We are given a table with the following information:

Bag P has 37 marbles and 10.8% of those marbles are blue. So there are 0.108(37) = 3.996, or 4, blue marbles in bag P.

Bag Q has X marble and 66.7% of those marbles are blue. Recall that the fraction 2/3 is approximately 66.7%, so there are (2/3)X blue marbles in bag Q.

Bag R has 32 marbles and 50% of those marbles are blue. So, there are 0.5(32) = 16 blue marbles in bag R.

We are also given that 1/3 of the total marbles in the 3 bags are blue. Thus, we can create the following equation:

1/3(37 + X + 32) = 4 + (2/3)X + 16

Multiplying the equation by 3, we have:

37 + X + 32 = 12 + 2X + 48

X + 69 = 2X + 60

X = 9

There are 9 marbles in bag Q.

Answer: B
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I have approximated 10.8% as 1/9, which is 11.1%

If 1/3 of the total number of marbles in the three bags listed in the table above are blue, how many marbles are there in bag Q?

A) 5
B) 9
C) 12
D) 23
E) 46

Total has to be multiple of 3 since 1/3rd of them are blue and 1st and 3rd bag have total 69 marbles so (69 + x)/3 is blue total hence x also must be divisible by 3

hence narrow down to B) 9 or C) 12

66.67 % is nothing but 2/3 rd (ie 33.33 * 2 ie 1/3 * 2)

so if x is 9 then blue marbles in 2nd bag is 6 (2/3rd of 9 = 6)
or if x is 12 then blue marbles in 3rd bag is 8 (2/3rd of 12 = 8)

do that math : if x = 9 then total = 69 + 9 = 78
and total blue = 16 + 4 + 6 = 26
26/78 = 1/3 -----> fits (most likely that B is the ans)

cross check
if x = 12 then total = 81
total blue = 28
28/81 != 1/3 hence (B)
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No. of blue marbles in each bag is
Bag P: 10.8% of 37 = 4
Bag Q:66.7% of X = 2/3 X
Bag R: 50% of 32 = 16

Total Blue marbles = 4 + 2/3 X + 16 = 20 + 2/3 X

Also Total marbles count = 37 + X + 32 = 69 + X

So, 1/3 (69+ X ) = 20 + 2/3 X
23 + 1/3 X = 20 + 2/3 X
1/3 X = 3
X = 9

Answer B

hi dave13

Pls check the calculations in the highlighted part above. How come 37+32 become 23? and when you are opening the bracket then each element of the bracket has to be multiplied by 1/3 or 0.33

dave13
The equation is correct. The arithmetic is off.

(EDIT As niks18 points out), check your math here:

Try using \(\frac{67}{100}=\frac{2}{3}\)

\(\frac{1}{3}x - \frac{2}{3}x\) = ????

Calculate again and see what you get.

Tip: use fractions with fractions and decimals with decimals. IMO fractions are easier.

Another way to think about setup and solving of the equation . . . (I do not know whether either suggestion will be easier for you - they're just possibilities)

After you replace \(.67\) with \(\frac{2}{3}\)

Change the setup and cross multiply.

(1) Setup: If blue marbles equal \(\frac{1}{3}\) of three bags' total marbles:
\(\frac{Blue}{Total}=\frac{1}{3}\)
\(\frac{16+4+\frac{2}{3}x}{37+32+x}=\frac{1}{3}\)

(2) Solve. Cross multiply.

Just one arithmetic mistake, in other words.

Your second solution is correct. If you use decimals (IMO, harder here!) what you calculated is correct.

Hope that helps.
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If we let \(x\) be the number of marbles in Bag \(Q\),

To start, the total of number of marbles in the three bags is \((37+ 32+ x) = 69 + x\) marbles.

Next, we need to find the number of blue marbles in the 3 bags. To do so, we multiply each percentage by the number of marbles in each bag (converting each percentage to decimal as needed):

The number of blue marbles present in each bag would be:

Bag P: \(0.108 \times 37 =3.996\) or \(4\) since we want the nearest whole number marbles
Bag R:\(0.50 \times 32 =\)\(16\) marbles
Bag Q:\(0.667(x) = 0.667x\) marbles or \(\frac{2x}{3}\) marbles (Note that 0.667 is equivalent to 2/3 in fractions)

The prompt states that \(\frac{1}{3}\) of the total marbles equals the total number of blue marbles. We can now form the equation below.

1/3 x Total Number of Marbles = Total Number of Blue Marbles

\(1/3 (69 + x) = (4 + 16 + 2x/3)\)
Distribute \(\frac{1}{3}\) into the parenthesis,
\(23 + \frac{x}{3} = 20 + \frac{2x}{3}\)
\(23 – 20= \frac{2x}{3} – \frac{x}{3}\)
\(3 = \frac{x}{3}\)
\(x = 3(3) = 9\)

Hence, there are 9 marbles in Bag Q. The final answer is .

There are 2 ways to do this question quickly.

1. is detailed by several other ppl.
10.8% of 37 = 4
66.7% of x = 2x/3
50% of 31 = 16

4 + 16 + 2x/3 = 1/3(32 + 37 + x)
20 + 2x/3 = 1/3(69 +x) => x = 9

2. Alternatively, use the answer choices and information given. Since Bag Q has 66.7% (2/3) blue marbles, and you can't really have fractions of marbles, the answer must be a multiple of 3: either (B) 9 or (C) 12.

Plug in the answers to the question.
9: 10.8%(37) + 66.7%(9) + 50%(32) = 26 AND 37 + 9 + 32 = 78 => WORKS

12: 10.8%(37) + 66.7%(12) + 50%(32) = 28 AND 37 + 12 + 32 = 81 => DOESN'T WORK

Total Marbles: 32+37+x, 69+x
Q Blue: 2/3*x
P,R Blue: 4+16, 20
Total Blue: 1/3*(69+x)=20+2/3*x
…69+x=60+2x, x=9

Ans (B)

Shortcut: plug in the answers into 37+x+32 and has to be a multiple of 3 since the blue balls are one third of the total.

Only option that fits is B.

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