aviva has a total of 52 coins, all of which are either dimes or nickels. The total value of the coins is $4.05. Find the number of each type of coin.
10x + 5(52-x) = 405
x = 29
so, 29 dimes, 23 nickels
To solve this problem, we can use a system of equations.
Let's assume that Aviva has "x" dimes and "y" nickels.
Since Aviva has a total of 52 coins, we can express this relationship as:
x + y = 52 (Equation 1)
The value of a dime is 10 cents, and the value of a nickel is 5 cents. So, the total value of dimes would be 10x cents, and the total value of nickels would be 5y cents.
Given that the total value of the coins is $4.05 (405 cents), we can express this relationship as:
10x + 5y = 405 (Equation 2)
Now we have a system of two equations with two variables. Let's solve it using the substitution method.
1. Solve Equation 1 for x:
x = 52 - y
2. Substitute the value of x in Equation 2:
10(52 - y) + 5y = 405
Expand and simplify the equation:
520 - 10y + 5y = 405
520 - 5y = 405
3. Move the constant term to the other side:
-5y = 405 - 520
-5y = -115
4. Divide by -5 to solve for y:
y = (-115) / (-5)
y = 23
We have found that y = 23, which represents the number of nickels. Now, substitute this value back into Equation 1 to solve for x.
x + y = 52
x + 23 = 52
x = 52 - 23
x = 29
We have found that x = 29, which represents the number of dimes.
Therefore, Aviva has 29 dimes and 23 nickels.