Mrs Smith pays kate for babysitting by giving her 10 cents for the first-half hour, 20 cents for the second half-hour, 40 cents for the third half-hour etc. How long does she work to earn $12.70?

1st half-hr = 10 cents = 5*2^1.

2nd half-hr = 20 cents = 5*2^2.

3rd half-hr = 40 cents = 5*2^3.

t = 5*2^n = 1270 cents.
5*2^n = 1270,
Divide both sides by 5:
2^n = 254,
nlog2 = log254,
n = log254 / log2 = 7.99 half-hrs =
3.99 hrs or 4 hrs.

To find out how long Kate works to earn $12.70, we need to determine the pattern in the payments and then calculate the total time worked.

First, let's observe the pattern in the payments:
- For the first half-hour, Kate receives 10 cents.
- For the second half-hour, she receives 20 cents, which is double the amount of the first half-hour.
- For the third half-hour, she receives 40 cents, which is double the amount of the second half-hour.

We can notice that the amount received doubles for each additional half-hour.

Now, let's calculate the total time worked to earn $12.70.

To do this, we can work backward by looking at the last half-hour payment. If Kate received 40 cents for the last half-hour, then we can divide $12.70 by 40 cents to determine how many sets of half-hours she worked.

$12.70 / 0.40 = 31.75 sets of half-hours

Since there are 2 half-hours in each set, we need to multiply 31.75 sets by 2 to find the total number of half-hours worked.

31.75 sets * 2 half-hours/set = 63.5 half-hours

Therefore, Kate works for 63.5 half-hours to earn $12.70.