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A mixture of sand and cement contains, 3 parts of sand and 5

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A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 01 Jun 2014, 11:14
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A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8

Source: Indian CAT
[Reveal] Spoiler: OA

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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 01 Jun 2014, 14:05
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ConnectTheDots wrote:
A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8

Source: Indian CAT


We have total of 8 parts: 3 parts of sand and 5 parts of cement.

In order there to be half sand and half cement (4 parts of sand and 4 parts of cement), we should remove 1 part of cement. With 1 part of cement comes 3/5 parts of sand, so we should remove 1 + 3/5 = 8/5 part of the mixture, which is (8/5)/8 = 1/5 of the mixture.

Answer: C.
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 02 Jun 2014, 12:32
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ConnectTheDots wrote:
A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8


This is just a weighted average question, so we can apply the formula for that: with \(C\) as the concentrations and \(V\) as the volumes...


\(C_1*\frac{V_1}{V_1 + V_2} + C_2*\frac{V_2}{V_1 + V_2} = C_{final}\)

If sand:cement=3:5, then the concentration of sand in the initial mixture is \(\frac{3}{8}\). Since we are asked for the proportion of the mixture that should be replaced, we can assume a total volume of \(1\) and let \(x\) be the "amount" of the mixture to be replaced by pure sand (i.e., concentration of 1). We can then write the following equation:

\((\frac{3}{8})*(1-x) + (1)*(x)=\frac{1}{2}\)

Thus, \(x=\frac{1}{5}\)
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 02 Jun 2014, 20:15
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ConnectTheDots wrote:
A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8

Source: Indian CAT


You can use the scale method here too.

A mix has 3/8 of sand. Another is all sand so fraction of sand is 1. You have to mix them to get 1/2 sand.

w1/w2 = (1 - 1/2)/(1/2 - 3/8) = 4/1

So the mix should be 4 parts and only sand should be 1 part. Hence 1/5 of the mix must have been replaced by sand.

Answer (C)
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 04 Jun 2014, 01:08
Initial

Sand ............... Cement ............... Total

\(\frac{3}{8}\) ..................... \(\frac{5}{8}\) ...................... 1

Requirement

\(\frac{4}{8} ..................... \frac{4}{8} ..................... 1\)

Just focus on cement:

To have \(\frac{4}{8}\) cement, we require to remove

\(\frac{5}{8} - \frac{4}{8} = \frac{1}{8}\)cement

In Mixture, the portion of cement is \(\frac{5}{8};\)so to remove \(\frac{1}{8}\) cement, mixture required to be removed

\(= \frac{\frac{1}{8} * 1}{\frac{5}{8}}\)

\(= \frac{1}{5}\)

Answer = C
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 25 Jun 2014, 15:42
Bunuel wrote:
ConnectTheDots wrote:
A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8

Source: Indian CAT


We have total of 8 parts: 3 parts of sand and 5 parts of cement.

In order there to be half sand and half cement (4 parts of sand and 4 parts of cement), we should remove 1 part of cement. With 1 part of cement comes 3/5 parts of sand, so we should remove 1 + 3/5 = 8/5 part of the mixture, which is (8/5)/8 = 1/5 of the mixture.

Answer: C.


Bunuel: I do not quite understand the highlighted section above. 1 part of the mixture will contain 3/8 sand and 5/8 cement. How did you come up with "1 part of cement contains 3/5 parts of sand"?
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 25 Jun 2014, 20:01
One more method:

Sand ............. Cement .............. Total

\(\frac{3}{8}\) .................. \(\frac{5}{8}\) ........................ 1

Let "x" quantity of mixture be removed;

\(\frac{3}{8} - \frac{3x}{8}\) ........... \(\frac{5}{8} - \frac{5x}{8}\) ............... 1 - x

Adding "x" quantity of sand

\(\frac{3}{8} - \frac{3x}{8} + x\) .......... \(\frac{5}{8} - \frac{5x}{8}\) .............. 1-x+x

Resultant should be half sand & half cement

Two options to set up the equation

Option I

\(\frac{3}{8} - \frac{3x}{8} + x = \frac{1}{2}\)

\(x = \frac{1}{5}\)

Option II

\(\frac{5}{8} - \frac{5x}{8} = \frac{1}{2}\)

\(x = \frac{1}{5}\)

Answer = C
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Re: A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 25 Jun 2014, 20:09
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Bunuel wrote:
ConnectTheDots wrote:
A mixture of sand and cement contains, 3 parts of sand and 5 parts of cement. How much of the mixture must be substituted with sand to make the mixture half sand and half cement?

A. 1/3
B. 1/4
C. 1/5
D. 1/7
E. 1/8

Source: Indian CAT


We have total of 8 parts: 3 parts of sand and 5 parts of cement.

In order there to be half sand and half cement (4 parts of sand and 4 parts of cement), we should remove 1 part of cement. With 1 part of cement comes 3/5 parts of sand, so we should remove 1 + 3/5 = 8/5 part of the mixture, which is (8/5)/8 = 1/5 of the mixture.

Answer: C.


Bunuel: I do not quite understand the highlighted section above. 1 part of the mixture will contain 3/8 sand and 5/8 cement. How did you come up with "1 part of cement contains 3/5 parts of sand"?



Not Bunuel, but seems I can explain it

Mixture .................. Sand ............... Cement

1 ............................ \(\frac{3}{8}\) ................... \(\frac{5}{8}\)

Multiply by \(\frac{8}{5}\) to all above

\(\frac{8}{5}\) ........................ \(\frac{3}{8} * \frac{8}{5}\) ............. \(\frac{5}{8} * \frac{8}{5}\)


\(\frac{8}{5}\) ........................ \(\frac{3}{5}\) ....................... 1

From the above, we can say that 1 part of cement comes with \(\frac{3}{5}\) parts of sand in \(\frac{8}{5}\) quantity of mixture.
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A mixture of sand and cement contains, 3 parts of sand and 5 [#permalink] New post 25 Jun 2014, 20:19
S1 - 3 parts element 1 , 5 parts element 2
S2 - x parts of mixture removed i.e, -(3/8 * x) parts element 1, -(5/8 * x) parts element 2
S3 - x parts of element 1 added

We thus have 3 - (3*x)/8 +x / 5 - (5*x)/8 = 1/1
x=1.6 . i.e, 1.6/8 = 1/5 of the mixture removed.
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A mixture of sand and cement contains, 3 parts of sand and 5   [#permalink] 25 Jun 2014, 20:19
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