If candies have a cost of x cents less per dozen . . . What is x?

Question

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

Answer 1

$p$ 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 $x=3$