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# A paint store mixes 3/4 pint of red paint and 2/3 pint

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A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  10 Jan 2018, 10:44
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A paint store mixes $$\frac{3}{4}$$ pint of red paint and $$\frac{2}{3}$$ pint of blue paint to make a new paint color called Perfectly Purple. How many pints of red paint would be needed to make 34 pints of Perfectly Purple paint?

A. 1.42

B. 18.0

C. 36.13

D. 47.5

E. 48.12

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Re: A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  12 Jan 2018, 02:57
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1 pint of Perfectly purple paint contains
3/4 pint of red paint and 2/3 pint of blue paint
3/4 + 2/3 = 17/12 pint of R+B = 1 pint of Perfectly purple
34 pints of Perfectly purple contains 17/12*a(where a is a scale factor) of both Red and Blue paints
solve for a, a = 24. In total 24 pints of Red and blue paints is used for 34 pints of Perfectly purple
Red paint in pints = 3/4*24=18 (B)
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Re: A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  12 Jan 2018, 06:52
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Carcass wrote:

A paint store mixes $$\frac{3}{4}$$ pint of red paint and $$\frac{2}{3}$$ pint of blue paint to make a new paint color called Perfectly Purple. How many pints of red paint would be needed to make 34 pints of Perfectly Purple paint?

A. 1.42

B. 18.0

C. 36.13

D. 47.5

E. 48.12

Kudo for right answer and solution

Let R = # of pints of red paint needed
Let B = # of pints of blue paint needed

A paint store mixes 3/4 pint of red paint and 2/3 pint of blue paint to make a new paint color called Perfectly Purple
This tells us the RATIO of the volumes of paint needed.
That is: R/B = (3/4)/(2/3)
Cross-multiply to get: (2/3)(R) = (3/4)(B)
Eliminate the fractions by multiplying both sides by 12 to get: 8R = 9B
Rearrange to get: 8R - 9B = 0

How many pints of red paint would be needed to make 34 pints of Perfectly Purple paint?
We want the TOTAL volume to be 34 pints
So, we can write: R + B = 34

We now have the following system that we must solve for R:
8R - 9B = 0
R + B = 34

Take bottom equation and multiply both sides by 9 to get the following EQUIVALENT equation:
8R - 9B = 0
9R + 9B = 306

ADD the two equations to get: 17R = 306
Solve to get: R = 18

Cheers,
Brent
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Re: A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  15 Sep 2018, 01:44
May I know why when I use the equation 3/4R+2/3B=1 (i.e. 3/4 red plus 2/3 blue to make one pint of purple) and solve simultaneously with R+B=34, I get a different answer?
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Re: A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  15 Sep 2018, 12:16
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Expert's post
Runnyboy44 wrote:
May I know why when I use the equation 3/4R+2/3B=1 (i.e. 3/4 red plus 2/3 blue to make one pint of purple) and solve simultaneously with R+B=34, I get a different answer?

$$\frac{3}{4}R+\frac{2}{3}B=1$$. This equation is incorrect. 1 pint cannot contain $$\frac{3}{4}$$ Red and $$\frac{2}{3}$$ Blue.

The correct equation is:

$$\frac{3}{4}R + \frac{2}{3}B = \frac{17}{12}$$

or $$9R + 8B=17$$ multiplying 12 on both sides.
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Re: A paint store mixes 3/4 pint of red paint and 2/3 pint [#permalink]  06 Aug 2019, 09:23
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Sometimes, ratio problems can be solved using not just part to part ratios, but part to whole ratios ( see mathisfun ratios section https://www.mathsisfun.com/numbers/ratio.html)

In this case, we are given 3/4 pint red to 2/3 pint blue. We add them together to get a whole. Like if we had 3 cats and 2 dogs, the "whole" would be 5 dogs and cats.

So, if we add the 3/4 pint red to 2/4 pint blue we get 17/12 pints of the "whole" mixture.

So, our ratio of part to whole, if we use just red as a part is 3/4 red to 17/12 whole.

We can also think of this as a conversion problem.

We are given 34 pints of a whole mixture (desired result) we can convert that quantity to red pints using our ratio of red parts to whole parts.

We can convert those whole pints to red pints by multiplying by the ratio of red pints to whole pints [ 3/4 pints red/(17/12 whole) ]

the "whole" or purple units cancel leaving the "red" units
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daum_equation_1565111894558_pints.png [ 15.89 KiB | Viewed 122 times ]

Re: A paint store mixes 3/4 pint of red paint and 2/3 pint   [#permalink] 06 Aug 2019, 09:23
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# A paint store mixes 3/4 pint of red paint and 2/3 pint

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