCD) in order to add/subtract fractions doesn’t require the extended way it’s been typically taught. In fact, I have seen some texts introducing a method which doesn’t require finding an LCD at all. Certain mixture problems can more readily be solved with an 1864 method Mixing it up with Alligation, posted earlier in this blog.

By the way, the 5 texts in which I found this procedure are all arithmetic texts, which indicates to me that this somewhat sophisticated idea of compound proportion was taught in elementary school. I’ll give you example problems from an old text to indicate that, in my view, it was a very handy procedure for the real world experience at that time. Today we call these “application” problems.

Here’s the rule as stated in Greenleaf’s 1881 The Complete Arithmetic, page 235 (the other 4 books are cited below and present the same rule).

Rule for Compound Proportions

“Make that number which is like the answer the third term. Form a ratio of each pair of the remaining numbers of the same kind according to the rule for simple proportion, as if the answer depended on them alone. Divide the product of the means by the product of the given extreme, and the quotient is the fourth term, or answer.”

Embedded in this is reference to “…the rule for simple proportion …” which Greenleaf provides on page 233 and it is:

Rule for Simple Proportions

“Make that number which is of the same kind as the answer the third term. If from the nature of the question the answer is to be larger than the third term, make the larger of the remaining numbers the second and the smaller the first term; but if the answer is to be smaller than the third term, make the second term smaller than the first. Divide the product of the means by the given extreme, and the quotient is the fourth term, or answer.”

Students had to be able to apply this latter rule for simple proportion before being presented compound proportion. There is no conflict between the two rules; in fact, there is some overlap. For simple proportions, the rule directs the student to understand “the nature of the question …” and use that to determine which values go in which of the 4 places in the proportion. The students had to be able to assess and estimate if the answer was going to be larger or smaller and place the correct terms in the first and second places. Wow! There is a lot of estimating and juggling of values and basically it seems that all of this effort is aimed at what we would say today as determining whether it’s a direct or inverse proportion. With problems with simple values, this is a somewhat manageable issue.

For example, a problem from the text is “If a man travel 319 miles in 11 days, how far will he travel in 47 days?” Using the rule for simple proportion, the setup would be:

11/47 = 319/x    (the rule doesn’t use “x”, but I did for demonstration purposes)

The solution is (47×319) ÷ 11 = 1363

However, in today’s approach to simple proportion, the setup (in most cases) simply follows from the order of the information in the problem, giving:

319/11 = x/47

This gives the same answer but notice that the rule states “Divide the product of the means by the given extreme …” and that doesn’t apply here. So, the 1881 rule is quite constraining when it comes to writing the proportion, when indeed there are several ways to set up the proportion for the problem.

Again, there is nothing wrong about the simple or compound proportion rules as provided by Greenleaf. The issue is that the rules are somewhat convoluted and constraining. If a student doesn’t learn this algorithm and follow it precisely, the likelihood is that the correct answer won’t be found. There are a lot of words referring to the terms and judgements that a student must make about which terms go where in the proportion. Further, look at what happens with a compound proportion problem, again from Greenleaf (#67, page 236):

“If 12 men in 15 days can build a wall 30 feet long, 6 feet high, and 3 feet thick, working 12 hours a day, in what time will 30 men build a wall 300 feet long, 8 feet high, and 6 feet thick, working 8 hours a day?”

Now, where does a student begin sorting through all this information if they use the rule above for simple proportion? What’s the “nature of the question”? For example, the rule states “…make the larger of the remaining numbers …” and how is a student to know which number is to be selected? I can visualize the instructor explaining in excruciating terms how all this works. Again, it’s not impossible to apply the rules as stated in 1881 but I urge you to look at Sheldon’s Compound Proportions in this blog and see how much more direct the rule is by framing information as cause and effect.

Briefly, Sheldon’s 1886 statement of the procedure:

“The solution of every example in proportion proceeds on the assumption that effects are in the same ratio as the causes that produce them. Every proportion is the comparison of two causes and two effects. In the method known as Cause and Effect, the causes form one ratio, and the effects the other. The first cause and the first effect are antecedents; the second cause and second effect consequents.”

