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The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.

"Freeman Dyson: Mathematician, Physicist, and Writer". Interview with Donald J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.

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# Using Historical Problems in the Middle School

## Problems from 19th Century American Textbooks

The second group of problems is from old American textbooks.

1.   If a barrel of ale will last a family of 6 persons for 2 months, how many persons would drink 9 barrels in a year?

Solution .

First state this: 1 barrel per 6 people per two months. If you divide 1 by 6 and by 2 you get .0833333...Then you state that nine barrels per X people per 12 months, or 9 divided by 12 equals .75. Then divide .75 by .0833333. By doing this you get 9 persons, which is the answer.

2.   If by paying a teacher $3.25 per week a certain school would be kept 11 1/2 weeks, how long might it be kept if she be paid only$2.875 per week?   (Question:  When would a teacher have been earning somewhere near $3 per week?) Solution: First I multiplied$3.25 which she was paid by 11 1/2 weeks, which gave me $37.75. Then I divided that by$2.875 to see the number of weeks, which was 13.

3.   A teacher receives 77 pieces of money for his month’s salary.  The number of half dimes (nickels) received is 85 5/7% of the number of dimes; the number of dimes equals 87 1/2% of the number of quarters; the number of quarters equals 40% of the number of halves; and the number of halves equals 55 5/9 % of the number of dollars; what was his salary?

Solution:

First I converted the percents to fractions.

Nickels = 6/7 dimes

Dimes = 7/8 quarters

Quarters = 2/5 halves

Halves = 5/9 dollars

Total 77 pieces.

I first guessed 19 dollars and did try and check. My answer was way off because I needed more pieces of money. I tried higher numbers of dollars and finally got the correct answer. I guessed 36 dollars and multiplied that by 5/9. I got 20 which was the number of halves. I multiplied 20 by 2/5 and got 8 which was the number of quarters. I multiplied 8 by 7/8 and got 7 which was the number of dimes. Then I multiplied 7 by 6/7 and got 6 which was the number of nickels. I added all of the pieces of money together and got 77. The salary was $49.00. (Note how much higher this was than in the previous problem, where the teacher made somewhere near$12 for a month.  Does this have to do with the fact that here the teacher is male, while in the previous problem the teacher is female?)

4.   A man bequeathed $900 to three friends; the first must have a certain portion, the second must have twice as much as the first, the third$28 more than the first. How much did each person receive?

Solution.

X + 2X + (X + 28 )  = 900

4X + 28  = 900

4X = 872

X = $218 Thus: First gets$218

Second gets 2 x 218 or $436 Third gets$218 + $28 =$246

5.   If 1 pig is worth $3 and 5 pigs are worth 2 sheep and 5 sheep are worth 2 cows and 10 cows are worth 3 horses, what is the value of 12 horses? Solution: First state the problem simply: 1 pig =$3, 5 pigs = 2 sheep, 5 sheep = 2 cows and 10 cows = 3 horses.  12 horses ?

I then calculated:  1 x 3 = 3; 3 x 5 = 15; 15/2 = 7.5; 7.5 x 5 = 37.5; 37.5/2 = 18.75; 18.75 x 10 = 187.5; 187.5/3 = 62.5; 62.5 x 12 = \$750, the answer.