Tashana has 16 coins that total to $1.85. She has only nickels, dimes and quarters. How many of each coin could

she have?

She could actualy do it several different combinations.

5 quarters, 1 dime, and 10 nickels
4 quarters, 5 dimes, and 7 nickels
3 quarters, 9 dimes, and 4 nickels
2 quarters, 13 dimes, and 1 nickel

nickels --- x

dimes --- y
quarters --- 16-x-y

5x + 10y + 25(16-x-y) = 185
5x + 10y + 400 - 25x - 25y = 185
-20x - 15y = -215
divide by -5
4x + 3y = 43

but x and y must be positive integers
we know the x-intercept is < 11
and the y-intercept < 14

y = (43 - 4x)/3
by inspection, if x = 4 , y = 9 is the only combination that will work
so she has 4 nickels, 9 dimes and 16-4-9 or 3 quarters.

check:
does she have 16 coins ?
4+9+3 = 16 , yes

does she have $1.85 ?
4(5) + 9(10) + 25(3) = 185 , yes!!

Should not have stopped after I found one.

To solve this problem, we can use a system of equations. Let's assign variables to the number of nickels, dimes, and quarters Tashana has.

Let:
N = number of nickels
D = number of dimes
Q = number of quarters

Given the following information, we can formulate two equations:

1. The total number of coins: N + D + Q = 16
2. The total value of the coins: 0.05N + 0.10D + 0.25Q = 1.85

We can now use these equations to solve for N, D, and Q.

To solve these equations, we can use a variety of methods such as substitution or elimination. One approach is to use the substitution method.

Let's solve equation 1 for N and substitute it into equation 2:

From equation 1, we have N = 16 - D - Q

Substituting this value into equation 2, we get:
0.05(16 - D - Q) + 0.10D + 0.25Q = 1.85

Now we can simplify and solve for D:

0.8 - 0.05D - 0.05Q + 0.10D + 0.25Q = 1.85
0.05D + 0.20Q = 1.05

Next, let's solve equation 1 for N and substitute it into equation 2:

From equation 1, we have N = 16 - D - Q

Substituting this value into equation 2, we get:
0.05(16 - D - Q) + 0.10D + 0.25Q = 1.85

Now we can simplify and solve for D:

0.8 - 0.05D - 0.05Q + 0.10D + 0.25Q = 1.85
0.05D + 0.20Q = 1.05

To eliminate the decimal, we will multiply the entire equation by 100:

5D + 20Q = 105

From equation 1, we have N = 16 - D - Q

Now we can plug in D into equation 1:

N = 16 - D - Q
N = 16 - (105 - 20Q) - Q
N = 16 - 105 + 20Q - Q
N = -89 + 19Q

We now have three equations:

N = -89 + 19Q (equation 3)
5D + 20Q = 105 (equation 4)
N + D + Q = 16 (equation 5)

Next, we can substitute equation 3 into equation 5:

(-89 + 19Q) + D + Q = 16
D + 20Q = 105

Simplifying, we get:
D + 20Q = 105

Now, we have a system of two equations with two variables:
D + 20Q = 105
5D + 20Q = 105

We can solve this system of equations to find the values of D and Q.

Multiplying the first equation by 5 to eliminate D, we get:
5D + 100Q = 525

Subtracting this equation from the second equation, we get:
(5D + 20Q) - (5D + 100Q) = 105 - 525
-80Q = -420
Q = 5.25

Since Q represents the number of quarters, and we cannot have a fraction of a coin, this means that the number of quarters must be a whole number. Therefore, there is no solution that satisfies all the given conditions. Tashana cannot have a combination of nickels, dimes, and quarters that would total 16 coins and $1.85.