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Senior Manager  Joined: 16 May 2007
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There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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Question Stats: 64% (02:40) correct 36% (02:55) wrong based on 667 sessions

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There are 2 bars of gold-silver alloy; one piece has 2 parts of gold to 3 parts of silver and another has 3 parts of gold to 7 parts of silver. If both bars are melted into 8 kg bar with the final gold to silver ratio of 5:11. What was the weight of the first bar?

A. 1 kg
B. 3 kg
C. 5 kg
D. 6 kg
E. 7 kg
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You can easily use the scale method here. The scale method is explained here:
tough-ds-105651.html#p828579

Focus on any one of the two elements say Gold.
First bar has 2/5 gold. Second bar has 3/10 gold and mixture has 5/16 gold.
Make the fractions comparable for easy calculation i.e. give them the same denominator. LCM of 5, 10 and 16 is 80.
First bar has 32/80 gold. Second bar has 24/80 gold and mixture has 25/80 gold.
Attachment: Ques3.jpg [ 5.9 KiB | Viewed 28889 times ]

So first bar weight:second bar weight should be in the ratio 1:7. Out of 8 total kgs, first bar must have been 1 kg.
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Karishma
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8
X is weight of first bar and (8-x) is weight of second bar.
(2/5)x + (3/10)(8-x) = (5/16)*8

=>x = 1
##### General Discussion
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I cannot give a generic solution to the mixture problems. These may vary and hence must be assessed on individual merit (some require systematic work, others can be solved by considering the LCMs).

Here is how you can solve the given problem
Let X and Y be the weights of 1st and 2nd bar respectively.
Thus, X+Y=8 ..................(1)

To get a second relation, recognize that 2/5 of first and 3/10 of second bar are gold. Similar 3/5 and 7/10 respectively for silver.
Use this relation
(2/5)X+(3/10)Y
-------------------
(3/5)X+(7/10)Y

The above ratio is equal to 5:11

Two equations and 2 variables. Solve to get X=1.

best,
parsifal
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First bar is x kg, then second bar is (8-x)kg

First bar:
Amt of gold = 2x/5kg
Amt of silver = 3x/5 kg

Second bar:
Amt of gold = 3(8-x)/10 kg
Amt of silver = 7(8-x)/10 kg

Total amt of gold = 2x/5 + 3(8-x)/10 = (4x+24-3x)/10 = (x+24)/10 kg
Total amt of silver = 3x/5 + 7(8-x)/10 = (6x+56-7x)/10 = (56-x)/10 kg

Ratio of gold/silver = 5/11 = (x+24)/10 * 10/(56-x)

5/11 = (x+24)/(56-x)
5(56-x) = (x+24)(11)
x = 1kg
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First bar has 40% gold and second bar has 30% gold by weight.

The resulting 8 kg bar has 31.25% of gold by weight.

So by obviously we know that maximum percent of second bar would go to make the final bar because 31.25 % is close to 30%. Working with options we get 1 kg of bar one as the answer. (we can use interpolation)

Hope this helps if time is short.
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Bar 1 is x kg
Bar 2 is y kg

First equation (Gold):
2/5 * x + 3/10 * y = 5/16 * 8

Second equation (Silver):
3/5 * x + 7/10 * y = 11/16 * 8

Two equations, solve for x
X=1kg
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5/16 * 8 = Final weight of gold

2/5 * W1 = G1

3/10 * W2 = G2

W1 + W2 = 8

G1 + G2 = 5/16 * 8

2/5 * W1 + 3/10 * W2 = 5/16 * 8

2/5 * W1 + 3/10 * (8 - w1) = 5/16 * 8

W1(2/5 - 3/10) = 5/2 - 24/10

W1 * (0.4-0.3) = 2.5 -2.4 = 0.1

=> W1 = 1

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I solve mixture problems like simultaneous equations.
Look first at the amount of gold. In the final amount you have 8 kg, and since the ratio is 5:11, this means there must be 5/16 amount gold, or 5/2kg of gold in the end result.
So let x be the amount of the first gold bar and y the amount of y gold bar added to create an equation of gold amounts:
2/5X + 3/10Y = 5/2
You also know however that:
X + Y = 8
Since you want to know the value of X, rearrange to:
Y = 8 - X
Now substitute into the first equation to solve for X = 1
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Shortest way to solve this Problem is to analyze it for about 30 secs and you have an answer in the 31st.

