document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=819639>Question 819639</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Sammy decided to go Christmas shopping at the mall. He had won a pass that allowed him to take an additional 20% off all final price purchases at any store, which he used for each purchase. First Sammy went to Game Stop to by his friend a video game. Game stop was having a 35% off store-wide sale. Then, he headed off to Payless to take advantage of their 25% off everything sale, to buy a pair of shoes. His last stop was at Moosejaw to buy a pullover, because they were having a one-day sale where everything was 15% off. As Sammy was leaving the mall, he stopped at a gift wrapping booth. They were charging $2.75(tax-included) for each gift wrapped. Sammy had all three gifts wrapped.<BR>\n" );
document.write( "At the end of the month Sammy opened his credit card statement and noticed the following three charges:<BR>\n" );
document.write( "Game Stop: $35.28<BR>\n" );
document.write( "Payless: $55.72<BR>\n" );
document.write( "Moosejaw: $96.89<BR>\n" );
document.write( "What was the original price of each item AND what was the average dollar amount Sammy spent on each purchase?<BR>\n" );
document.write( "Using MICHIGAN sales tax! 6% <BR>\n" );
document.write( "What I already solved:<BR>\n" );
document.write( "1. Work backwards<BR>\n" );
document.write( "2. Undo the tax<BR>\n" );
document.write( "3. $55.72 - $3.15 (tax)= $52.57<BR>\n" );
document.write( "$35.28 - $2.54 (tax)= $32.74<BR>\n" );
document.write( "$96.89 - $5.48 (tax)= $91.41<BR>\n" );
document.write( "4. Undo the percent off<BR>\n" );
document.write( "5. Total of 45% saved at Payless (25% + 20%)<BR>\n" );
document.write( "Total of 55% saved at Game Stop  (35% + 20%)<BR>\n" );
document.write( "Total of 35% saved at Moosejaw   (15% + 20%)<BR>\n" );
document.write( "*This is the part of the process I don't understand </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #493178 by </B><A HREF=/tutors/aboutme.mpl?userid=KMST><B>KMST(5328)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=KMST><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=KMST><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=493178\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ORIGINAL PRICE ON EACH ITEM:<BR>\n" );
document.write( "The game at Game Stop was originally priced at $64.00.<BR>\n" );
document.write( "For the sale, the store reduced the price by 35%, so the reduced price ended up being<BR>\n" );
document.write( "100% - 35% = 65% of the original price<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=64.00%2A0.65 ALIGN=MIDDLE  ALT=\"64.00%2A0.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=41.60 ALIGN=MIDDLE  ALT=\"41.60\"   DISABLED_style= \"max-width:90%; height:auto;\" > .<BR>\n" );
document.write( "The extra 20% from Sammy's pass reduced the price further by 20% calculated based on the $41.60 sale price.<BR>\n" );
document.write( "That is a reduction of<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=41.60%2A0.20 ALIGN=MIDDLE  ALT=\"41.60%2A0.20\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=8.32 ALIGN=MIDDLE  ALT=\"8.32\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So Sammy ended up paying $41.60 - $8.32 = $33.28 (plus tax).<BR>\n" );
document.write( "That $33.28 can be calculate directly as 80% of $41.60:<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=41.60%2A0.8 ALIGN=MIDDLE  ALT=\"41.60%2A0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33.28 ALIGN=MIDDLE  ALT=\"33.28\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "That twice reduced price is not 55% less than $64.00.<BR>\n" );
document.write( "You cannot add percentages that are not applied to the same (original price.<BR>\n" );
document.write( "Finally, the tax was calculated based on the final $33.28 price, and that increased the amount paid for the game by<BR>\n" );
document.write( "$<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=33.28%2A0.06 ALIGN=MIDDLE  ALT=\"33.28%2A0.06\"   DISABLED_style= \"max-width:90%; height:auto;\" >= $<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.00 ALIGN=MIDDLE  ALT=\"2.00\"   DISABLED_style= \"max-width:90%; height:auto;\" > (rounded) for a total paid of<BR>\n" );
document.write( "$33.28 + $2.00 = $35.28, which is<BR>\n" );
document.write( "100% + 6% = 106% of the final price,<BR>\n" );
