Capacity (Volume) Word Problems

Posted on January 16, 2013 by Sheryl

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

Thursday, January 16, 1913:  We had an examination in Geometry this morning. I think I will make a better mark than what I did the other time.

Source: Kimball's Commercial Arithmetic (1911)

Source: Kimball’s Commercial Arithmetic (1911)

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

What was the Geometry test about—proofs? . . . angles? . . . shapes? . . . capacities?

The directions for doing capacity problems in a hundred-year-old textbook (I think it was called volume by the time I was in school. Is capacity the same thing as volume?) seem very different from what I remember doing when I was a student:

The method of finding the contents of any regular vessel in gallons, bushels, barrels, etc. is called gauging.

The capacity of tanks, cisterns, etc. is usually expressed in gallons or barrels. In every liquid gallon there are 231 cu. in.

To find the exact number of gallons in any vessel, divide the number of cubic inches in the vessel by 231.

To find the number of gallons in a cylindrical vessel, multiply the square of the diameter by the height, and this product by 5 7/8.

To find the approximate number of gallons in a cistern, multiply the number of cubic feet by 7 1/2 and from the product, subtract 1/400 of the product.

The capacity of bins, etc. is usually expressed in bushels. The standard bushel in the United States is a measure 8 inches deep, 18 1/2 inches in diameter, and contains 2150.42 cubic inches.  Hence, to find the number of bushels in any bin, divide the number of cubic inches in the bin by 2140.42.

Kimball’s Commercial Arithmetic (1911)

Got that?  Want to try some problems?

  1. Find the contents in gallons of a tank 4 ft. square and 5 ft. deep.

  2. The water in a cistern 8 ft. square is 2 ft. deep, how many gallons does it contain?

  3. A bin 8 ft. by 4 ft. by (?) contains 90 bushels of grain. Find the missing dimension.

  4. How many tons of water will fill a tank 11 ft. 8 in. by 3 ft. 6 in. by 2 ft. 3 in., if the weight of a cubic foot of water is 1,000 ounces?

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Two Old Mental Math Tricks for Adding Fractions

Posted on September 24, 2012 by Sheryl

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

Tuesday, September 24, 1912:  It is raining now. I guess or was. Had an exam in Geometry. Took up Arithmetic today. Didn’t have to but I chose to do so.

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

High school courses apparently were only a month of so long a hundred years ago.

I wonder why Grandma decided to take Arithmetic if it wasn’t required. Maybe she enjoyed doing mental math.

Here are two mental math tricks for adding fractions that I found in a hundred year old textbook:

Example 1: Add two fractions whose numerators are 1.

Solution: Add the denominators, and place the sum over the product of the denominators.

Example 2: Add two fractions whose numerators are alike and greater than 1.

Solution: Add the denominators and multiply the sum by the numerator of either of the fractions, and write the product over the product of the denominators.

Source: Kimball’s Commercial Arithmetic (1911)

If you like  math, you might also enjoy these previous posts:

An Old Mental Math Trick

Odd, Unusual, and Strange Math Problems

More Odd, Unusual, and Strange Math Problems

Cube Root Word Problems

1911 Algebra Problems: The Lusitania and Molasses

Old Math Problems

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More Odd, Unusual, and Strange Math Problems

Posted on October 16, 2011 by Sheryl

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

Monday, October 16, 1911: Nothing new at school or at home. Read several stories after I had worked some problems. Still have some for tomorrow though.

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

I wonder what type of problems Grandma was working on.  I enjoy looking at hundred-year-old math books. The problems are so different from the ones in today’s books.

I’ve previously shared some problems with you.  Here are some more odd, unusual, and strange problems from 1911:

1. If 44 cannons, firing 30 rounds an hour for 3 hours a day, consume 300 barrels of powder in 5 days, how long will 400 barrels last 66 cannons, firing 40 rounds an hour for 5 hours a day?

2. A ditch 80 yards long, 10 ft. deep, and 9 ft. wide was dug by 20 men in 12 1/2 days of 10 hours each; and a ditch 76 yards long and 12 ft. wide was dug by 30 men in 7 1/2 days of 9 1/2 hours each. How deep was the latter ditch?

3. A speculator bought 10 village lots, and gave a 4-months’ note in payment. This note was immediately discounted in the bank at 8%, and the bank discount was $192. What was the average price of the lots?

4. A druggist bought 6 pounds of quinine at $11 per pound, avoirdupois weight, and sold it in 2-grain capsules at 10 cents per dozen. What was his profit?

Kimball’s Commercial Arithmetic: Prepared for Use in Normal, Commercial and High Schools and the Higher Grades of the Common School (1911)

A hundred years ago prescriptions weren’t required and druggists made their own medicines, men actually dug ditches by hand, and labor laws about how many hours a day a person could work had not yet been enacted.

