Fractions in Old Algebra Book

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, November 23, 1911: Am working at my algebra in the evening so I can make a better mark than I did last month. If it isn’t any better I will be beyond all hope.

 Her middle-aged granddaughter’s comments 100 years later:

In October Grandma struggled mightily with algebra—the topic was least common multiples (L.C.M.) and highest common factors (H.C.F.)—and she ended up getting a 68% on the exam.

I’m not sure what Grandma was working on in November—but in one early 20th century algebra book—Durrell’s School Algebra, the chapter after L.C.M. and H.C.F. was Fractions.

The book says:

In algebra, a fraction is often useful in expressing a general formula

Here are a couple of exercises from the book:

1. If three boys weigh a, b, c pounds respectively, what is their average weight?

2. If sugar is worth a cents a pound, how many pounds can be obtained in exchange for b pounds of butter worth c cents a pound?

3. If coal is worth c dollars a ton, how many tons can be obtained in exchange for f bushels of wheat worth h cents a bushel and for w bushels of corn worth y cents a bushel?

Lowest Common Multiples and Highest Common Factors

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, October 26, 1911: Have such difficult algebra problems. So much work it is to find the H.C. F. and L.C.M. Good bye for me if we happen to get one of these in examination.

Her middle-aged granddaughter’s comments 100 years later:

Important: If you aren’t into math—skip my comments today and come back tomorrow.  Suffice it to say that Grandma was doing some fairly difficult algebra.

But, if you enjoy math here’s my take on what this diary entry is talking about–

First I’ll give an example of the L.C.M. (lowest common multiple) and H.C.F. (highest common factor) of two whole numbers (integers);  then I’ll explain how it’s done for algebraic expressions.

Integers

The L.C.M. is the smallest integer that two whole numbers can be divided by.  1 would always be the L.C.M.

For example, for 8 and 12  the L.C.M. would be 1.

The H.C.F.(highest common factor) is the largest integer that two whole numbers can be divided by.

For the same two numbers (8 and 12), the H.C.F. would be 4.

If the H.C. F. is 1, it is a prime number.

Algebraic Expressions

The basic idea is the same as for algebraic expressions. For example, for H.C.F. of 2ab and 4a2b is 2a.

But it quickly gets complicated. I’m going to give you directions and examples from a 1911 algebra textbook below for H.C.F. [An aside:  If you really want to understand this concept you might find the information on the algebrahelp.com website helpful.]

Now, here are the directions for finding the H.C.F. in  Durrell’s School Algebra (1912):

The method of finding the H.C.F. is to:

Factor the given expressions, if necessary:

Take the H.C.F. of the numerical coefficients:

Annex the literal factors common to all of the expressions, giving to each factor the lowest exponent which it has in any expression.

Ex. 1:  Find the H.C. F. of 6x2y – 12xy2 + 6y3 and 3x2y2 + 9xy3 – 12y4

6x2y – 12xy2 + 6y3 = 6y(x – y)2

3x2y2 + 9xy3 – 12y4 = 3y2(x2 + 3xy – 4y2) = 3y2(x + 4y)(x – y)

H.C.F. = 3y(x – y)

Whew, I’m getting a headache just typing these expressions. But if you’re still with me, here’s a couple problems you could try from the 1911 textbook:

Find the H.C.F.

1. 4a2b , 6ab2

2. x2 – 3x , x2 – 9

3. x2 + x , x2 – 1 , x2 – x – 2

4. 4a3x – 4ax3 , 8a2x3 – 8ax4 , 4a2x2(a – x)2

5. 3a2 – 10a + 3 , 9a – a3 , (3 – a)3

Old Math Problems

16-year-old Helena Muffly wrote exactly 100 years ago today: 

Wednesday, September 6, 1911: Have to study in the evenings now, instead of sitting around, reading or doing nothing. I got stuck on an algebra problem this evening. Don’t know whether I’ll get it yet or not. I know how to work the problems of that kind but this is a bulky one.