Notice the simplicity of identifying cause and effect and then the causes forming one ratio and the effects the other. The words” antecedents” and “consequents” could be updated to 1st and 3rd term and 2nd and 4th term, respectively.

Taking the above compound problem the 1st causes are 12 men, 15 days 12 hours a day and the 1st effect is to build the wall 30 feet long, 6 feet high, and 3 feet thick. The 2nd causes are 30 men working 8 hours a day and the 2nd effect is to build a wall 300 feet long, 8 feet high, and 6 feet thick. You are to find “…in what time…” which is a 2nd cause. There is a shortcut that can be used but let me show you – in what I call slow-motion-math – one way to make sure the terms get placed correctly. I typically use the labels and then replace it with the values (for a lot of different types of problems, not just compound proportions). The proportion following Sheldon’s procedure is:

Causes                     Effects

1st       men, days, hours         length, height, thickness

2nd       men, x, hours               length, height, thickness

I used “x” for days in the second cause. If the numbers are substituted, we have:

12•15•12 = 30•6•3
30•x•8     300•8•6

Cross-multiply and divide, solving for x and the answer is 240.

Again, a detailed description of the “cause and effect” is in Sheldon’s Compound Proportions in this blog.

The essence of this posting is to demonstrate the importance of a well thought-out procedure expressed in easily understood language. If you are an instructor, you likely have done this kind of “simplifying” of the algorithm because as stated in the text, it seemed too fussy for students to follow. Not every algorithm can be simplified but I believe it’s an instructor’s responsibility to make math more accessible to students by removing the fog of awkwardly phrased rules and algorithms. Give it a try.

Posted in algebra, basic math operations, Historical Math, math instruction, mathematics, proportion, Proportions, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Metric to Metric Conversion: Ultimately, it’s a Proportion!

Posted by mark schwartz on July 27, 2016


This discussion came about because one student in one class simply asked “Why does this work”? He was referring to the procedure for converting metric units to other metric units, for example, “how many centimeters are there in 10 kilometers?” He could see that the “moving the decimal point” procedure worked but he kept insisting that there must be more to it; that somehow someone had figured this out and he wanted to know how it had been figured out. I had no clear answer to this and told him (and the class) that I would think about it. What I came up with isn’t necessarily the reality of the derivation of that procedure, but it did start with something that we had already discussed in class – proportions – and he was willing to accept this as a demonstration of why it works but wasn’t about to consider it a true explanation. Loved this guy!

The Story

In Colaw and Ellwood’s 1900 School Arithmetic: Advanced Book (page 252) is a discussion of the metric system. Among other interesting things, they note that kilo hecta and deka are Greek, while deci, centi, milli are Latin. In reading through their discussion, I got to thinking about how they, as we do today, convert one metric unit to another: a 7-point scale and simply “move the decimal point” … but some of their commentary made me think about how and why this 7-point scale works.

If asked how many decimeters are in .04 kilometers, one has a variety of strategies to use. If the 7-point scale (kilo hecta, deka, unit, deci, centi, milli) is known, one can write .04 at the kilo point on the scale and then visualize moving from kilo to deci, which would give a move of 4 places to the right. If the decimal point is moved four places to the right, this shows that .04 kilos is 400 decis. Typically, students are accepting of this ‘shortcut’ because it is much more manageable than other systems. But, the question was “why does it work?”

I believe the underpinning for the move-the-decimal method is to do the problem by first converting all the units to the amount at each point on the scale that equals one unit. This by no means is a rigid mathematical derivation but rather a way of demonstrating the relationships using a previously studied math relationship, namely proportions.

The traditional 7-point scale looks like this:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1000             100                  10                    1                      1/10                 1/100               1/1000

This scale shows the number of units in a named place-value. “Kilo” means 1000 units; “deci” means 1/10 of a unit, etc.. But let’s ask the question from the point of view of the unit: how many kilos would it take to make a unit? How many decis would it take to make a unit, etc.?

