The Method of False Position

Image source: hebbarskitchen.com

Question:

My grandma uses a household bowl to measure ingredients for most of her dishes. While making khichdi, she usually puts one full dish and its eighth of dal into an empty pot, which brings it to the 500 ml mark. I need to find the measurement of the bowl so that I can figure out her reference point for future recipes. What is the bowl's measurement in millilitres? 

Solution:

x + ^x⁄₈ = 500 

Try x = 80: 

80 + ⁸⁰⁄₈ = 90 

90 multiplied by what gives 500? 

90        1 

180      2

360      4 

720      8 

 45       ¹⁄₂

 15        ¹⁄₆

  5         1⁄₁₈

360 + 90 + 45 + 5 = 500 so we have 4 + 1 + 1⁄₂ + 1⁄₁₈ = 50⁄₉ multiplied by 90 gives 500. 

Thus x = 80 x 5 5⁄₉

80         1

160       2   

320       4

40         1⁄₂

20         1⁄₄

10         1⁄₈

 x = 80 + 320 + (80 + 320)/9

x = 400 + 400/9

9       1 

18     2

36     4 

72     8 

144    16

288   32

576    64

x = 400 + 32 + 8 + 4 +  4/9 = 444 4⁄₉  ml