If you want to do the quinine problem–and, for some reason never had a math class that taught you the conversion factors for apothecaries and avoirdupois weights :) – here is the information you need:

Apothecaries Weight

20 grains = 1 scruple

3 scruples = 1 dram

8 drams = 1 ounce

12 ounces = 1 pound

Avoirdupois Weight

16 ounces = 1 pound

In case you missed the previous posts that contained math problems, here are the links:

Odd, Unusual, and Strange Math Problems

1911 Algebra Problems: The Lusitania and Molasses

Old Math Problems

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An Old Mental Math Trick

Posted on September 28, 2011 by Sheryl

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

Thursday, September 28, 1911: With just about the same languor as last year, I pursue my studies. It is almost a review, nothing hardly new. Tomorrow we commence with mental arithmetic. Certainly is baby stuff, but we haven’t had it for two years and he thinks we need it. I am eager to take up something I haven’t yet had.

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

Grandma sounds bored. She must have been good at mental arithmetic and found it easy.

In the days before calculators it was important to know how to quickly do various math operations in your head—and students learned many math “tricks” and practiced mental math strategies.

Here is an example of a math trick that enables you to easily multiply certain two digit numbers together in your head:

To multiply together two numbers whose tens are alike, and the sum of whose units is ten.

RULE: Multiply the units together for the two right-hand figures of the product, and the remaining part of the multiplicand by the remaining part of the multiplier increased by 1.

Example: 64 X 66 =  ?

64

 66

4224

Solution: The 6 units X 4 units = 24 units which we write for the two right-hand figures of the product. Then 6 tens multiplied by 1 more than itself for the remaining figures. Thus, 6 X (6+1) = 42.

 Kimball’s Commercial Arithmetic (1911 )

Now you can try doing some mental math. Here are some oral exercises that were in the book:

Oral Exercises

1.  Multiply 25 by 25

2.  Multiply 35 by 35

3.  Multiply 75  by 75

4.  Multiply 17 by 13

5.  Multiply 43 by 47

6;  Multiply 56 by 54

7;  Multiply 15 by 15

8.  Multiply 22 by 28

9.  Multiply 65 by 65

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Odd, Unusual, and Strange Math Problems

Posted on January 31, 2011 by Sheryl

15-year-old Helena wrote a hundred years ago today:

Tuesday, January 31, 1911.  If anything of real importance happened today I would write it down, but as nothing has it will not be here to read. This is the last day of the first month. What do you think of it? Vice versa.

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

No mention of arithmetic problems in today’s diary entry. Maybe it went better today than yesterday.

I’m still fascinated by the problems in the 1911 high school arithmetic textbook that I found. The book contains some really strange problems–including some that deal with topics that probably would be considered unacceptable today.  

1. If 44 cannons, firing 30 rounds an hour for 3 hours a day consume 300 barrels of powder in 5 days, how long will 400 barrels last 66 cannons, firing 40 rounds an hour for 5 hours a day?

2. Bought by avoirdupois weight, 20 pounds of opium at 40 cents an ounce, and sold the same by Troy weight at 50 cents an ounce; did I gain or lose, and how much?

3. A wine merchant imported 1000 dekaliters of wine, at a cost of 75 cents a liter, delivered. At what price per gallon must he sell the same to clear $2000 on the shipment?

4. A certain number of men, twice as many women, and three times as many boys, earn $123.80 in 5 days; each man earned $1.20, each woman 66 1/3 cents, and each boy 53 1/3 cents per day. How many were there of each?

Kimball’s Commercial Arithmetic: Prepared for Use in Normal, Commercial and High Schools and the Higher Grades of the Common School (1911)

Remember that a hundred years ago patent medicines containing opium were legal, child labor laws were just being enacted, and it was way before woman had equal rights.

If you want to do the opium problem here are a couple of definitions:

Avoidupois weight (The usual system used in the U.S.):  16 ounces = 1 pound

Troy weight:  12 ounces = 1 pound

Filed under: Education | Tagged: , , | 8 Comments »

Arithmetic Problems in 1911 High School Text Book

Posted on January 30, 2011 by Sheryl

15-year-old Helena wrote a hundred years ago today:

Monday, January 30, 1911. My! How the wind did blow today, smashed some panes in the school house windows with a deafening crash and alarmed us all, fortunately we escaped uninjured. Boo hoo I haven’t got all my arithmetic problems for tomorrow. Boo hoo. I’m getting stupid.

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

It was either a damn strong wind–or the windows weren’t very strong. I wonder if the panes blew out when the class was working on the dreaded arithmetic.  To get a sense of what the problems were like I found a high school arithmetic book that was published in 1911. Here are the written exercises in the chapter titled The Equation:

1. To four times a certain number I add 16, and obtain as a result 188. What is the number?

2. A man having $100 spent a part of it; he afterwards received five times as much as he had spent, and then his money was double what it was at first. How much did he spend?

3. A farmer had two flocks of sheep, each containing the same number. He sold 21 sheep from one flock and 70 from the other, and then found that he had left in one flock twice as many as in the other. How many had he in each?

4. Divide 100 into two such parts that a fourth of one part diminished by a third of the other part may be equal to 11.

5. Find the area of a square field whose diagonal is 50 rods.

Kimball’s Commercial Arithmetic: Prepared for Use in Normal, Commercial and High Schools and the Higher Grades of the Common School (1911)

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