Here’s how the “unitized” 7-point scale would look:

Kilo                 hecta                deka                unit                  deci                 centi                milli

1/1000          1/100                 1/10                 1                      10                    100                  1000

It appears as though the scale has been reversed, and it has because we are viewing the information from the perspective of what it takes to make one unit. For example, it can be read as “1/1000 of a kilo equals 1 unit” or “10 decis equals 1 unit”, etc. The point of this is that all of the place-value names are now on the same scale and having them on the same scale permits one to establish proportions.

For example, on this scale 1/1000 of a kilo equals 10 decis because they both equal 1 unit. Another way of stating this relationship is to state that “1/1000 kilos is to 10 decis”, which is a phrase describing the first rate of two rates that would make up a proportion. What would be the second rate? The original problem was “how many decimeters are in .04 kilometers?”

In this case, being consistent with the idea in proportions that the numerators are all the same type of units and the denominators are all the same type of units, what is seen is the relationship of kilos to decis, is:

kilo  1/1000  =  .04  = 400 decis
deci    10        x

It is this proportional relationship which provides the basis for conversions from one metric unit to another, as long as the units used are those that “equate” them to 1. Students must be comfortable knowing how many ‘dekas’ it takes to make one unit (since a ‘deka’ is 10 units, it takes 1/10 of a ‘deka’ to equal a unit). This may seem counterintuitive since we typically say, for example, that a kilo is a thousand units, which is true but the focus here is with how many kilos it takes to make a unit.

Given this discussion of the two methods, it seems most likely that students would tend toward the ‘move the decimal point’ system. It doesn’t require any computation. But the point of presenting both of these it to bring out the reality that the ‘easier’ system is based on a proportional system. Just another example of the power of proportions based on an interested student’s insightful inquiry.

Posted in algebra, basic math operations, math instruction, mathematics, proportion, remedial/developmental math | Tagged: , , , | Leave a Comment »

What? That Much Percent Increase?

Posted by mark schwartz on July 8, 2016


I like coincidences. Particularly when they provide learning opportunities for my students. We had just spent time learning about percent and percent increase and decrease. The problems in the text were good but not really challenging. The coincidence was that I was reading John McPhee’s The Curve of Binding Energy (1974) and I’ll start the story with what he said on page 18.

The Story

“Thousands of miles of tubes, pipes, and other conduits were needed to create a network of flow wherein the gas could now go through a membrane, now return to try again, now go on to a new membrane, gradually advancing, in a process of separation and elimination, until what had begun as seven-tenths of one percent U-235 was more than ninety percent U-235 – fully enriched, weapons-grade uranium.”

I’d never heard this detailed an explanation of how weapons-grade uranium was made. But what really got my attention was that his statement could be a percent increase problem. I worked it out before I gave it to the class, rounding off the initial “more than ninety percent” to a manageable 90%.

Further, I decided that it would be an in-class extra credit exercise and allowed that the students had to first work within their assigned group, but once they had an answer they could discuss it with other groups.

I did this because the percent increase is 12,757% and this size percent increase would cause the groups to question what they did, even if they got that number. There were occasional answers to the problems in the text that resulted in percent increases of more than 100% but nothing quite like this. Once I gave them the problem and answered any preliminary questions and they got to work, I roamed the room listening to the strategies they came up with to do the problem.

The first issue was how to numerically express seven-tenths of one percent. One group asked if they could talk to another group to get help expressing it. So, I stopped the class and said that if they are willing to accept the following condition, they can work as a class to get the answer. The condition was that everyone in the class would get the same grade. They accepted. There was an eruption of conversation and as I roamed around, I was asked if what they got was right. I just referred them to other members of the class.

Once there was consensus on how to represent the initial percent, they simply continued with what they had learned about setting up percent increase problems. By the way, I taught a somewhat non-traditional method that doesn’t use a formula, rather it uses a somewhat modified percent proportion approach. You can look at it in this blog at Percent Proportion.

Several groups quietly called me over to show me their result, asking if they were right. Some were and some weren’t but I wouldn’t say yes or no, reminding them of the condition under which they were working. So, more talk, discussion and exchange of how to set up the problem.

