document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209197>Question 1209197</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Megan wants to buy an item worth $400. She has two coupons of 10% and 15% off. What is the total discount rate, as a percentage, if she used both coupons? Hint: The answer is not 25% </SPAN>\n" );
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document.write( "Answer: <font color=red>23.5%</font>\r<BR>\n" );
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document.write( "Work Shown<BR>\n" );
document.write( "a = 0.10 represents the 10% discount rate<BR>\n" );
document.write( "b = 0.15 is the 15% discount<BR>\n" );
document.write( "The order of a,b doesn't matter.\r<BR>\n" );
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document.write( "(1-a)*(1-b) = proportion Megan spends<BR>\n" );
document.write( "1-(1-a)*(1-b) = proportion Megan saves<BR>\n" );
document.write( "1-(1-a)*(1-b) = 1-(1-0.10)*(1-0.15) = 0.235 = <font color=red>23.5%</font> is total discount rate as a percentage. This value is exact and hasn't been rounded.\r<BR>\n" );
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document.write( "This idea can be extended to more than two discounts.<ul><li>3 discounts: 1-(1-a)*(1-b)*(1-c)</li><li>4 discounts: 1-(1-a)*(1-b)*(1-c)*(1-d)</li><li>5 discounts: 1-(1-a)*(1-b)*(1-c)*(1-d)*(1-e)</li><li>And so on.</li></ul>a,b,c,d,e,... represent the individual discount rates in decimal form between 0 and 1 excluding both endpoints.<BR>\n" );
document.write( "The final result will be in decimal form between 0 and 1. Multiply by 100 to convert to a percentage. This is equivalent to moving the decimal point two spots to the right (eg: 0.1234 converts to 12.34%)\r<BR>\n" );
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document.write( "Another approach.\r<BR>\n" );
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document.write( "Megan spends 400*(1-0.10)*(1-0.15) = 306 dollars <BR>\n" );
document.write( "She saves 400-306 = 94 dollars. <BR>\n" );
document.write( "Then 94/400 = 0.235 = <font color=red>23.5%</font> is the combined discount rate.<BR>\n" );
document.write( "Note that the 400 can be changed to any other positive number to get the same result at the end.<BR>\n" );
document.write( "Therefore the starting price will not affect the answer.\r<BR>\n" );
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document.write( "This section will discuss yet another approach. <BR>\n" );
document.write( "It's a bit slower, but it's always a good idea to be able to look at math problems from multiple viewpoints.<BR>\n" );
document.write( "The item starts off at $400<BR>\n" );
document.write( "Let's say Megan uses the 10% discount first.<BR>\n" );
document.write( "She saves 0.10*($400) = $40 and the price is now $400 - $40 = $360<BR>\n" );
document.write( "Or as a slight shortcut we could say 0.90*($400) = $360 since saving 10% means she pays the remaining 90%.\r<BR>\n" );
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document.write( "Now she'll use the 15% discount.<BR>\n" );
document.write( "She saves an extra 0.15*($360) = $54 and the final price is $360 - $54 = $306<BR>\n" );
document.write( "Overall Megan saves $400 - $306 = $94  (or add the discounts 40+54 = 94)<BR>\n" );
document.write( "Divide this total savings over the original starting price to get the combined discount rate<BR>\n" );
document.write( "94/400 = 0.235 = <font color=red>23.5%</font> which is the final answer.\r<BR>\n" );
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document.write( "The 23.5% is somewhat close to 25% which was mentioned in the hint.<BR>\n" );
document.write( "A first-time student's approach would likely say 10% + 15% = 25%<BR>\n" );
document.write( "But that is <u>not</u> correct.<BR>\n" );
document.write( "Why can't we add the percentages? Because the second discount does NOT apply to the original $400; instead it applies to the reduced cost of some value less than $400.<BR>\n" );
document.write( "I've seen many students make the common mistake of adding the percentages when trying to find the total discount rate. Be careful to avoid this trap.\r<BR>\n" );
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document.write( "What if Megan swapped the order of the discounts?<BR>\n" );
document.write( "Let's say she used the 15% discount first.<BR>\n" );
document.write( "She saves 0.15*($400) = $60 and the price is now $400 - $60 = $340<BR>\n" );
document.write( "Next she'll use the 10% discount.<BR>\n" );
document.write( "She saves another 0.10*($340) = $34 and the final price is $340 - $34 = $306<BR>\n" );
document.write( "We arrive at the same final price of $306. <BR>\n" );
document.write( "It turns out the order of the discounts does <u>not</u> matter. <BR>\n" );
document.write( "Refer back to the template 1-(1-a)*(1-b) to see that we can multiply (1-a) and (1-b) in either order.<BR>\n" );
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