In how many ways is it possible to spend exactly $200 on pumpkins if large pumpkins cost $5 and small $3
well, 40 $5 pumpkins cost exactly $200
LCM(5,3) = 15
So, for every 15/5=3 fewer $5/pumpkins, you get 15/3=5 more $3 pumpkins.
So, we have
$5 $3
----------
40 0
37 5
34 10
31 15
28 20
25 25
22 30
19 35
16 40
13 45
10 50
7 55
4 60
1 65
To find the number of ways to spend exactly $200 on pumpkins, we can use a combination of large and small pumpkins. Let's denote the number of large pumpkins as "L" and the number of small pumpkins as "S."
Given that large pumpkins cost $5 and small pumpkins cost $3, we can set up the following equation to represent the total cost:
5L + 3S = 200
To solve this equation, we can use a method called "diophantine equation." Let's step through it to determine the possible combinations:
Step 1: Solve for one variable in terms of the other.
Rearrange the equation to get:
5L = 200 - 3S
L = (200 - 3S) / 5
Step 2: Check the divisibility condition.
The right-hand side of the equation needs to be divisible by 5 for L to be an integer. Since the denominators are 3 and 5:
- If S is a multiple of 5, then the equation will yield an integer solution for L.
- If S is not a multiple of 5, there will be no integer solution for L, and that combination will be discarded.
Step 3: Test different values for S.
To find the possible combinations, substitute values for S that are multiples of 5 and solve for L. Note that both L and S must be non-negative integers.
S = 0:
L = (200 - 3(0)) / 5 = 200 / 5 = 40 --> (L, S) = (40, 0)
S = 5:
L = (200 - 3(5)) / 5 = 185 / 5 = 37 --> (L, S) = (37, 5)
S = 10:
L = (200 - 3(10)) / 5 = 170 / 5 = 34 --> (L, S) = (34, 10)
Continue this process until S = 65 (since 5L + 3S = 200, and 65 is the largest multiple of 5 less than 200).
Step 4: Calculate the number of combinations.
Count the number of combinations obtained in step 3.
So, there are 14 possible ways to spend exactly $200 on pumpkins if large pumpkins cost $5 and small pumpkins cost $3.
To determine the number of ways to spend exactly $200 on pumpkins, given that large pumpkins cost $5 and small pumpkins cost $3, we need to find all possible combinations of large and small pumpkins that sum up to $200.
One approach to solving this problem is by using a technique known as "coin change" or "making change." We can solve this problem using dynamic programming. Here's how we can proceed:
1. Define a list, DP, of size 201, initialized with all zeros. This list will store the number of ways to make change for each dollar amount from 0 to $200.
2. Set DP[0] = 1, indicating that there is one way to make change for $0 (by using no pumpkins).
3. Iterate through each possible quantity of large and small pumpkins. Let's call the quantity of large pumpkins "L" and the quantity of small pumpkins "S".
- L ranges from 0 to (200 / 5) = 40, representing the maximum number of large pumpkins we can buy.
- S ranges from 0 to (200 / 3) = 66, representing the maximum number of small pumpkins we can buy.
4. For each combination of L and S, calculate the total cost, which is given by 5L + 3S.
5. If the total cost is less than or equal to $200, update DP[total_cost] by adding the value of DP[total_cost - 5L - 3S].
- This step accounts for the fact that we can reach the current total cost by using the previously calculated number of ways to make change for a smaller total cost.
6. After completing the iteration, DP[200] will contain the number of ways to make change for $200.
To summarize, we can solve this problem using dynamic programming by calculating the number of ways to make change for each dollar amount from $0 to $200, considering the combinations of large and small pumpkins.