# Ratio word problem with eggs?

The eggs in a certain basket are either white or brown. If the ratio of the number of white eggs to the number of brown eggs is $2/3$ , each of the following could be the number of eggs in the basket except.....

A 10

B 12

C 15

D 30

E 60

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Do you have any guess? – Berci Mar 2 '13 at 23:44
Hint: the smallest scenario is 2 white eggs and 3 brown eggs, for a total of 5 eggs. – alex.jordan Mar 2 '13 at 23:46

$$\dfrac 23 = \dfrac 46 = \dfrac 69 = \dfrac{8}{12} = \dfrac{10}{15} = \dfrac {12}{18} = \dfrac {14}{21} = \dfrac {16}{24} = \dfrac{18}{27} = \dfrac{20}{30} = \dfrac{22}{33} = \dfrac{24}{36}$$

For each ratio, the total number of eggs is given by the sum of the numerator with the denominator, ordered as follows $$\text{sum}\;\dfrac {2n}{3n}: 2n + 3n = 5n: \quad 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, ...$$

Note that $12$ does not appear among them.

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Oh I see then $4/6$ is 10 eggs in basket which is equal to $2/3$ Cannot be 12 eggs because you can't get a 2 to 3 ratio. – Little Jon Mar 2 '13 at 23:58
Exactly, Little Jon! – amWhy Mar 2 '13 at 23:58

Hint: For every $2$ white eggs, there are $3$ brown eggs (which is what that ratio means), so the total number of eggs can be divided evenly into groups of $5$. Which of the given options can't be divided evenly in this way?

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The smallest case will be of $2$ white and $3$ brown eggs, i.e $5$ eggs in total. Can you see how does this imply that the total number of eggs will always be of the form $5x, x\in \mathbb{N}?$

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