By: Tao Steven Zheng (郑涛)

【Problem】

The earliest “linear equation” found on Egyptian papyri were called “Ahau problems”. Although they were essentially linear equations, the ancient Egyptians solved it using a method called regula falsi or “false-position method”. Here is problem 19 from the :

A number taken 1 and 1/2 times and added to 4 to make 10. What is the number?

【Solution】

Let represent the unknown number. According to the problem, we get

The ancient Egyptian scribe would subtract 4 on both sides to get

The scribe makes an initial guess  based on the denominator of the fraction 1/2, and then proceeds to calculate the left hand side of the equation.

The scribe then notices that the initial guess gives a result that is half of 6. He then concludes that the correct answer must be double the initial guess.

Therefore, the desired number is 4.

【Historical Note】

The Moscow Papyrus, also called the Golenishchev Papyrus, is an ancient Egyptian mathematical document that was written around 1850 BC. In 1893, the text was bought by the Russian Egyptologist Vladimir Golenishchev in 1893. In 1912, the papyrus entered the collection of the Pushkin State Museum of Fine Arts in Moscow. The entire document contains 25 problems on arithmetic, algebra, and geometry.