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A lady came to the market to sell eggs. Her first customer bought 1/2 of all her eggs and 1/2 of an egg. Her second customer bought 1/2 of all the remaining eggs and 1/2 of an egg. Finally, a third customer bought 1/2 of all of her remaining eggs and 1 2 of an egg. At this point she had sold all of the eggs she had brought to the market so she went home. How many eggs did the lady have when she first came to the market?

Answer :

Answer:

total eggs = 7

Step-by-step explanation:

We can start by [tex]x[/tex] number of eggs.

Total eggs = [tex]x[/tex]

  • 1st buyer

the first buyer buys 1/2 of all eggs and another 1/2

[tex]x-\left(\dfrac{x}{2}+\dfrac{1}{2}\right)[/tex]

we can say that the remaining eggs are equal to this above equation and call it [tex]A[/tex].

[tex]A = x-\left(\dfrac{x}{2}+\dfrac{1}{2}\right)[/tex]

we can simplify it further

[tex]A = x-\dfrac{x}{2}-\dfrac{1}{2}[/tex]

[tex]A = \dfrac{x}{2}-\dfrac{1}{2}[/tex]

[tex]A = \dfrac{1}{2}(x-1)[/tex]

  • 2nd buyer

the second buyer takes 1/2 of the remaining eggs and another 1/2 egg.

remaining eggs from the first buyer were [tex]A[/tex]

remaining eggs from the second buyer are:

[tex]A-\left(\dfrac{A}{2}+\dfrac{1}{2}\right)[/tex]

we can call these remaining eggs as [tex]B[/tex].

[tex]B = A-\left(\dfrac{A}{2}+\dfrac{1}{2}\right)[/tex]

now simplify (similar to how equation [tex]A[/tex] was simplified)

[tex]B = \dfrac{1}{2}(A-1)[/tex]

we can further simplify it by substituting the equation [tex]A[/tex], making equation [tex]B[/tex] in terms of [tex]x[/tex]

[tex]B = \dfrac{1}{2}\left(\dfrac{x-1}{2} - 1\right)[/tex]

[tex]B = \dfrac{x}{4} -\dfrac{1}{4} - \dfrac{1}{2}[/tex]

[tex]B = \dfrac{x}{4} -\dfrac{3}{4}[/tex]

this is the amount of the remaining eggs after the 2nd buyer.

  • 3rd buyer

This is where things get interesting. The third buyer takes half of what remaining, i.e [tex]B[/tex].

[tex]B-\left(\dfrac{B}{2}+\dfrac{1}{2}\right)[/tex]

but this time, there are no remaining eggs!

So the equation is equal to zero!

[tex]0 = B-\left(\dfrac{B}{2}+\dfrac{1}{2}\right)[/tex]

simplify (as we were doing for previous equations)

[tex]\dfrac{1}{2}(B-1) = 0[/tex]

now we can substitute the equation of [tex]B[/tex]

[tex]\dfrac{1}{2}\left(\dfrac{x}{4} -\dfrac{3}{4}-1\right) = 0[/tex]

all we need to do now is just solve for x!

[tex]0 = \dfrac{x}{4} -\dfrac{3}{4}-1[/tex]

[tex] \dfrac{x}{4} -\dfrac{3}{4} = 1[/tex]

[tex] x-3 = 4[/tex]

[tex] x = 7[/tex]

The total number of eggs when the lady first came to the market were 7!

Hope this helps! :)

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