Final answer:
The problem involves solving for the number of nickels, dimes, and quarters given their total value and ratios. A system of equations using the coin values is set up, leading to the answer: 6 nickels, 16 dimes, and 10 quarters.
Step-by-step explanation:
To solve the problem involving a collection of nickels, dimes, and quarters worth $6.90, we'll use the given ratios of coins and their respective values to set up a system of equations. To do this, let's assume the number of nickels is 3x, the number of dimes is 8x, and using the second ratio, if there are 4y dimes, there will be 5y quarters.
Since each nickel is worth 5 cents (or $0.05), each dime is worth 10 cents (or $0.10), and each quarter is worth 25 cents (or $0.25), our coin value chart will have the values:
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- Nickels (3x) = $0.05 × 3x
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- Dimes (8x) = $0.10 × 8x
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- Quarters (5y) = $0.25 × 5y
The total value is therefore $0.05 × 3x + $0.10 × 8x + $0.25 × 5y = $6.90.
Because we have two ratios (nickels to dimes and dimes to quarters), set 8x equal to 4y to reflect the second ratio (8x = 4y or x = y/2). Now, substituting x for y/2 in the equation:
$0.05 × 3( y/2 ) + $0.10 × 8( y/2 ) + $0.25 × 5y = $6.90. By solving for y, we can find the number of quarters, and then the number of dimes and nickels.
After calculating, we find the total number of each type of coin: 6 nickels, 16 dimes, and 10 quarters.