You have a total of 52 dimes and quarters. You have 10 more quarters than dimes. Which system of equations can you use to find the number x of quarters and number y of dimes you have? Use the system to determine how much money you have in quarters and dimes.
To solve this problem, we can set up a system of equations. Let's say x represents the number of quarters, and y represents the number of dimes.
1. The total number of coins is 52:
x + y = 52
2. You have 10 more quarters than dimes:
x = y + 10
So, the system of equations representing this scenario would be:
x + y = 52
x = y + 10
To determine how much money you have in quarters and dimes, we need to consider their values. A dime is worth $0.10, while a quarter is worth $0.25.
The amount of money in dimes is found by multiplying the number of dimes (y) by $0.10:
Amount in dimes = 0.10 * y
The amount of money in quarters is found by multiplying the number of quarters (x) by $0.25:
Amount in quarters = 0.25 * x
To calculate the total amount of money, we add the amounts in dimes and quarters:
Total amount = Amount in dimes + Amount in quarters
Now, we can substitute the value of x and y from the system of equations into the equations for the total amount:
Total amount = 0.10 * y + 0.25 * x
We already have the system of equations:
x + y = 52
x = y + 10
Substitute the value of x from the second equation into the total amount equation:
Total amount = 0.10 * y + 0.25 * (y + 10)
This equation gives us the total amount of money you have in terms of the number of dimes (y). To find the specific amounts, you need to solve the system of equations by substitution or elimination.
Alternatively, if you want to find the numerical values of x and y, you can solve the system of equations as follows:
From the second equation, substitute the value of x into the first equation:
(y + 10) + y = 52
Simplify the equation:
2y + 10 = 52
Subtract 10 from both sides:
2y = 42
Divide both sides by 2:
y = 21
Substitute the value of y back into the second equation:
x = 21 + 10
x = 31
Therefore, you have 31 quarters and 21 dimes. To find the total amount of money, substitute the values into the equation we derived earlier:
Total amount = 0.10 * 21 + 0.25 * 31
Calculate it:
Total amount = $2.10 + $7.75 = $9.85
So, you have a total of $9.85 in quarters and dimes.
number of dimes ---- x
number of quarters ---- x+10
x + x+10 = 52
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