30 coins consisting of nickels and dimes are worth $2.40. How many of each kind are there?
number of nickels --- x
number of dimes ---- 30-x
solve
5x + 10(30-x) = 240
Step 1: Let's assume the number of nickels as 'x' and the number of dimes as 'y'.
Step 2: The value of 'x' nickels is 5x cents.
Step 3: The value of 'y' dimes is 10y cents.
Step 4: According to the problem, the value of 30 coins in cents is 240 cents.
Step 5: Writing this equation by combining the values of nickels and dimes, we have:
5x + 10y = 240.
Step 6: We also know that the total number of coins is 30, so we can express this as another equation:
x + y = 30.
Now we have a system of two equations:
5x + 10y = 240,
x + y = 30.
We can solve this system of equations by substitution or elimination method.
Using the substitution method:
Step 7: Solve the second equation for 'x', which gives us:
x = 30 - y.
Step 8: Substitute the value of 'x' in the first equation:
5(30 - y) + 10y = 240.
Step 9: Simplify and solve for 'y':
150 - 5y + 10y = 240,
5y = 240 - 150,
5y = 90,
y = 90/5,
y = 18.
Step 10: Now, substitute the value of 'y' back into the second equation to find 'x':
x + 18 = 30,
x = 30 - 18,
x = 12.
Therefore, there are 12 nickels and 18 dimes.
To solve this problem, we can create a system of equations.
Let's represent the number of nickels as 'n' and the number of dimes as 'd.'
We are given two pieces of information:
1) There are a total of 30 coins: n + d = 30
2) The total value of the coins is $2.40: 0.05n + 0.10d = 2.40
Now, we can solve this system of equations using different methods, such as substitution or elimination.
Let's solve it using the substitution method:
Step 1: Solve the first equation for n in terms of d:
n = 30 - d
Step 2: Substitute the value of n in the second equation:
0.05(30 - d) + 0.10d = 2.40
Step 3: Simplify and solve for d:
1.50 - 0.05d + 0.10d = 2.40
0.05d = 2.40 - 1.50
0.05d = 0.90
d = 0.90 / 0.05
d = 18
Step 4: Substitute the value of d back into the first equation to find n:
n + 18 = 30
n = 12
Therefore, there are 12 nickels and 18 dimes in the given 30 coins.