document.write( "which can be calculated directly by multiplying the final price by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.06 ALIGN=MIDDLE  ALT=\"1.06\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "So, with all those percentages the original price was multiplied by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.65 ALIGN=MIDDLE  ALT=\"0.65\"   DISABLED_style= \"max-width:90%; height:auto;\" >, then by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.8 ALIGN=MIDDLE  ALT=\"0.8\"   DISABLED_style= \"max-width:90%; height:auto;\" >, and finally by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.06 ALIGN=MIDDLE  ALT=\"1.06\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the final billed amount.<BR>\n" );
document.write( "If rounding errors from each step did not add up, the result would be the same  as multiplying times<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.65%2A0.8%2A1.06=0.5512 ALIGN=MIDDLE  ALT=\"0.65%2A0.8%2A1.06=0.5512\"   DISABLED_style= \"max-width:90%; height:auto;\" >,<BR>\n" );
document.write( "which means that we could undo all the multiplications by dividing by 0.5512:<BR>\n" );
document.write( "$35.28 ÷ 0.5512 = $64.01 (rounded)\r<BR>\n" );
document.write( "\n" );
document.write( "For the shoes at payless the original price was reduced by 25%, to<BR>\n" );
document.write( "100% - 25% = 75% of the original price.<BR>\n" );
document.write( "That means that the original price was multiplied times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75 ALIGN=MIDDLE  ALT=\"0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get the sale price.<BR>\n" );
document.write( "Sammy's pass meant that the sale price was reduced by a further 20% to<BR>\n" );
document.write( "100% - 20% = 80% of the already reduced sale price.<BR>\n" );
document.write( "That means that the sale price was multiplied times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.80 ALIGN=MIDDLE  ALT=\"0.80\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get Sammy's final price.<BR>\n" );
document.write( "Then 6% sales tax was added, increasing the amount Sammy paid to<BR>\n" );
document.write( "100% +65 = 106% of the final special price Sammy got.<BR>\n" );
document.write( "That means Sammy's special price was multiplied times <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.06 ALIGN=MIDDLE  ALT=\"1.06\"   DISABLED_style= \"max-width:90%; height:auto;\" >.<BR>\n" );
document.write( "To undo all those calculations, we can divide, first by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.06 ALIGN=MIDDLE  ALT=\"1.06\"   DISABLED_style= \"max-width:90%; height:auto;\" > to undo the tax, then by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.80 ALIGN=MIDDLE  ALT=\"0.80\"   DISABLED_style= \"max-width:90%; height:auto;\" > to undo the special 20% discount, and finally by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75 ALIGN=MIDDLE  ALT=\"0.75\"   DISABLED_style= \"max-width:90%; height:auto;\" > to undo the sale price reduction.<BR>\n" );
document.write( "We could even round at each step.<BR>\n" );
document.write( "We can undo it all at once by dividing by<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.75%2A0.8%2A1.06=0.636 ALIGN=MIDDLE  ALT=\"0.75%2A0.8%2A1.06=0.636\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "It should give the same result, or almost the same if roundings add up.<BR>\n" );
document.write( "$55.72 ÷ 0.636 = $87.61<BR>\n" );
document.write( " <BR>\n" );
document.write( "For the pullover bought at Moosejaw, the factor for the one-day sale 15% price reduction applied first is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.85 ALIGN=MIDDLE  ALT=\"0.85\"   DISABLED_style= \"max-width:90%; height:auto;\" > , and after that the same <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.80 ALIGN=MIDDLE  ALT=\"0.80\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1.06 ALIGN=MIDDLE  ALT=\"1.06\"   DISABLED_style= \"max-width:90%; height:auto;\" > factors were applied to calculate Sammy's special discount and the price plus tax respectively.<BR>\n" );
document.write( "The overall factor is<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=0.85%2A0.80%2A1.06=0.7208 ALIGN=MIDDLE  ALT=\"0.85%2A0.80%2A1.06=0.7208\"   DISABLED_style= \"max-width:90%; height:auto;\" > and the original price was<BR>\n" );
document.write( "$96.89 ÷ 0.7208 = $134.42 </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );