DISCOVERY PROJECT

Equations through the Ages

Equations have been used to solve problems throughout recorded history, in every civilization. Here is a problem from ancient Babylon (ca. 2000 B.C.).

I found a stone but did not weigh it. After I added a seventh, and then added an eleventh of the result, I weighed it and found it weighed $1$ mina. What was the original weight of the stone?


The answer given on the cuneiform tablet is ${\frac 23}$ mina, $8$ sheqel, and $22{\frac 12}$ se, where $1$ mina = $60$ sheqel, and $1$ sheqel = $180$ se.

In ancient Egypt, knowing how to solve word problems was a highly prized secret. The Rhind Papyrus (ca. 1850 B.C.) contains many such problems. Problem 32 in the Papyrus states

A quantity, its third, its quarter, added together become $2$. What is the quantity?

The answer in Egyptian notation is $1 + \overline 4$ + $\overline {76}$, where the bar indicates "reciprocal," much like our notation $4^{-1}$.

The Greek mathematician Diophantus (ca. 250 A.D.) wrote the book Arithmetica, which contains many word problems and equations. The Indian mathematician Bhaskara (12th century A.D.) and the Chinese mathematician Chang Ch'iu-Chien (6th century A.D.) also studied and wrote about equations. Of course, equations continue to be important today.

  1. Solve the Babylonian problem and show that their answer is correct.
  2. Solve the Egyptian problem and show that their answer is correct.
  3. The ancient Egyptians and Babylonians used equations to solve practical problems. From the examples given here,do you think that they may have enjoyed posing and solving word problems just for fun?
  4. Solve this problem from 12th-century India.

    A peacock is perched at the top of a $15$-cubit pillar, and a snake's hole is at the foot of the pillar. Seeing the snake at a distance of $45$ cubits from its hole, the peacock pounces obliquely upon the snake as it slithers home. At how many cubits from the snake's hole do they meet, assuming that each has traveled an equal distance?

  5. Consider this problem from 6th-century China.


    If a rooster is worth $5$ coins, a hen $3$ coins, and three chicks together one coin, how many roosters, hens, and chicks, totaling $100$, can be bought for $100$ coins?

    This problem has several answers. Use trial and error to find at least one answer. Is this a practical problem or more of a riddle? Write a short essay to support your opinion.
  6. Write a short essay explaining how equations affect your own life in today's world.