# What was the number of blue marbles in the bag before any changes were made?

I don't know whether I am wrong or the answer sheet is wrong. Here is how I solve the problem: First, list equations according to the problem. Then solve equations to find B. In the equations, G and B represent green and blue marbles, respectively.

$$\frac{G}{B+G-3G}=\frac{2}{5} -----------(1)$$ $$\frac{B}{B+G+7B}=\frac{5}{8}---------(2)$$

A bag contains only blue and green marbles. If three green marbles are removed from the bag, the probability of drawing a green marble from the remain marbles would be 2/5. If, instead, seven more blue marbles are added to the bag, the probability of obtaining a blue marble would be 5/8. What was the number of blue marbles in the bag before any changes were made? The answer sheet says 16.

Let $x=$ number of blue marbles originally, and $y=$ number of green marbles originally.
The correct equations should be:$$\frac25=\frac{y-3}{x+y-3}$$$$\frac58=\frac{x+7}{x+y+7}$$ The answer I got is $x=18,y=15$

A bag contains only blue and green marbles.

• Set $b:=$ original number of blue marbles and $g:=$ original number of green marbles. Total number of marbles is thus $b+g$.

If three green marbles are removed from the bag, the probability of drawing a green marble from the remain marbles would be 2/5.

• Changed number of green marbles $=g-3$. Changed total marble count $=b+g-3$.

$$\frac{g-3}{b+g-3} = \frac{2}{5}$$

If, instead, seven more blue marbles are added to the bag, the probability of obtaining a blue marble would be 5/8.

• Changed number of blue marbles $=b+7$. Changed total marble count $=b+7+g$.

$$\frac{b+7}{b+7+g} = \frac{5}{8}$$

What was the number of blue marbles in the bag before any changes were made?

\begin{align} 5(g-3) &= 2(b+g-3) \\ \implies 5g -15 &= 2b +2g -6 \\ \implies 3g -2b &= 9 \tag{1}\\ 8(b+7) &= 5(b+7+g) \\ \implies 8b+56 &= 5b +5g + 35 \\ \implies 5g - 3b &= 21 \tag{2}\\ \implies 2g - b &= 12 \tag{3; from (2)-(1)}\\ \implies g &= 15 \tag{2$\times$(3)-(1)}\\ \text{and}\quad b &=18 \\ \end{align}

$\to$ Originally $18$ blue marbles in the bag.

Check: $\frac{12}{18+12} = \frac{12}{30} = \frac 25 \checkmark$ and $\frac{25}{25+15} = \frac{25}{40} = \frac 58 \checkmark$

• Why is it $g-3$, not just g? – learning Jun 17 '16 at 3:43
• @learning $g$ is the original number of green marbles. $g-3$ is what's left after $3$ are removed. – Joffan Jun 17 '16 at 3:46