It was interesting to watch the evolution of the shared work – people got up and moved around the room; some asked to and did use the white board; I heard a lot of “show me” and “why did you do that?” and “that doesn’t seem right”. But, ultimately there was class consensus on the right answer.

They did, however, insist that I walk through how I thought about it even though they got it. So, I put on what I call my “slow-motion-math” hat and gave them the following:

Ninety percent is 90/100, so the amount of increase is 90 – .7, or 89.3%. Seven-tenth of one percent is (7/10)(1/100) or 7/1000 (I did this because I saw a lot of questioning on how to express it). This in percent is .7/100 (or if you were sure, you could have just written .7 over 100). So the question can be put in a percent increase frame. First, the amount of increase is 89.3 and since it started at .7, the amount of increase relative to the beginning point can be expressed as 89.3/.7, or 893/7 (they questioned if doing this would give the same answer and we discussed this). Using the proportion statement 893/7 = x/100 gives a percent increase of 12,757%, rounded. So, doing the original problem led to some other related talk about fractions, decimals and rounding. Neat.

After all was said and done, I got questioned about this exercise because there was a sense that it was a trick question. I have noticed that when students feel uncertain about a math problem, the frequently asked question is just that. I then heard stories from the class about their prior math experiences where trick questions unfortunately were used to presumably teach them something about math, but the only learning was frustration because a lot of the tricks were beyond the bounds of what had been taught and in essence they quit. Given what I heard, I may have quit too. Somehow they concluded that math is just knowing the right tricks.

But once they were accepting that it was an interesting problem, I noted to them that as they read books, magazines, watch TV or come across “mathy” stuff, they might play with it as we did with this problem. And of course they noted to me to record the “A” for all of them.

Posted in basic math operations, Historical Math, math instruction, mathematics, percent, proportion, Proportions, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Percent Problems from 1868

Posted by mark schwartz on June 28, 2016


After presenting percent and having students work many problems from the text, I decided to give them a set of problems from 1868. I did this for several reasons: first, I gave this as an in-class quiz in which I allowed the entire class to discuss strategies, compare answers, and work with others. I roamed the room to watch things happen. Second, I gave these problems because they all had fractions in them as well as related information that they had to decide where it fit in the problem. Problems like these are rare in today’s texts. It was a real challenge for the class but some of the students actually said it was fun! The story begins with a copy of what I handed to them. I’d like to note that I taught a visual percent proportion method only (no formulae, no short cuts) – Percent Proportion – which you can see in this blog. It’s a slower procedure but gives students a better understanding of percent. They said so.

The Story

Here’s the assignment I gave the class …

The 5 problems below are from The Progressive Practical Arithmetic by Daniel Fish. (Ivison, Phinney, Blakeman & Co, Chicago, 1868, pg. 228). There are some minor punctuation and word changes.

Please show all your setups and solutions. Consider carefully how to handle the fractions in the problems. Consider that only non-repeating decimals will give you a precise and accurate answer, so you may have to use improper fractions in your calculations, or use the “fraction” key if your calculator has one. Feel free to work with anyone (or everyone) in the class. Be careful handling dollars, cents and decimal points, and fractional answers. ALL ANSWERS ARE MONEY – DOLLARS AND CENTS.

Before you start, any questions?

  1. A miller bought 500 bushels of wheat at $1.15 a bushel, and he sold the flour at % advance (profit) on the cost of the wheat. What was his gain?

2. A grocer bought 3 barrels of sugar, each containing 230 pounds, at cents a pound, and sold it at percent profit. What was his whole gain?

3. A sloop, freighted with 3840 bushels of corn, encountered a storm, when it was found necessary to throw percent of her cargo overboard. What was the loss, at cents a bushel?

4. A gentleman bought a store and contents for $4720. He sold the same for percent less that he gave, and then lost 15 percent of the selling price in bad debts. What was his entire loss?

5. A man commenced business with $3000 capital. The first year he gained percent which he added to his capital. The second year he gained 30 percent on the whole sum, which gain he also put into his business. The third year he lost percent of his entire capital. How much did he make in the 3 years?


