Jim's Market couldn't keep Crunchy Critter Crackers in stock. Jim started with 300 boxes but everyone wanted them. The first day Jim sold 6 boxes, and on the second day he sold 14 boxes. Each day 8 more boxes were sold than the day before. So after two days, he had sold 20 boxes. If he kept selling the crackers at this rate, when would Jim run out of Crunchy Critter Crackers?
2/2 (2*8 + (72-1)*4) ?
To find out when Jim would run out of Crunchy Critter Crackers, let's break down the information given:
- On the first day, Jim sold 6 boxes.
- On the second day, Jim sold 14 boxes.
- Each day, 8 more boxes were sold than the day before.
To determine the total number of boxes sold after two days, we can add up the number of boxes sold on each day:
Total boxes sold after two days = 6 boxes + 14 boxes = 20 boxes.
Now, let's figure out the pattern of the number of boxes sold each day:
- Day 1: 6 boxes
- Day 2: 14 boxes
- Day 3: 14 boxes + 8 boxes = 22 boxes
- Day 4: 22 boxes + 8 boxes = 30 boxes
- Day 5: 30 boxes + 8 boxes = 38 boxes
- Day 6: 38 boxes + 8 boxes = 46 boxes
- Day 7: 46 boxes + 8 boxes = 54 boxes
As we can see, each day the number of boxes sold increases by 8. So, after two days, the number of boxes sold was 20, after three days it would be 22 + 8 = 30, and so on.
To find out when Jim would run out of Crunchy Critter Crackers, we need to solve the equation: 300 - (total number of boxes sold) = 0.
Let's set up the equation using the pattern we identified:
300 - (20 + 30 + 38 + 46 + 54 + ... + x) = 0.
We subtract the total number of boxes sold from 300 (the starting number of boxes) and set it equal to zero because if there are no boxes left, Jim has run out of stock.
Now, we need to determine the value of x, which represents the number of days it takes for Jim to run out of Crunchy Critter Crackers.
To solve the equation, we can use the formula for the sum of an arithmetic series:
Sn = n/2 * (a1 + an),
where Sn is the sum of the series, n is the number of terms in the series, a1 is the first term, and an is the last term.
In this case, an represents the number of boxes sold on the nth day. We need to find the value of n when we add up all the terms until the sum reaches 300 (Jim runs out of stock).
Let's find the number of terms (n) needed to reach a sum (Sn) of 300 boxes:
300 = n/2 * (20 + (20 + (n - 1) * 8)).
Now, we can solve this equation to find the value of n:
300 = n/2 * (40 + 8n - 8).
300 = n/2 * (32 + 8n).
600 = n * (32 + 8n).
600 = 32n + 8n^2.
8n^2 + 32n - 600 = 0.
Solving this quadratic equation will give us the value of n when Jim runs out of stock. Once we find the positive root, we will have the number of days it takes for Jim to run out of Crunchy Critter Crackers.
you have an AP with
a = 6
d = 8
and you want Sn = n/2 (2*8 + (n-1)*4) >= 300
so solve that for n