Here we go-:

We have been given the Gold (G) to Silver (S) ratio for the first alloy which is 2:3.

We can formulate the question as 2G+3S= What?

And the question stem says that, when we have merged both first and the second piece of the alloys, the ratio for the merged piece is 5:11.

The second equation can be written as 5G+11S=8

Now, our job is to apply our mind a little

If 5G+3S=8
From, the first equation, it can be inferred that, in the second equation G is 2.5 times and S is approx. 3.7 times of what they are in the first equation 2*2.5= 5 and 3*3.7= 11 approx.

Since, 2.5 + 3.7= 6.2 approx.; which is an addition to the existing first alloy

And 8 - 6.2= 1.8, which could have been the weight for the first alloy. Nearest figure in the answer choices available to us is 1 and that's our answer. Veritas Prep GMAT Instructor V
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VeritasPrepKarishma wrote:
You can easily use the scale method here. The scale method is explained here:
tough-ds-105651.html#p828579

Focus on any one of the two elements say Gold.
First bar has 2/5 gold. Second bar has 3/10 gold and mixture has 5/16 gold.
Make the fractions comparable for easy calculation i.e. give them the same denominator. LCM of 5, 10 and 16 is 80.
First bar has 32/80 gold. Second bar has 24/80 gold and mixture has 25/80 gold.
Attachment:
Ques3.jpg

So first bar weight:second bar weight should be in the ratio 1:7. Out of 8 total kgs, first bar must have been 1 kg.

Responding to a pm:

Think of what the formula is:

w1/w2 = (C2 - Cavg)/(Cavg - C1)

First bar: C1 = 32/80, w1 = weight of first bar
Second bar: C2 = 24/80, w2 = weight of second bar
Cavg = 25/80

Simply plug these values in the formula.

w1/w2 = (C2 - Cavg)/(Cavg - C1) = (24/80 - 25/80)/(25/80 - 32/80) = 1/7
w1:w2 = 1:7

You don't need to worry about anything else.

When using the scale method, we flip the ratio because we calculate (Cavg - C1) first and (C2 - Cavg) later. This is opposite to the way it is in the formula so we flip the ratios.

Above, when I made the scale, I put the second bar first and the first bar later. The reason was that it is more intuitive that way on the number line since C2 = 24/80 is smaller than C1 = 32/80. Since I was finding Cavg - C2 first and C1 - Cavg later, I didn't need to flip the ratios.

My advice would be to simply identify one element as Element1, another as Element2 and figure out C1, w1 and C2, w2 for the 2 of them and simply plug in the formula. There will be no confusion in that case.
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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For anybody who is used to solving mixture probs using the cross method

Attachment: IMG.JPG [ 11.34 KiB | Viewed 25512 times ]

Hence, the ratio of the two gives 1/7 i.e 1 part of 1st bar and 7 parts of second.
Please notice that the question can be easily twisted to ask for the weight of first bar for minimum possible integer weight of final bar (Figure 8kg not provided) in which case this method would certainly help.
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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can some one solve it using cross method ?

A portion of the 85% solution of chemicals was replaced with an equal amount of 20% solution of chemicals. As a result, 40% solution of chemicals resulted. What part of the original solution was replaced?
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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smartmanav wrote:
can some one solve it using cross method ?

A portion of the 85% solution of chemicals was replaced with an equal amount of 20% solution of chemicals. As a result, 40% solution of chemicals resulted. What part of the original solution was replaced?

