Question 1201833
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Answer: 
<font color=red>70 dimes
33 quarters</font>


70 dimes + 33 quarters = 70*10 + 33*25 = 1525 cents = 15.25 dollars


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Work Shown:


d = number of dimes
q = number of quarters
These variables are placeholders for nonnegative whole numbers 0,1,2,3,...


d+q = 103 coins total
q = 103-d
It will be used in a <font color=blue>substitution</font> step later


10d = value of the dimes only (cents)
25q = value of the quarters only (cents)
10d+25q = total value (cents)
10d+25q = 1525
10d+25( q ) = 1525
10d+25( <font color=blue size=4>q</font> ) = 1525
10d+25( <font color=blue size=4>103-d</font> ) = 1525   ... <font color=blue>substitution</font>
10d+25*103+25*(-d) = 1525
10d+2575-25d = 1525
-15d+2575 = 1525
-15d = 1525-2575
-15d = -1050
d = -1050/(-15)
d = <font color=red>70 dimes</font>


q = 103-d
q = 103-70
q = <font color=red>33 quarters</font>


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Another approach:


The 25 cents in $15.25 tells us that there's at least one quarter. 
It is impossible to reach this subtotal amount with dimes alone.


If we had 1 quarter and 103-1 = 102 dimes, then we have: 
1 quarter + 102 dimes = 1*25 + 102*10 = 1045 cents = $10.45
We come up short of the goal $15.25


Let's replace 1 dime with 1 quarter.
Difference in coin value = 25-10 = 15 cents
Each time we do this replacement of "1 dime for 1 quarter", the total value goes up by 15 cents aka $0.15
This is the net change.


Because we want the $10.45 to go to $15.25, we must increase by $4.80
(since 15.25-10.45 = 4.80 )


n = number of increases of 15 cents each
0.15n = total increase needed
0.15n = 4.80
n = 4.80/0.15
n = 32
We need 32 increases of 15 cents each to go from $10.45 to $15.25
Meaning we need to do the replacement of "1 dime for 1 quarter" a total of 32 times.


Add 32 to the quarter count, and subtract 32 from the dime count.
1+32 = <font color=red>33 quarters</font>
102-32 = <font color=red>70 dimes</font>
which are the final answers.


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Check:
70 dimes + 33 quarters
= 70*10 + 33*25
= 700 cents + 825 cents
= 1525 cents
= 1525/100
= 15.25 dollars
The answers are confirmed.
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