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Two uniform cleaning solutions contain only cleaning concentrate and water. In solution A, the ratio of cleaning concentrate to water is 1 to 3. In solution B, the ratio of cleaning concentrate to water is 2 to 3. If 24 liters of solution A are poured into solution B, what is the ratio of cleaning concentrate to water in the resulting solution B?
(1) After the 24 liters are added, solution B contains 18 liters more water than cleaning concentrate.
(2) Before the 24 liters are added, solution B contains 6 liters more water than cleaning concentrate.
(1) After the 24 liters are added, solution B contains 18 liters more water than cleaning concentrate.
(2) Before the 24 liters are added, solution B contains 6 liters more water than cleaning concentrate.
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Official Solution:
Two uniform cleaning solutions contain only cleaning concentrate and water. In solution A, the ratio of cleaning concentrate to water is \(1:3\). In solution B, the ratio of cleaning concentrate to water is \(2:3\). If 24 liters of solution A are poured into solution B, what is the ratio of cleaning concentrate to water in the resulting solution B?
(1) After the 24 liters are added, solution B contains 18 liters more water than cleaning concentrate.
Since in solution A, the ratio of cleaning concentrate to water is \(1:3\), the 24 liters from solution A contain 6 liters of cleaning concentrate and 18 liters of water.
Let the amounts of cleaning concentrate and water in solution B be \(2x\) and \(3x\), respectively.
After the addition:
Cleaning concentrate \(= 2x + 6\)
Water \(= 3x + 18\)
So:
\((3x + 18) - (2x + 6) = 18\)
\(x + 12 = 18\)
\(x = 6\)
Thus, after the addition, solution B contains 18 liters of cleaning concentrate and 36 liters of water. Therefore, the ratio of cleaning concentrate to water is:
\(18:36 = 1:2\)
Sufficient.
(2) Before the 24 liters are added, solution B contains 6 liters more water than cleaning concentrate.
Since solution B has cleaning concentrate and water in the ratio \(2:3\), solution B contains 12 liters of cleaning concentrate and 18 liters of water.
The 24 liters from solution A contain 6 liters of cleaning concentrate and 18 liters of water. So after the addition, solution B contains:
\(12 + 6 = 18\) liters of cleaning concentrate
and
\(18 + 18 = 36\) liters of water.
Therefore, the ratio of cleaning concentrate to water is:
\(18:36 = 1:2\)
Sufficient.
Answer: D
Two uniform cleaning solutions contain only cleaning concentrate and water. In solution A, the ratio of cleaning concentrate to water is \(1:3\). In solution B, the ratio of cleaning concentrate to water is \(2:3\). If 24 liters of solution A are poured into solution B, what is the ratio of cleaning concentrate to water in the resulting solution B?
(1) After the 24 liters are added, solution B contains 18 liters more water than cleaning concentrate.
Since in solution A, the ratio of cleaning concentrate to water is \(1:3\), the 24 liters from solution A contain 6 liters of cleaning concentrate and 18 liters of water.
Let the amounts of cleaning concentrate and water in solution B be \(2x\) and \(3x\), respectively.
After the addition:
Cleaning concentrate \(= 2x + 6\)
Water \(= 3x + 18\)
So:
\((3x + 18) - (2x + 6) = 18\)
\(x + 12 = 18\)
\(x = 6\)
Thus, after the addition, solution B contains 18 liters of cleaning concentrate and 36 liters of water. Therefore, the ratio of cleaning concentrate to water is:
\(18:36 = 1:2\)
Sufficient.
(2) Before the 24 liters are added, solution B contains 6 liters more water than cleaning concentrate.
Since solution B has cleaning concentrate and water in the ratio \(2:3\), solution B contains 12 liters of cleaning concentrate and 18 liters of water.
The 24 liters from solution A contain 6 liters of cleaning concentrate and 18 liters of water. So after the addition, solution B contains:
\(12 + 6 = 18\) liters of cleaning concentrate
and
\(18 + 18 = 36\) liters of water.
Therefore, the ratio of cleaning concentrate to water is:
\(18:36 = 1:2\)
Sufficient.
Answer: D
