To solve the problem using the guess and check method, we start by defining the two types of coins Rupert has: nickels (N) and pennies (P).
We know:
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The total value of the coins is 77 cents: \[ 5N + P = 77 \] (since each nickel is worth 5 cents and each penny is worth 1 cent)
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The total number of coins is 21: \[ N + P = 21 \]
Now we can use these two equations to set up our checks. First, we can express P in terms of N using the second equation: \[ P = 21 - N \]
Now we substitute this expression for P into the first equation: \[ 5N + (21 - N) = 77 \] Simplifying gives: \[ 5N + 21 - N = 77 \] \[ 4N + 21 = 77 \] \[ 4N = 77 - 21 \] \[ 4N = 56 \] \[ N = 14 \]
Now, substituting \(N=14\) back into the equation for \(P\): \[ P = 21 - N = 21 - 14 = 7 \]
Thus, Rupert has 14 nickels and 7 pennies.
Let's confirm this:
- Value from nickels: \(14 \times 5 = 70\) cents
- Value from pennies: \(7 \times 1 = 7\) cents
- Total value: \(70 + 7 = 77\) cents
The total number of coins is \(14 + 7 = 21\) coins.
So, we have the correct solution: 14 nickels and 7 pennies.
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