Project 1. Method of False Position.
In this project, we are going to learn about an ancient algebraic technique that is built around correcting guesses. We may gain greater appreciation for the value of wrong guesses and what we can gain from them.
(a)
Solve the following equation any way you would like.
\begin{equation*}
x \left( 1+\frac{1}{3}+\frac{1}{4} \right)=14.
\end{equation*}
Check your answer using technology.
(b)
Notice that 12 is the least common multiple of 3 and 4: the denominators. Distribute 12 in the following expression.
\begin{equation*}
12 \left( 1+\frac{1}{3}+\frac{1}{4} \right).
\end{equation*}
Is this bigger, equal to, or smaller than 14?
(c)
We multiplied by a convenient number, which is not quite right. Because it is multiplication we can scale (multiply) our not quite right guess to make it right. Consider
\begin{equation*}
y \cdot 12 \left( 1+\frac{1}{3}+\frac{1}{4} \right) = 14.
\end{equation*}
Replace 12 times the sum with your result from the previous step.
Solve the resulting equation for y.
(d)
Note that y is the correction to our guess of 12. Calculate \(y \cdot 12\text{.}\)
This will match your original solution. If not, check your calculations.
(e)
This method is called false position because it guesses a convenient number which is typically false then corrects it. One of the original motivations for this method was the lack of a useful notation for fractions (it dates to the Sumerians and ancient Egyptians).
Many people today use similar methods when dealing with fractions. What is a reason people might distribute a convenient number before doing the solving?
