11. a collection of nickels and dimes is worth $9.45. if the number of dimes is doubled, the value is
$16.65. find the number of each coin.

45 nickels and 72 dimes

Step-by-step explanation:

We need to find the number of dimes and the number of nickels, therefore those are our unknowns, that we name N (for the number of nickels), and D (for the number of dimes).

Using these letters to represent the unknowns, let's write an equation for each of the word statements:

a) "A collection of nickels and dimes is worth $9.45" which we translate to math terms using the fact that each nickel is worth $0.05, and each nickel is worth $0.1. Therefore, the actual value of the collection can be written as the number of a coin type times its value:

Collection of nickels and dimes = $9.45

value of collection of N nickels + value of collection of D dimes = $9.45

0.05 * N + 0.1 * D = 9.45 (we dropped the dollar signs to make the equation look simpler)

0.05 N + 0.1 D = 9.45

b) "If the number of dimes is doubled, the value is  $16.65"

We double just the number of dimes from "D" to "2D", and the result should be $16.65. Remember to drop the dollar signs to make the equation easier to read algebraically.

0.05 * N + 0.1 * 2D = 16.65

0.05 N + 0.2 D = 16.65

Now we subtract term by term the equation we obtained in b) minus the equation we obtained in a):

0.05 N + 0.2 D = 16.65

-                             -

0.05 N + 0.1 D  =  9.45

Notice that:

1) 0.05 N minus 0,05 N gives zero (the two terms cancel out)

2) 0.2D minus 0.1D results in 0.1D

3) 16.65 minus 9.45 results in 7.2

0 + 0.1 D = 7.2

0.1 D = 7.2

therefore D is 7.2 divided by 0.1 which equals 72 (that is: the collection has 72 dimes)

Now we use this result in the first equation we found in part a), to solve for the number of nickels:

0.05 N + 0.1 D  =  9.45

0.05 N + 0.1 (72) = 9.45

0.05 N + 7.2 = 9.45

0.05 N = 9.45 - 7.2

0.05 N = 2.25

N = 2.25/0.05

N = 45

that is: the collection has 45 nickels

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