Roaming the room and hearing the discussions and debates was a lot of fun. I occasionally answered questions with questions on how to handle the values and the relationships. As it turned out, when the work was turned in, everybody got it! And, of course, the class wanted this format for all future quizzes, but I noted to them that I had other formats in mind. Someone pointed out how valuable this learning experience was and why would I not want to repeat it? Ah, logic, but as it turned out the other quiz formats turned out to be good learning experiences too.

Below are the answers if you want to play with these 5 problems.

  1.    $95.83 &1/3 (I accepted $95.83; our money system stops with pennies)
  2.    $10.35
  3.    $900
  4.    $1209.50
  5.    $981.25

Posted in fractions, Historical Math, math instruction, mathematics, percent, proportion, remedial/developmental math | Tagged: , , , , | Leave a Comment »

Sheldon’s Compound Proportions

Posted by mark schwartz on April 13, 2016


Here’s another one of those delightful conceptions from one of my old texts, this one dated 1886. It addresses simple and compound proportions. I want to note that compound proportion is presented in many of the other texts in my collection, yet this topic seems non-existent in today’s math curriculum. Also, the authors at that time had a variety of approaches to compound proportion, but I find this one to be the clearest conception, primarily because he uses the terms “cause” and “effect”, which are common terms in today’s conversations and it seems to make it easier to visualize the relationships. Some of the other methods introduced special terms and relationships that I found cumbersome or somewhat vague.

I have on occasion presented it in Algebra classes that collectively were interested in pursuing the idea. They expressed interest because in some of the “application” problems in the text, students had to do 2 proportions serially in order to get an answer, and they found this troublesome. Examining the problems made me aware that Sheldon’s compound proportion method would work.

The Story:

In Sheldon’s Complete Arithmetic (1886) he presents concepts about simple and compound proportions. His method for simple proportions is more complex than today’s method so it won’t be discussed here. If you’re interested, contact me.

Again, as far as I can tell, most current texts that introduce proportions do not include compound proportions. This is an interesting historical shift in the depth of topics covered in mathematics curriculum in the 19th century compared to the 20 and 21st centuries. Other texts of Sheldon’s era routinely address compound proportions using a variety of approaches.

Sheldon has two proposals for working with compound proportions. The first is labelled “Compound Proportion” and the second is labelled “Cause and Effect”. I believe the second one he presents is the most interesting and useful. I worked problems from both sets and interestingly, some of the problems presented for “Compound Proportion” can be solved readily with Cause and Effect but problems in the set for Cause and Effect present difficulties for his “Compound Proportion” method.

So, only his Cause and Effect will be presented here and I believe a good way to introduce compound proportions is to show you the problem he used to demonstrate it, which is:

“If 18 men in 10 days can cut 540 cords of wood, in how many days can 15 men cut 630 cords?”

And now Sheldon’s statement of the procedure:

“The solution of every example in proportion proceeds on the assumption that effects are in the same ratio as the causes that produce them. Every proportion is the comparison of two causes and two effects. In the method known as Cause and Effect, the causes form one ratio, and the effects the other. The first cause and the first effect are antecedents; the second cause and second effect consequents.”

I found that a visual representation of the setup of the proportion helped students understand his method. This includes not using the terms antecedents and consequents, rather beginning the discussion of the method with words, not numbers. Consistent with his statement, the proportion is:

1st cause  =   1st effect
2nd cause      2nd effect

We discussed this in class, emphasizing the relationship of causes to effects.

Although Sheldon’s statement does not include how to solve the proportion, it will be solved just as with simple proportions – cross-multiply and divide. I believe the examples below will clarify how to solve a compound proportion.

Restating the problem:

“If 18 men in 10 days can cut 540 cords of wood, in how many days can 15 men cut 630 cords?”

The first cause is “18 men in 10 days” and the first effect is “540 cords” and the second cause is “in how many days can 15 men” with the effect being “cut 630 cords”.

