A man exchanged a dollar bill for change and received 7 coins, none of which were half dollars. How many of these coins were dimes?

A man exchanged a dollar bill for change and received 7 coins, none of which were half dollars. So the coins which were dimes (d) = 3.67.
In this explanation, we will solve a problem involving exchanging a dollar bill for change and receiving seven coins, with none of them being half dollars. We'll use mathematical expressions to represent the number of each type of coin, and we aim to find the number of dimes among these coins.
Let's use variables to represent the number of each type of coin. Let:

"d" be the number of dimes,

"q" be the number of quarters,

"n" be the number of nickels,

"p" be the number of pennies.


We are given that a man exchanged a dollar bill for change, so the total value of the coins he received should equal one dollar. In mathematical terms, we can write this as an equation:
1 dime (10 cents) × d + 1 quarter (25 cents) × q + 1 nickel (5 cents) × n + 1 penny (1 cent) × p = 100 cents.
The problem also states that he received seven coins. So, we have another equation :
d + q + n + p = 7.
Now, we need to consider the constraint that none of the coins are half dollars. In other words, there are zero half dollars (0 × 50 cents). This gives us another equation:
0.5 × 0 = 0.
Now, we have a system of three equations with four unknowns (d, q, n, p):

10d + 25q + 5n + p = 100,

d + q + n + p = 7,

0 = 0.


This system has multiple solutions, but we need to find the number of dimes (d). To do this, we can use substitution or elimination . Let's use substitution:
From the second equation, we can rearrange it to express p in terms of the other variables :
p = 7 - d - q - n.
Now, substitute this expression for p into the first equation:
10d + 25q + 5n + (7 - d - q - n) = 100.
Simplify the equation:
10d + 25q + 5n + 7 - d - q - n = 100.
Combine like terms:
9d + 24q + 4n + 7 = 100.
Now, isolate d on one side of the equation:
9d = 100 - 24q - 4n - 7,
9d = 93 - 24q - 4n,
d = (93 - 24q - 4n) / 9.
Since we're dealing with whole numbers, the values of d, q, and n must be integers. This means that for d to be an integer , the numerator (93 - 24q - 4n) must be divisible by 9.
As we have multiple unknowns (q and n) and only two equations left, we might have multiple solutions. For instance, let's try with q = 1 and n = 0:
d = (93 - 24(1) - 4(0)) / 9,
d = (93 - 24 - 0) / 9,
d = 69 / 9,
d = 7.67.
Since d should be an integer representing the number of dimes, this solution doesn't work. However, there are other possible values for q and n that may give us a whole number solution for d. For instance, if we take q = 2 and n = 3:
d = (93 - 24(2) - 4(3)) / 9,
d = (93 - 48 - 12) / 9,
d = 33 / 9,
d = 3.67.
Again, this doesn't give us an integer solution for d. You can try other combinations of q and n, but in this case, there might not be a solution that fulfills all the conditions . Therefore, it seems there might be some missing information or constraints to find a unique solution for the number of dimes.
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Answered by Anshuyadav | 2024-06-18

The problem can be solved using equations that represent the total number of coins and their total value. By testing combinations of coins, we can find various solutions for how many dimes are in the collection. The method involves setting up equations based on the conditions provided and solving for the variables systematically.
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Answered by Anshuyadav | 2024-12-20