Rick has a box of coins .The box currently contains 51 coins, consisting of pennies, dimes and quarters. The numbers of pennies is equal to the numbers of dimes and the total value is $5.73. How many of each denominations of coins does he have?

the number of pennies which is equal to the number of dimes is?
the number of quarters is?

p = d

p + d + q = 51

p + 10 d + 25 q = 573 cents
so

2 p + q = 51
11 p + 25 q = 573

multiply the first equation by 25 and subtract

50 p + 25 q = 1275
11 p + 25 q = 573
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39 p = 702
p = 18 etc

To solve this problem, we can set up a system of equations. Let's represent the number of pennies as "x," the number of dimes as "x," and the number of quarters as "y."

Equation 1: x + x + y = 51 (the total number of coins is 51)
Equation 2: 0.01x + 0.10x + 0.25y = 5.73 (the total value of the coins is $5.73)

Now, let's solve these equations.

From Equation 1, we can simplify it as 2x + y = 51.

From Equation 2, we can simplify it as 0.11x + 0.25y = 5.73.

Now we can use one of two methods to solve this system of equations: substitution or elimination. Let's use the substitution method.

1. Solve Equation 1 for y:
y = 51 - 2x

2. Substitute the value of y in Equation 2 with (51 - 2x):
0.11x + 0.25(51 - 2x) = 5.73

3. Simplify and solve for x:
0.11x + 12.75 - 0.5x = 5.73
-0.39x = -7.02
x ≈ 18

Now that we have found the value of x, we can substitute it back into Equation 1 or Equation 2 to find the value of y:

Using Equation 1:
2(18) + y = 51
36 + y = 51
y = 15

Therefore, Rick has 18 pennies and 18 dimes, and the number of quarters is 15.