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Laura Farnam Techniques developed wherever math was studied Rhind Papyrus A quantity; its half and its third are added to it. It becomes 10. x + (1/2)x + (1/3)x = 10 False Position Double False Position Guessing method Expect wrong answer Make computations easy Example 1: A quantity; its fourth is added to it. It becomes 15. Example 1: A quantity; its fourth is added to it. It becomes 15. x + (1/4)x = 15 4 + (1/4)(4) = 4+1 = 5 15/5 = 3 4 x 3 = 12 Example 2: A quantity; its third and its fifth are added to it. It becomes 46. Example 2: A quantity; its third and its fifth are added to it. It becomes 46. x + (1/3)x + (1/5)x = 46 15 + 5 + 3 = 23 46/23 = 2 15 x 2 = 30 Why Does it Work? Ax = B Multiply x by a factor A(kx) = k(Ax) = kB Used in textbooks until 19th century Daboll’s Schoolmaster’s Assistant (early 1800s) Guessing method Example 3: A purse of 100 dollars is to be divided among four men A, B, C, and D, so that B may have four dollars more than A, and C eight dollars more than B, and D twice as many as C; what is each one’s share of the money? Steps Guess Find error Repeat Cross-multiply guesses and errors Take the difference (if similar) or the sum (if different) Divide by the difference/sum of the errors y = mx +b Plot the line using two points “Rise over Run” 100 – 70 = 100 – 80 x–6 x-8 False Position (Ax = B) Double False Position (Ax + C = B) Linear A constant ratio “the change in the output is proportional to the change in the input”