Answer:
Our friend is incorrect.
# of Quarters = 10
# of Dimes = 3
Explanation:
For this problem, we will need to write two equations to find the amount of quarters and dimes we have to test our friend's hypothesis.
Let d represent the number of dimes and q represent the number of quarters.
Then, we can build the following equations:
0.25q + 0.1d = 2.8
q = 8 + d
Note, the value of a quarter is $0.25 and the value of a dime is $0.10.
Now, we can plug the value in for q and solve for d to test our friend's hypothesis.
0.25q + 0.1d = 2.8
0.25(8 + d) + 0.1d = 2.8
2 + 0.25d + 0.1d = 2.8
2 + 0.35d = 2.8
2 + 0.35d + -2 = 2.8 + -2
0.35d = 0.8
0.35d * ( 1 / 0.35 ) = 0.8 * ( 1 / 0.35 )
d = 2.857
Note, we cannot have part of a dime so if we were to follow conventionally rounding rules you would say d = 3; however, in terms of currency you would want to round down (Use a floor function) so d = 2.
If d = 2, we can say the following:
q = 8 + d
q = 8 + 2
q = 10
Hence, we would have 10 quarters and 2 dimes, but wait, that would only be 10 * 0.25 and 2 * 0.1 which is 2.7. That is not our 2.8 value for the amount of money we had. Why might this be?
Due to the nature that coins cannot be partial, we had to round down. What if we said we had 3 dimes ( d = 3 ) ?
If d = 3, we can say the following:
q = 8 + d
q = 8 + 3
q = 11
Hence, we would have 11 quarters and 3 dimes, but wait, that would be 11 * 0.25 and 3 * 0.1 which is 3.05 which is greater than our value of 2.8.
From these two possibilities, we can say that our friend's hypothesis could be plausible if we were allowed to have fractional coins; however, since coins are whole, the friend was incorrect.
To find the actual value of coins, we will modify our formula to be based on a ceiling function (Rounded up to the whole number) for dimes.
q = 8 + (d - 1)
The reason we are using d - 1, is to account for the extra dime we are adding by using the ceiling function. Hence, let's solve given d = 3.
If d = 3, we can say the following:
q = 8 + ( d - 1 )
q = 8 + ( 3 - 1 )
q = 8 + 2
q = 10
Hence, we can say that 10 quarters and 3 dimes, ($2.50 and $0.30) will be equivalent to our original $2.80.
Cheers.