When you remove a portion of 85% solution and replace it with 20% solution, you are basically just mixing 85% and 20% solution to get 40% solution.

w1/w2 = (A2 - Aavg)/(Aavg - A1) = (85 - 40)/(40 - 20) = 45/20 = 9/4
(This is just the formula representing the scale. Check here if it's unclear: http://www.veritasprep.com/blog/2011/03 ... -averages/)

So 20% solution : 85% solution = 9:4

So out of a total 13 lts, 9 lts is 20% solution and 4 lts is 85% solution. This means 9/13 of the original solution was replaced by the 20% solution.

For more on this, check out the first example here: http://www.veritasprep.com/blog/2012/01 ... -mixtures/
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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stevie1111 wrote:
For anybody who is used to solving mixture probs using the cross method

Attachment:
IMG.JPG

Hence, the ratio of the two gives 1/7 i.e 1 part of 1st bar and 7 parts of second.
Please notice that the question can be easily twisted to ask for the weight of first bar for minimum possible integer weight of final bar (Figure 8kg not provided) in which case this method would certainly help.

Did in the same way, with one addition

Took LCM of 5, 10 & 16 which is 80

Multiplied 80 with all three fractions to get:

32 |||||||||||||||| 24

|||||| 25 ||||||||||||||

(25-24) |||||||| (32-25)

= 1:7 = 1/8 * 8 = 1kg = Answer = A
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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Aligation came to rescue.

X: 2:3
Y: 3:7

Mixing X and Y we get 8KG, so the ratio should be Y-8/8-X.

(2X+3Y)/(3X+7Y) = 5/11

X/Y = 2/7

2/7 = Y-8/8-X

8-X=7

X=1
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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Hi,

Lets do it by a different approach:

Lets take two Bars of Gold-Silver alloy and name them as A and B.

Now

let the total alloy content in A : 2x+3x
let the total alloy content in B : 3y+7y

now total weight of the mixture is 8 kg. That means its A+B

so the equation will be: 2x+3x+3y+7y=8

5x+10y=8 …….. a)

Also its given that the ration of the constituent mixture is 5:11( Gold: Silver)

Now that equates as:

2x+3y 5
———— = --------
3x+7y 11

i.e., 7x=2y……b)

now solving equations a and b gives x= 1/5 and y= 7/10

So the total weight of Alloy A= 2(1/5)+3(1/5)= 1

Therefore, the total weight of Alloy A is 1 Kilograms.
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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trahul4 wrote:
There are 2 bars of gold-silver alloy; one piece has 2 parts of gold to 3 parts of silver and another has 3 parts of gold to 7 parts of silver. If both bars are melted into 8 kg bar with the final gold to silver ratio of 5:11. What was the weight of the first bar?

A. 1 kg
B. 3 kg
C. 5 kg
D. 6 kg
E. 7 kg

8kg bar total -> 5/2 kg gold total and 11/2 kg silver total

solve for this system of equation

2/5 x1 + 3/10 x2 = 5/2
3/5 x1 + 7/10 x2 = 11/2

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There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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let X=weight of first bar
let Y= weight of second bar
2X/5+3Y/10=5/16(X+Y)
X/Y=1/7
X+Y=8
X/(X+Y)=1/8
X=1kg

Originally posted by gracie on 28 Sep 2015, 15:21.
Last edited by gracie on 30 Sep 2015, 14:28, edited 1 time in total.
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Re: There are 2 bars of gold-silver alloy; one piece has 2 parts  [#permalink]

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I think this question is very ambiguous!

First, When you say 2 parts of Gold and 3 Parts of silver, are you talking about their Volume or Weight?
From the answers given here, you've obviously taken it as the weight. But common sense says that when you use 'parts' you usually mean volume.
Second, it does not just say that the ratio in the first bar was 2/3 and that the ratio in the second bar was 3/7. Instead, it says 2 parts of gold in the first bar and 3 parts of gold in the second bar. What if the 'parts' referred to an actual measurement. This would mean that in all there were 5 parts of Gold and 10 parts of Silver.

Going by these two ambiguities, one can solve for the densities of Gold and Silver and the answer for the question would be something around 2.5 kgs. Re: There are 2 bars of gold-silver alloy; one piece has 2 parts   [#permalink] 09 Feb 2016, 03:09

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