find the 5 diff. ways a collection of 100 coins-pennies, dimes, and quarters- can be worth exactly worth $4.99.
p + d + q = 100
p + 10d + 25q = 499
subtract them to get 9d + 24q = 399 or
3d + 8q = 133
we need values of d and q which are positive whole numbers.
After a few trials I got
d = 7 and q = 14
the "slope" of 3d + 8q = 133 is - 3/8
that is, if we increase the value of d by 8 and decrease the value of q by 3 we get another whole number solution.
so I have the following ordered pairs of (d,q)
7 14
15 11
23 8
31 5
39 2
sub each of these into my first equation to find p (pennies)
I will do one of them, you can do the rest
if d=23 q=8 then
p+23+8 = 100
p = 69
check: 23+8+69 = 100
69 + 10(23) + 25(8) = 499
To find the five different ways a collection of 100 coins (pennies, dimes, and quarters) can be worth exactly $4.99, we'll need to explore different combinations of coins. We can break down the problem into small steps:
Step 1: Determine the maximum number of quarters that can be used.
Since a quarter is worth 25 cents, the maximum number of quarters that can be used to reach $4.99 is 499/25 = 19 quarters.
Step 2: Explore combinations with different numbers of quarters.
We'll go through the different possibilities, starting with the highest number of quarters and decreasing it incrementally until reaching zero quarters:
- 19 quarters + dimes and pennies
- 18 quarters + dimes and pennies
- 17 quarters + dimes and pennies
- 16 quarters + dimes and pennies
- 15 quarters + dimes and pennies
Step 3: Determine the maximum number of dimes and explore combinations with different numbers of dimes.
A dime is worth 10 cents, so the maximum number of dimes that can be used is (499 - 25 * number of quarters) / 10.
We'll go through the different possibilities, starting with the highest number of dimes and decreasing it incrementally until reaching zero dimes for each number of quarters.
Step 4: Calculate the number of pennies required.
Since a penny is worth 1 cent, the number of pennies required can be calculated as (499 - 25 * number of quarters - 10 * number of dimes).
Step 5: Repeat steps 3 and 4 until all possible combinations are found.
Let's go through the steps to find the five different ways a collection of 100 coins can be worth exactly $4.99:
Combination 1: 19 quarters + 0 dimes + 4 pennies
Combination 2: 17 quarters + 7 dimes + 14 pennies
Combination 3: 14 quarters + 14 dimes + 24 pennies
Combination 4: 11 quarters + 21 dimes + 34 pennies
Combination 5: 8 quarters + 28 dimes + 44 pennies
These are the five different ways a collection of 100 coins (pennies, dimes, and quarters) can be worth exactly $4.99.
To find the different ways a collection of 100 coins, consisting of pennies, dimes, and quarters, can be worth exactly $4.99, we need to use a systematic approach. Here's how you could do it:
1. Start by assigning variables for the number of pennies, dimes, and quarters: Let's say p represents the number of pennies, d represents the number of dimes, and q represents the number of quarters.
2. Set up the equations based on the given information:
- Equation 1: p + d + q = 100 (the total number of coins should be 100)
- Equation 2: 0.01p + 0.10d + 0.25q = 4.99 (the total value of the coins should be $4.99)
3. Rewrite Equation 2 in terms of a single variable:
- Multiply both sides of Equation 2 by 100 to eliminate decimals: 1p + 10d + 25q = 499
4. Use substitution or elimination to solve the system of equations:
- Choose one variable and solve for it in terms of the other(s):
- For example, if we choose to solve for p, we can rewrite Equation 1 as p = 100 - d - q
- Substitute this expression for p into Equation 2:
- 1(100 - d - q) + 10d + 25q = 499
- Simplify the equation to solve for d or q
5. Substitute the obtained values back into Equation 1 to find the remaining variable.
6. Repeat steps 4 and 5, making sure to keep track of all the possible integer solutions.
7. Continue this process until you have exhausted all possible combinations of pennies, dimes, and quarters, keeping in mind any constraints.
Using this systematic approach, you can find all the different ways a collection of 100 coins can be worth exactly $4.99.