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If candies have a cost of x cents less per dozen, it would have cost 3 cents less for x + 3 candies than if they had cost x cents more per dozen. What is x?

I didn't get far with what I was able to do:

  • Cost x less per dozen
  • (x+3)-3 ?
  • What is x
kingW3
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Daniel Marksman
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1 Answers1

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$p$ is the initial price per dozen

If candies costed $x$ cent less per dozen, the total cost is $\dfrac{x+3}{12}\left(p-x\right)$

If candies costed $x$ cent more per dozen it is $\dfrac{x+3}{12}\left(p+x\right)$

The problem says that in the first case the cost is $3$ cents less so we have the equation

$$\frac{x+3}{12}(p-x)=\frac{x+3}{12}(p+x)-3$$ Least common denominator $$(x+3) (p-x)=(x+3) (p+x)-36$$ Expand $$p x+3 p-x^2-3 x=p x+3 p+x^2+3 x-36$$ move everything in the RHS

$2 x^2+6 x-36=0$ simplify dividing all by $2$

$x^2+3x-18=0$ which gives $x_1=-6;\;x_2=3$

The actual solution is $x=3$

Siong Thye Goh
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Raffaele
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    Typo: $x^2+3x-12$ in the second last line is supposed to be $x^2+3x-18$. Credit: user471693 for pointing out the mistake. – Siong Thye Goh Aug 13 '17 at 18:49
  • @SiongThyeGoh how did the -3 become -36. I know you multiplied -3 by 12, but then wouldn't you also have to multiply the (p+x) and (p-x) by 12 too? – Daniel Marksman Aug 14 '17 at 04:51
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    @DanielMarksman those terms have a $\frac{x+3}{12}$ as their coefficient. the denominator handles that for us. – Siong Thye Goh Aug 14 '17 at 05:16