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Session 6 Part A Part B Part C Homework
 
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Session 6, Part B:
False Position and Backtracking (30 minutes)

In This Part: False Position | Backtracking

The solutions to mathematical problems often involve writing and solving equations. We generally think of solving equations by representing quantities with variables. Early Egyptians did not have symbolic notation, however, so they invented a technique called false position to solve mathematical problems.
Note 4

The Egyptians used false postition to solve number puzzles. A modified version of false position was used by Diophantus to solve problems with squared variables and was still taught in the Renaissance for proportion problems.

False position begins by selecting a convenient answer or making an educated guess, one that makes the calculations of the problem simpler. It does not have to be the correct answer. After calculating the result from the convenient answer, a false position problem is solved by using the result to determine how to adjust the convenient answer to make it correct.

Here's an example from the Rhind papyrus, an ancient Egyptian scroll containing mathematical tables and calculations:

Find a quantity such that when it is added to 1/4 of itself, the result is 15.

To use the method of false position, start by selecting a convenient answer. Let's pick 4. Why 4? It simplifies the calculation: 4, plus 1/4 of itself (which is 1), equals 5.

Next, use the result to determine how to adjust the convenient answer. We got 5 as our result, and we wanted 15. The number to multiply by 5 (the result we got) to get 15 (the result we want) is 3.

So, multiply 4 (the convenient answer we started with) by 3 to get the correct answer, which is 12.

There are a number of similar false position problems in the Rhind papyrus. Try to solve the following problem using the method of false position:


 

Problem B1

Solution  

A quantity and its 1/7 added together become 32. What is the quantity?


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Start by assuming a "convenient" value for the solution, one that makes the "1/7" portion of the problem easier. Then, decide how to adjust the answer you got to make 32, the answer you want. "Its 1/7" means 1/7 of the original quantity.   Close Tip

 

Problem B2

Solution  

Consider the following equation: 4[3(2n - 4) / 6] = 8. Think of as many strategies as you can for solving this equation.
Note 5


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
False position will work here. You might also try writing the steps of the equation as an algorithm of function machines, and use a diagram to find the value of n. Have a look at Session 3, Part C.   Close Tip

Next > Part B (Continued): Backtracking

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