Kate's parents decided it was time to start giving her pocket money. They were not sure how much to give her so they started with $1 for the first week and doubled it every week. How many weeks until Kate gets over $20 pocket money?
week 1: $1
week 2: 2 x $1 = $2
week 3: 2 x $2 = $4
week 4: 2 x $4 = $8
week 5: 2 x $8 = $16
week 6: 2 x $16 = $32
Assuming that you mean what she gets that week, I agree.
could this question also be interpreted as how many weeks until Kate gets over $20 in pocket money total? as in the answer being 5 weeks?
Please help! thank you.
To find out how many weeks until Kate gets over $20 pocket money, we can start by listing out Kate's pocket money for each week.
Week 1: $1
Week 2: $1 * 2 = $2
Week 3: $2 * 2 = $4
Week 4: $4 * 2 = $8
Week 5: $8 * 2 = $16
Week 6: $16 * 2 = $32
From the list, we can see that Kate gets over $20 in week 6.
Now, let me explain how this can be calculated using a formula. Since Kate's parents are doubling her pocket money each week, we can represent her pocket money for a given week, n, using the formula:
Pocket Money = $1 * 2^(n - 1)
Here, n represents the week number. Starting from week 1, we substitute n = 1 into the formula:
Pocket Money = $1 * 2^(1 - 1) = $1 * 2^0 = $1 * 1 = $1
In week 2, n = 2:
Pocket Money = $1 * 2^(2 - 1) = $1 * 2^1 = $1 * 2 = $2
Similarly, we can calculate the pocket money for each week. When the pocket money reaches over $20, we have:
$1 * 2^(n - 1) > $20
Simplifying the inequality:
2^(n - 1) > 20
To solve for n, we can take the logarithm of both sides. Assuming base 2 logarithm:
log2(2^(n - 1)) > log2(20)
Simplifying:
n - 1 > log2(20)
n > log2(20) + 1
Using a calculator, we find that log2(20) ≈ 4.32. Adding 1, we get:
n > 4.32 + 1 = 5.32
Since n represents the week number and we can't have a fraction of a week, we round up to the next whole number:
n > 6
Therefore, it takes Kate 6 weeks to receive over $20 in pocket money.