Her middle-aged granddaughter’s comments 100 years later:

I suppose Grandma forgot some math over the summer.

Here are some problems in the first chapter of an algebra book that was published in 1911. Maybe the problems Grandma was struggling with were similar to these.

  1. A bicycle and suit cost $54. How much did each cost, if the bicycle cost twice as much as the suit?
  2. Two boys dug 160 clams. If one dig 3 times as many as the other, how many did each dig?
  3. The average length of a fox’s life is twice that of a rabbit’s. If the sum of these averages is 21 years, what is the average length of a rabbit’s life?
  4. The water and steam in a boiler occupied 120 cubic feet of space and the water occupied twice as much space as the steam. How many cubic feet did each occupy?
  5. Canada and Alaska together annually export furs worth 3 million dollars. If Canada exports 5 times as much as Alaska, find the value of Alaska’s export.
  6. The poultry and dairy products of this country amount to 520 million dollars a year, or 4 times the value of the potato crop. What is the value of the potato crop?

First Year Algebra (1911) by William J. Milne

For additional 1911 math problems see these previous posts:

Odd, Unusual, and Strange Math Problems

1911 Algebra Problems: The Lusitania and Molasses

1911 Algebra Problems: The Lusitania and Molasses

15-year-old Helena Muffly wrote exactly 100 years ago today: 

Thursday, March 2, 1911:  Dear me, what shall I write? Mrs. Hester was out this afternoon. I intended to work thirty-one algebra problems this evening or rather tonight but instead of that I only worked one. Perhaps I may get the remaining thirty tomorrow, but it is only perhaps.

Her middle-aged granddaughter’s comments 100 years later: 

Sometimes I have a vague idea about what I might say about a diary entry—and then I discover something interesting that sends my post in a totally different direction. Today is one of those days—

I found a high school algebra textbook published in 1911 at the library and idly flipped through the pages while pondering—Should I include some example problems from the chapter on Simple Equations  . . . or from the chapter on Quadratic Equations? And then I saw the problem on the Lusitania:

4. One ton of coal will make 8.7 tons of steam. If the Lusitania requires 1200 tons of coal a day for this purpose, how many tons of steam are required for an hour?

First Year Algebra (1911, page 157) by William J. Milne

Lusitania

Wait—Isn’t the Lusitania famous because it was sunk  during World War I by the Germans  in 1915? Why was the Lusitania in a textbook published in 1911?

And, as I sought answers, this post  headed in a totally different direction.

The Lusitania was a British ship that made its first trans-Atlantic trip in 1907—and it periodically held the world record as the fastest ship to make the crossing. For example, in October 1907, it held the record for an eastbound trip with a time of 4 days, 19 hours, and 53 minutes. The average speed was 24 knots/hr. (27.6 miles/hr.).

(Cruise ships today don’t cross the Atlantic as quickly as they did a hundred years ago. It now takes at least 6-7 days to make the crossing. I guess that if  someone wants to cross quickly they just fly.)

In the early 1900s there were several very fast ships that held the record at one time or another. They informally competed with one each other and the newspapers regularly reported on when the ships entered the New York harbor –or  the harbors in England on eastward trips– since there was the potential with every trans-Atlantic voyage that the world record would be broken.

A hundred years ago the general public across the US knew about the Lusitania and were following its story even before it was sunk by a German torpedo. (And, the Lusitania was apparently considered a good topic for an algebra problem since it was a timely, high-interest topic that might motivate students ).

Algebra problems provide lots of hints about what was common knowledge a hundred years ago. For example, would you ever find a problem about molasses pumps and tubing in a text today? Well, it provided the context for the word problem that followed the Lusitania problem in the 1911 textbook:

5. A grocer paid $8.50 for a molasses pump and 5 feet of tubing. He paid 12 times as much for the pump as for each foot of tubing. How much did the pump cost? the tubing?

First Year Algebra (1911, page 157) by William J. Milne

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