The “x” will represent “…in how may days”… in the second cause. In “slow-motion math”, here’s how the proportion is set up first with labels, not numbers, and then solved using cross-multiply and divide when the numbers are substituted. The students need not always use the labels but it was found, at first, to be quite helpful in visualizing Sheldon’s method.

Causes     Effects

men*days = 1st cords
men*x      2nd cords

18*10 =    540
15*x       630

540*15x = 18*10*630

x = 18*10*630

x = 14

A brief aside here. Remember that in 1886, there were no calculators, so that most of Sheldon’s examples and problems allowed for a lot of cancellation in the initial ratios, or if the values are carried through rather than doing the indicated multiplications. This isn’t necessary today given students can use calculators. Indeed, from the setup of the problem the student could simply enter (18⦁10⦁630) / (540⦁15). Again, note that the term with the “x” is the divisor.

Look at this and realize that there are two entries in both the first and in the second cause. As in this problem, the conditions of a problem may require multiple entries in cause, or effect, or both (an example is below). When making these entries it’s just a matter of slowly and carefully doing it.

Reflect on the setup and solution for a while before moving onto the next example. I intentionally picked one that on the surface appears very messy but if Sheldon’s method is followed, it becomes clear (at least I think it does!)

The problem: How many men working 10 hours a day will be required to build a wall 160 feet long, 40 feet high and 3 feet thick in 15 days, if 75 men in 12 days of 9 hours each can build a wall 120 long, 30 feet high and 2 feet thick?

I did say it was messy, didn’t I? Some of this can be simplified before putting it in the compound proportion, although all these numbers could be placed according to Sheldon’s method. But, let’s do some preliminary calculations. The 1st cause includes what is being looked for (how many men?), so there is the “x”. The 1st cause also has 10 hours a day for 15 days, or 150 hours. The volume of the wall in the 1st effect is 160⦁40⦁3 = 19,200 cubic feet. The 2nd cause has 75 men and also 12 days of 9 hours each, or 108 hours. The volume of the wall in the 2nd effect is 120⦁30⦁2 = 7200 cubic feet.

So, here’s how the information is placed in the compound proportion …

150*x = 19200
108*75  7200

Cross-multiplying and dividing … (75⦁108⦁19200) / (150⦁72000) … gives 144.

If all the cause and effect values were entered separately – no preliminary calculations – here’s what it would look like (again recall that in 1886, there weren’t calculators but there was simplifying each fraction in the proportion before doing the calculation).

10*15*x  =  160*40*3
12*9*75     120*30*2

Again, the “x” is not included in the calculation but recall that the term with the “x” is the divisor. Below are a few more problems, with answers, for playing with Sheldon’s idea. Just take your time identifying the causes, effects and what you’re looking for. Have fun!

1. If 10 horses in 14 weeks eat 5.65 tons of hay, how long will 11.3 tons last 7 horse?     Answer:   40 weeks

2.  If the freight on 150 cattle averaging 900 pounds is $250 for 100 miles, what should be the freight on 275 cattle averaging 1200 pounds, for 150 miles?   Answer: $916.67

3. If a party of 15 persons pay $690 for 8 weeks’ board, how much should be paid by a party of 12 persons for 16 weeks’ board?      Answer:  $1104

4. If 12 men working 8 hours a day can saw 340 cords of wood in 10 days, how long will it take for 16 men working 9 hours a day to saw 204 cords?              Answer:  4 days

5. When $280 is paid for 175 barrels of apples containing 2 bushels each, how much should be paid for 125 barrels of apples containing 2.5 bushels each?        Answer:   $250

6. … and from Brook’s 1877 The Normal Higher Arithmetic (pg. 432) comes the following problem, which requires you to carefully determine what are the causes and effects.   If 24 pipes, each delivering 6 gal. a minute, fill a cistern 8 ft. long, 6 ft. wide, and 5 ft. deep, in 12 and 36/77 min., how many pipes, each flowing 8 gal. a minute, will a cistern 10 ft. long, 7 ft. wide, and 9 ft. deep, in 21 and 9/11 minutes?         Answer:  27 pipes






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