Question 176729
The number of quarters: {{{q}}}


The number of 50c pieces: {{{f}}}


Given:  {{{q + f = 30}}}


The value of each quarter is 25 cents, so the value of all her quarters is {{{25q}}} in cents.


Similarly the value of all her 50c pieces is {{{50f}}}


The total value of her change is {{{20 - 9.25 = 10.75}}} dollars which is the same as {{{1075}}} cents.  The value of the quarters plus the value of the 50c pieces must equal the total value of the change, so:


{{{25q + 50f = 1075}}}


Solve the first equation for {{{q}}}:  {{{q + f = 30}}} → {{{q = 30 - f}}}.


Substitute this expression for {{{q}}} into the second equation:


{{{cartoon(25*red(q) + 50f = 1075, 25*red((30 - f)) + 50f = 1075)}}}


Solve for {{{f}}}


{{{25(30 - f) + 50f = 1075}}}


{{{750 - 25f + 50f = 1075}}}


{{{25f = 325}}}


{{{f = 13}}}


So there were 13 half-dollars, and therefore {{{30 - 13 = 17}}} quarters.


Check:

{{{17 * .25 = 4.25}}}


{{{13 * .50 = 6.50}}}


{{{4.25 + 6.50 = 10.75}}} Checks.