By receiving two successive discounts, a dealer bought merchandise for $9 that

was listed at $20. What were the discounts in percent if the first discount

was 5 times the second?

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There are two ways of expressing discounts, so we have to be careful not to

get them confused.  There are DOLLAR discounts and there are PERCENT

discounts.



Let the first DOLLAR discount be $x

Let the second DOLLAR discount be $y

>>...bought merchandise for $9 that was listed at $20...<<

Translation: $20 minus the 1st DOLLAR discount minus the 2nd DOLLAR discount

equals $9, or

$20 - $x - $y = $9

   20 - x - y = 9         

           11 = x + y 

        x + y = 11

>>...the first discount was 5 times the second...<<

Note:  This is badly worded because it is not clear whether that's talking

about the DOLLAR or the PERCENT discount.  I will assume it means the

DOLLAR discount and is to be taken as:

>>...the first DOLLAR discount was 5 times the second...<<

Translation:  x = 5y.

So we have this system of equation to solve:

   x + y = 11 

       x = 5y

From the second equation, substitute 5y for x in the first equation

  x + 5x = 11

      6x = 11

       x = 11/6 = $1.83 1/3

Then substitute 11/6 for x in the first equation

11/6 + y = 11

 11 + 6y = 66

      6y = 55

       y = 55/6 = $9.16 2/3

>>...What were the discounts in percent...?...<< 

So you are asked for PERCENT discounts, not DOLLAR discounts

So we need to ask:  "What percent of $20 is $1.83 1/3?" or

"What percent of 20 is 11/6?"

So we divide 11/6 by 20:  11/6 divided by 20 = (11/6)×(1/20) = .09 1/6 or

(9 1/6)%­.  That is the first PERCENT discount.  

After the first discount, the price was reduced to 

$20 - $x = 20 - 11/6 = 120/6 - 11/6 = 109/6 = $18.16 2/3

After the second discount, the price was reduced to

($18.16 2/3) - ($9.16 2/3) = $9.

So now we need to ask:  "What percent of $18.16 2/3 is $9.16 2/3?" or

"What percent of 109/6 is 55/6?"

So we divide 55/6 by 109/6:  55/6 divided by 109/6 = (55/6)×(6/109) = 55/109

or .50 50/109 which when changed to a percent becomes

(50 50/109)%   

That is the second PERCENT discount. 

Because this came out with terrible answers, it may have supposed

to have been 

>>...the first PERCENT discount was 5 times the second...<<

which would have changed the answer.  Check with your teacher as to which

it was supposed to be.

Edwin















 

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