How do I answer this without logarithms?

Question

A fat cat that was initially 12kg reduces its weight by 10% each month. How long does it take for the cat to be at least 6kg lighter than its original weight? Give your answer as a whole number of months.

Answer 1

If the can reduces its weight by 10% each month, then in any given

month the weight is 90% of the previous months.

Weight as a function of number of months is found by the equation:

W(m) = W₀(0.9)^m

 

If we start at 24 kg and we want to weigh "at least 6kg less than original weight"

we want w(m) ≤ 24-6 or w(m) ≤ 18 kg

24(0.9)^m ≤ 18

0.9^m ≤ 18/24

0.9^m ≤ 3/4

0.9^m ≤ 0.75

 

We want the smallest number for m that will cause 0.9^m ≤ 0.75

Since we don't want to use logarithms for an exact answer then let's

plug in a small number for m and compare the answer to 0.75

 

m=1:  0.9¹ = 0.9   not less than 0.75

m=2:  0.9² = 0.81 not less than 0.75

m=3:  0.9³ = 0.729  YES this is less than 0.75

 

Your answer is m = 3 months