Final answer:
To solve this problem, set up a system of equations based on the given information. Use the equations to solve for the variables n, d, and q to find the number of nickels, dimes, and quarters.
Step-by-step explanation:
To solve this problem, we need to set up a system of equations based on the given information. Let's use the variables n, d, and q to represent the number of nickels, dimes, and quarters, respectively.
Based on the information given:
- The number of nickels is four times the number of quarters, so n = 4q.
- There are twice as many quarters as there are dimes, so q = 2d.
- The total number of coins is 154, so n + d + q = 154.
- The total value of the coins is $14.00, which can be converted to 1400 cents.
- The value of a nickel is 5 cents, a dime is 10 cents, and a quarter is 25 cents.
Using these equations, we can solve for the variables n, d, and q to find the number of nickels, dimes, and quarters in the bag.
Substituting the second equation into the first equation, we get:
n = 4(2d).
Substituting this value of n into the third equation, we get:
4(2d) + d + 2d = 154.
Simplifying the equation, we have:
8d + d + 2d = 154.
Combining like terms, we get:
11d = 154.
Dividing both sides of the equation by 11, we get:
d = 14.
Substituting this value of d into the second equation, we get:
q = 2(14) = 28.
Finally, substituting the values of n, d, and q into the equation for the total value of the coins, we can solve for n:
5n + 10d + 25q = 1400.
Substituting the values, we get:
5n + 10(14) + 25(28) = 1400.
Simplifying the equation, we have:
5n + 140 + 700 = 1400.
Combining like terms, we get:
5n + 840 = 1400.
Subtracting 840 from both sides of the equation, we get:
5n = 560.
Dividing both sides of the equation by 5, we get:
n = 112.
Therefore, there are 112 nickels, 14 dimes, and 28 quarters in the bag.