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Math Expert V
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If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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67 00:00

Difficulty:   65% (hard)

Question Stats: 67% (02:43) correct 33% (03:02) wrong based on 1544 sessions

HideShow timer Statistics 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

Attachment: 2016-06-17_1618.png [ 22.59 KiB | Viewed 30031 times ]

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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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Bunuel wrote: 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

Attachment:
2016-06-17_1618.png

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.

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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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10
Bag P : 10.8% of 37 = 4
Bag Q : 66.7% of x = 2x/3
Bag R : 50% of 32 = 16.

Given, 4 + 2x/3 + 16 = 1/3 ( 37+x+32) => x = 9. Hence B.
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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FacelessMan wrote:
Bag P : 10.8% of 37 = 4
Bag Q : 66.7% of x = 2x/3
Bag R : 50% of 32 = 16.

Given, 4 + 2x/3 + 16 = 1/3 ( 37+x+32) => x = 9. Hence B.

I have approximated 10.8% as 1/9, which is 11.1%
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If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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This might look like it contains hard calculations, but the thing to notice here is that the number of balls are always going to be integers.

So, 10.8% of 37 will be 4, we know 50% of x is 32 and we can keep 66.7% as 2/3rds. Once you do this, the question can be solved to arrive at ~9 quickly.
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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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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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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manlog wrote:
I have approximated 10.8% as 1/9, which is 11.1%

10.8% of 37 -> 10% of 37 = 3.7.

3.7 is a little low and the number of balls must be an integer; can round up to 4.
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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Bunuel wrote: 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

Attachment:
2016-06-17_1618.png

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

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If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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many thanks generis and niks18

yeah working with fractions is easier , also good to know that $$\frac{2}{3}$$= 66% $$4+16+\frac{2}{3}x= \frac{1}{3} (37+32+x)$$

$$20+\frac{2}{3}x = \frac{1}{3}x+23$$

$$\frac{2}{3}x-\frac{1}{3}x = 3$$

$$\frac{1}{3}x =3$$

$$x =9$$

have a great weekend Originally posted by dave13 on 28 Apr 2018, 06:41.
Last edited by dave13 on 29 Apr 2018, 03:17, edited 1 time in total.
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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dave13 wrote:
Bunuel wrote: 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

Attachment:
2016-06-17_1618.png

generis, niks18, pushpitkc, hello here is my unique solution two solutions one is incorrect and next one seems more accurate unfortunately i cant solve 700 level q under 2 min, takes me more time befire i figure out how the problem can be approached $$4+16+0.67x= \frac{1}{3} (37+32+x)$$

$$20+0.67x =\frac{1}{3}x+23$$

$$20-23 = \frac{1}{3}x-\frac{67}{100}$$

$$-3 = \frac{33}{300}x$$

$$-3 = \frac{11}{100}x$$

the above solution is incorrect, i kept 1/3 as fraction, so i wonder what did i do wrong

in the next solution I converted 1/3 to decimal = 0.33%

$$4+16+0.67x=0.33 (37+32+x)$$

$$20+0.67x = 0.33x+23$$

$$0.67x - 0.33x = 23-20$$

$$3 = 0.34x$$

$$x = 8.8$$

round to the tenth $$8.8$$ and get $$9$$ have a great weekend 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
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If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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niks18 wrote:
dave13 wrote:
Bunuel wrote: 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

Attachment:
2016-06-17_1618.png

generis, niks18, pushpitkc, hello here is my unique solution two solutions one is incorrect and next one seems more accurate unfortunately i cant solve 700 level q under 2 min, takes me more time befire i figure out how the problem can be approached $$4+16+0.67x= \frac{1}{3} (37+32+x)$$

$$20+0.67x =\frac{1}{3}x+23$$

$$20-23 = \frac{1}{3}x-\frac{67}{100}$$

$$-3 = \frac{33}{300}x$$

$$-3 = \frac{11}{100}x$$

the above solution is incorrect, i kept 1/3 as fraction, so i wonder what did i do wrong

in the next solution I converted 1/3 to decimal = 0.33%

$$4+16+0.67x=0.33 (37+32+x)$$

$$20+0.67x = 0.33x+23$$

$$0.67x - 0.33x = 23-20$$

$$3 = 0.34x$$

$$x = 8.8$$

round to the tenth $$8.8$$ and get $$9$$ have a great weekend 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

niks18, hello there to answer your question - How come 37+32 become 23?

$$37+32 =69$$ so$$\frac{1}{3}*69 =23$$ so I get $$23+0,33x$$ $$20+0.67x = 0.33x+23$$

isnt it correct ? Senior SC Moderator V
Joined: 22 May 2016
Posts: 3560
If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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dave13 wrote:
Bunuel wrote: 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

Attachment:
2016-06-17_1618.png

generis, niks18, pushpitkc, hello here is my unique solution two solutions one is incorrect and next one seems more accurate unfortunately i cant solve 700 level q under 2 min, takes me more time befire i figure out how the problem can be approached $$4+16+0.67x= \frac{1}{3} (37+32+x)$$

$$20+0.67x = \frac{1}{3}x+23$$

$$20-23 = \frac{1}{3}x-\frac{67}{100}$$

$$-3 = \frac{33}{300}x$$

$$-3 = \frac{11}{100}x$$

the above solution is incorrect, i kept 1/3 as fraction, so i wonder what did i do wrong

dave13
The equation is correct. The arithmetic is off.

(EDIT As niks18 points out), check your math here:
Quote:
$$20-23 = \frac{1}{3}x-\frac{67}{100}$$

$$-3 = \frac{33}{300}x$$

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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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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dave13 wrote:
niks18 wrote:
dave13 wrote:
generis, niks18, pushpitkc, hello here is my unique solution two solutions one is incorrect and next one seems more accurate unfortunately i cant solve 700 level q under 2 min, takes me more time befire i figure out how the problem can be approached $$4+16+0.67x= \frac{1}{3} (37+32+x)$$

$$20+0.67x =\frac{1}{3}x+23$$

$$20-23 = \frac{1}{3}x-\frac{67}{100}$$

$$-3 = \frac{33}{300}x$$

$$-3 = \frac{11}{100}x$$

the above solution is incorrect, i kept 1/3 as fraction, so i wonder what did i do wrong

in the next solution I converted 1/3 to decimal = 0.33%

$$4+16+0.67x=0.33 (37+32+x)$$

$$20+0.67x = 0.33x+23$$

$$0.67x - 0.33x = 23-20$$

$$3 = 0.34x$$

$$x = 8.8$$

round to the tenth $$8.8$$ and get $$9$$ have a great weekend 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

niks18, hello there to answer your question - How come 37+32 become 23?

$$37+32 =69$$ so$$\frac{1}{3}*69 =23$$ so I get $$23+0,33x$$ $$20+0.67x = 0.33x+23$$

isnt it correct ? hi dave13
yes its correct. my mistake I got confused here. So in your method 1 the mistake is in highlighted part above. kindly check your calculation of 1/3x-67x/100. Rest is ok
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If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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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 .
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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Just want to clarify, should one always round up to the nearest integer if the variable is an actual object?

My error in solving this problem was assuming it was 3 marbles in Bag P, instead of 4. Due to the fact that, I didnt see it possible for there to be .994 of a marble, just like how in other problems 1.5 people is equivalent to just 1 person, as you cant have half a person.

Thanks!
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Re: If 1/3 of the total number of marbles in the three bags listed in the  [#permalink]

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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 Re: If 1/3 of the total number of marbles in the three bags listed in the   [#permalink] 25 Sep 2019, 21:03
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