Hi, Can someone help me solve this question without equation? Thanks
A can contains a mixture of two liquids A and B in ratio of 7:5. 9 liters of mixture is drawn off and the can is filled with B. The new ratio is 7:9. How many liters of A was there in the can initially?
A. 25
B. 21
C. 20
D. 10
E. 14
B
mixtures
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Let B = the pure liquid B, S = the original solution, and M = the mixture.ahahkhyati.j wrote:Hi, Can someone help me solve this question without equation? Thanks
A can contains a mixture of two liquids A and B in ratio of 7:5. 9 liters of mixture is drawn off and the can is filled with B. The new ratio is 7:9. How many liters of A was there in the can initially?
A. 25
B. 21
C. 20
D. 10
E. 14
The following approach is called ALLIGATION -- a very good way to handle MIXTURE PROBLEMS.
Alligation can be performed only with fractions or percentages.
Step 1: Convert any ratios to FRACTIONS.
B:
Here, B/total = 1/1.
S:
Since A:B = 7:5, and 7+5=12, B/total = 5/12.
M:
Since A:B = 7:9, and 7+9=16, B/total = 9/16.
Step 2: Put the fractions over a COMMON DENOMINATOR.
W = 1/1 = 48/48.
S = 5/12 = 20/48.
M = 9/16 = 27/48.
Step 3: Plot the 3 numerators on a number line, with the numerators for B and S on the ends and the numerator for M in the middle.
B 48-----------------27------------------20 S
Step 4: Calculate the distances between the numerators.
B 48-------21--------27---------7--------20 S
Step 5: Determine the ratio in the mixture.
The ratio of B to S in the mixture is equal to the RECIPROCAL of the distances in red.
B:S = 7:21 = 1:3 = 9:27.
Since B:S = 9:27, 9 liters of pure B was added to 27 liters of the original solution, implying that the can = 9+27 = 36 liters.
Since in the original solution B/total = 5/12, A/total = 7/12.
Thus:
Before the pure B was added, the amount of A in the can = (7/12)(36) = 21 liters.
The correct answer is B.
For two other problems that I solved with alligation, check here:
https://www.beatthegmat.com/ratios-fract ... tml#484583
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An alternate approach is to PLUG IN THE ANSWERS, which represent the original volume of A.ahahkhyati.j wrote:A can contains a mixture of two liquids A and B in ratio of 7:5. 9 liters of mixture is drawn off and the can is filled with B. The new ratio is 7:9. How many liters of A was there in the can initially?
A. 25
B. 21
C. 20
D. 10
E. 14
Since original A:B = 7:5, the original volume of A is likely to be a MULTIPLE OF 7.
Implication:
The correct answer choice is probably B or E.
Since new A:B = 7:9, and 7+9=16, (new A)/(total volume) = 7/16.
Implication:
After 9 liters are removed from the can and replaced with pure B, the remaining volume of A must constitute 7/16 of the total volume.
Answer choice B: 21
Since original A:B = 7:5 = 21:15, original A = 21 liters and original B = 15 liters, for a total of 36 liters.
When 9 of the 36 liters are removed, since 9/36 = 1/4, A's volume and B's volume are each reduced by 1/4.
Thus:
Remaining A = (3/4)(21) = 63/4.
(remaining A)/(total volume) = (63/4)/36 = 7/16.
Success!
The correct answer is B.
Answer choice E: 14
Since original A:B = 7:5 = 14:10, original A = 14 liters and original B = 10 liters, for a total of 24 liters.
When 9 of the 24 liters are removed, since 9/24 = 3/8, A's volume and B's volume are each reduced by 3/8.
Thus:
Remaining A = (5/8)(14) = 35/4.
(remaining A)/(total volume) = (35/4)/24 ≠7/16.
Eliminate E.
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If anyone reading this doesn't mind a *little* bit of algebra, I think there's a shorter way.
Suppose our initial ratio is 7x / 5x. That means that B is (5/12) of the mixture, that A is (7/12) of the mixture, and that the total amount in the mixture is 12x.
When we replace 9 liters of the mix with 9 liters of B, that's equivalent to replacing (7/12)*9 liters of A with (7/12)*9 liters of B. Our total remains 12x.
That gives us an equation of
New A = 7/(7 + 9) of New Total, or
7x - (7/12)*9 = (7/16)*12x, or
x = 3
Since the initial amount of A was 7x, our answer is 7*3, or 21.
Suppose our initial ratio is 7x / 5x. That means that B is (5/12) of the mixture, that A is (7/12) of the mixture, and that the total amount in the mixture is 12x.
When we replace 9 liters of the mix with 9 liters of B, that's equivalent to replacing (7/12)*9 liters of A with (7/12)*9 liters of B. Our total remains 12x.
That gives us an equation of
New A = 7/(7 + 9) of New Total, or
7x - (7/12)*9 = (7/16)*12x, or
x = 3
Since the initial amount of A was 7x, our answer is 7*3, or 21.
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Actually, even my last explanation made it harder than it needs to be!
Suppose our initial amounts of A and B are called a and b. From our ratios, we know that
Old Amount of A = (7/12) * Total
New Amount of A = (7/16) * Total
Since the total is the same both times, we have
Old A / (7/12) = New A / (7/16), or
Old A = (4/3) * New A
Since New A = Old A - (7/12)*9, we have
x = (4/3) * (x - 63/12), or
x = (4/3)x - 7, or
7 = (1/3)x, and x = 21.
Too easy!
Suppose our initial amounts of A and B are called a and b. From our ratios, we know that
Old Amount of A = (7/12) * Total
New Amount of A = (7/16) * Total
Since the total is the same both times, we have
Old A / (7/12) = New A / (7/16), or
Old A = (4/3) * New A
Since New A = Old A - (7/12)*9, we have
x = (4/3) * (x - 63/12), or
x = (4/3)x - 7, or
7 = (1/3)x, and x = 21.
Too easy!