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08 Apr 2020, 10:42
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B
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Difficulty:
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44% (02:46) correct
56%
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The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7. What is the ratio of the strength of D to that of A ?
A. 3 : 10
B. 1 : 3
C. 2 : 5
D. 1 : 4
E. 1 : 5
Are You Up For the Challenge: 700 Level Questions
A. 3 : 10
B. 1 : 3
C. 2 : 5
D. 1 : 4
E. 1 : 5
Are You Up For the Challenge: 700 Level Questions
08 Apr 2020, 11:12
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Let three salt solutions A, B, C have the strengths p%, q%, r% respectively. Using weights from ratio 1:2:3 we will have the equation
\(\frac{p}{6}+\frac{2q}{6}+\frac{3r}{6}=20\) or \(p+2q+3r=120\)
Using weights from ration 3:2:1 we have the next equation
\(\frac{3p}{6}+\frac{2q}{6}+\frac{r}{6}=30\) or \(3p+2q+r=180\)
If we subtract first equation from the second one, we obtain: 2p-2r=60 or r=p-30.
If we put it in the second equation, we get 3p+2q+p-30=180 or 2q=210-4p or q=105-2p
The question is to find the ratio: \((\frac{2q}{9}+\frac{7r}{9}):p=(210-4p+7p-210):9p=3p:9p=1:3\)
The correct answer is B.
\(\frac{p}{6}+\frac{2q}{6}+\frac{3r}{6}=20\) or \(p+2q+3r=120\)
Using weights from ration 3:2:1 we have the next equation
\(\frac{3p}{6}+\frac{2q}{6}+\frac{r}{6}=30\) or \(3p+2q+r=180\)
If we subtract first equation from the second one, we obtain: 2p-2r=60 or r=p-30.
If we put it in the second equation, we get 3p+2q+p-30=180 or 2q=210-4p or q=105-2p
The question is to find the ratio: \((\frac{2q}{9}+\frac{7r}{9}):p=(210-4p+7p-210):9p=3p:9p=1:3\)
The correct answer is B.
General Discussion
10 Apr 2020, 09:30
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Bunuel
Given:
1. The strength of a salt solution is p% if 100 ml of the solution contains p grams of salt.
2. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%.
3. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%.
4. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7.
Asked: What is the ratio of the strength of D to that of A ?
Let the strength of solutions A, B & C be x%, y% & z% respectively
2. If three salt solutions A, B, C are mixed in the proportion 1 : 2 : 3, then the resulting solution has strength 20%.
(x + 2y + 3z) / 6 = 20
x + 2y + 3z = 120 (1)
3. If instead the proportion is 3 : 2 : 1, then the resulting solution has strength 30%.
3x + 2y + z = 180 (2)
(1) + (2)
4x + 4y + 4z = 300
x + y + z = 75 (3)
2*(3) - (1)
x - z = 30
x = z + 30
z = x - 30 (5)
y = 75 - x - z = 75 - x - (x-30) = 105 - 2x (6)
4. A fourth solution, D, is produced by mixing B and C in the ratio 2 : 7.
Strength of D = (2y +7z)/9
The ratio of the strength of D to that of A = (2y + 7z)/9x = {2(105 -2x) +7(x-30)} / 9x = (210 - 4x + 7x - 210 )/9x = 3x/9x = 1/3
IMO B
