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# A certain mixture of paint requires blue, yellow, and red paints in

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I - states How much We need of yellow.
II - states How much We have of yellow.

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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

given : 2x + 3x + x = 6x

st[1] = 6x = 20 or x = $$\frac{10}{3}$$ : we do not know if we have this much Yellow paint==> Hence not sufficient

st[1] = ok so we have in total 10 quarts of yellow paint : we do not know what is the final mixture amount, if it is 60 then Yellow is sufficient but if the needed mixture is 120 then we will need 20 quarts of yellow which w do not have ==> hence not sufficient

st[1] + st[2] tells we need $$\frac{10}{3}$$ of yellow == which is less than the total yellow so its sufficient
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

Target question: Is there enough of the yellow paint to make the desired amount of the mixture?

Statement 1: Exactly 20 quarts of the mixture are needed.
Since the ratio of blue : yellow : red = 2 : 3 : 1, we can conclude that we need:
20/3 quarts of blue paint
10 quarts of yellow paint
20/6 quarts of red paint
Since we don't know how much yellow paint is available, we cannot answer the target question with certainty.
So, statement 1 is NOT SUFFICIENT

Statement 2: Exactly 10 quarts of the yellow paint are available.
In this case, we don't know the VOLUME of mixed paint are needed. Consider these two cases:
Case a: we need 6 quarts of the mixed paint, which means we need 2 quarts of blue paint, 3 quarts of yellow paint and 1 quarts of red paint. In this case, there IS enough yellow paint
Case b: we need 600 quarts of the mixed paint, which means we need 200 quarts of blue paint, 300 quarts of yellow paint and 100 quarts of red paint. In this case, there is NOT enough yellow paint
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that we NEED 10 quarts of yellow paint
Statement 2 tells us that we HAVE 20 quarts of yellow paint
So, YES, there IS enough yellow paint
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

b:y:r = 2:3:1
=> b = 2x
=> y = 3x
=> r = x
Total = 6x => y is half of total.
y - available = required ?

1) 20 Quartz needed => 10 Quartz of yellow needed
Yellow available ?
Insufficient

2) 10 Quartz of yellow available
Yellow needed ?
Insufficient.

1+2)
10 Needed = 10 Required.
Sufficient.
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

We are given the ratio:

blue : yellow : red = 2x : 3x : x

We need to determine if there is enough yellow paint available to allow the ratio to hold true.

Statement One Alone:

Exactly 20 quarts of the mixture are needed.

We can create the following equation:

2x + 3x + x = 20

6x = 20

x = 20/6 = 10/3

Thus, we see that 3 x 10/3 = 10 quarts of yellow paint are needed. However, since we do not know how much yellow paint is available, statement one alone is not sufficient to answer the question.

Statement Two Alone:

Exactly 10 quarts of the yellow paint are available.

Since we do not know how many quarts of paint are needed, we cannot answer the question. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using both statements together, we see that we need 10 quarts of yellow paint and we have exactly 10 quarts of yellow paint available. Thus, we have enough yellow paint.

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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
VeritasPrepKarishma wrote:
Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

Blue : Yellow : Red = 2 : 3 : 1

We need to find the amount of Yellow paint required. On the ratio scale, it is 3 parts and the other two combined are 2 + 1 = 3 parts. So half of the mix must be yellow paint.

To find whether we have enough yellow, we need two things - how much yellow we have and how much mix we need.
Both statements together answer these questions and are hence sufficient.

VeritasPrepKarishma is it safe to generalize as follows:

This question shows that in case of 3 or more ratios such as Blue : Yellow : Red = 2 : 3 : 1, you can combine any two ratios to get B&R : Y = 3:3 ?
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
FANewJersey wrote:
VeritasPrepKarishma wrote:
Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

Blue : Yellow : Red = 2 : 3 : 1

We need to find the amount of Yellow paint required. On the ratio scale, it is 3 parts and the other two combined are 2 + 1 = 3 parts. So half of the mix must be yellow paint.

To find whether we have enough yellow, we need two things - how much yellow we have and how much mix we need.
Both statements together answer these questions and are hence sufficient.

VeritasPrepKarishma is it safe to generalize as follows:

This question shows that in case of 3 or more ratios such as Blue : Yellow : Red = 2 : 3 : 1, you can combine any two ratios to get B&R : Y = 3:3 ?

Yes, that is correct.

Ratios give relative value of each of the constituents. So for 2 parts of Blue, you have 3 parts of Yellow and 1 part of Red. So Blue + Red will be 3 parts and Yellow will be 3 parts.
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A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

Blue:Yellow: Red:$$2:3:1$$

Total Paints are: $$2x:3x:1x$$

(1) Given $$2x+3x+x=20$$

$$⇒ 6x=20$$

$$⇒ x=\frac{20}{6}=\frac{10}{3}$$

∴ The Yellow Paint $$=3*\frac{10}{3}=10$$

Now, We know the quantity needed for yellow paint, but we don't whether we have this quantity. Insufficient.

(2) We don't know the total need for paint. Insufficient.

Considering both:
We know the total need and the quantity of yellow paint. Sufficient.

The answer is $$C$$.

Originally posted by MHIKER on 16 Sep 2020, 12:41.
Last edited by MHIKER on 08 Jan 2021, 11:13, edited 2 times in total.
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

St1 - 20 quarts of the mix are needed.
Insufficient b/c we don't know how much of Y we have.

St2 - Exactly 10 quarts of yellow are available
Insufficient b/c we don't know how much Y we need.

St1/St2
If B:Y:R = 2:3:1, then that means each "part" is 20/6 = 3.33. So, in a 20 quart mix we have B:Y:R = 6.66:9.99:3.33.
10 quarts is enough.
Sufficient.
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Bunuel wrote:
A certain mixture of paint requires blue, yellow, and red paints in ratios of 2:3:1, respectively, and no other ingredients. If there are ample quantities of the blue and red paints available, is there enough of the yellow paint to make the desired amount of the mixture?

(1) Exactly 20 quarts of the mixture are needed.
(2) Exactly 10 quarts of the yellow paint are available.

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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Hi Karishma,
I was confused with the language - "If there are ample quantities of the blue and red paints available"
From st2, we are given that Y = 10 quarts,
As it is given that ample quantities of the blue and red paints available, Why can't we conclude that blue = 20/3 and red paints=10/3 from ratio 2:3:1?

Thanks!
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
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Sneha2021 wrote:

Hi Karishma,
I was confused with the language - "If there are ample quantities of the blue and red paints available"
From st2, we are given that Y = 10 quarts,
As it is given that ample quantities of the blue and red paints available, Why can't we conclude that blue = 20/3 and red paints=10/3 from ratio 2:3:1?

Thanks!

Sure, but from stmnt 2, how do you know how much total paint you need?
Are you sure you need (10 + 20/3 + 10/3 = 20) quarts of paint?

What if we actually need 40 quarts of paint? Then yellow is not sufficient. What we need is to know whether we have enough yellow for our needs.
Blue and red are ample means there is unlimited supply of these two. But yellow is the one in short supply.
If we need 40 quarts of paint and we have only 10 quarts of yellow, then we do not have enough yellow.
If we need 20 quarts or less of paint then we have enough yellow.
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]
actually the question states the Blue, Yellow, Red in ratio terms?

so it might costs 200 Blue, 300 Yellow, and 100 Red to make a mixture?

Can anyone help?
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Re: A certain mixture of paint requires blue, yellow, and red paints in [#permalink]