Two solutions A and B having sugar concentrations of 65% and 85% respectively are required to make a mixture. In what ratio these two solutions be mixed to get a mixture of sugar concentration 73%
3 ∶ 2
Understanding Mixture Problems and Concentration Ratios
This question involves mixing two solutions with different sugar concentrations to achieve a desired intermediate concentration. This type of problem can be efficiently solved using the alligation rule, which is a method used to find the ratio in which two or more ingredients at given prices (or concentrations) must be mixed to produce a mixture at a desired price (or concentration).
Let the two solutions be A and B.
- Solution A has a sugar concentration of 65%.
- Solution B has a sugar concentration of 85%.
- The desired mixture concentration is 73%.
Using the alligation rule, we can set up a cross diagram to find the ratio of the quantities of solution A and solution B to be mixed. The differences in concentrations are calculated as follows:
Difference between the concentration of Solution B and the desired concentration:
85% - 73% = 12%
This difference represents the proportion of Solution A needed.
Difference between the desired concentration and the concentration of Solution A:
73% - 65% = 8%
This difference represents the proportion of Solution B needed.
The ratio in which Solution A and Solution B should be mixed is the inverse ratio of these differences.
Ratio of Quantity of A : Quantity of B = (Difference B - Desired) : (Desired - Difference A)
Ratio of Quantity of A : Quantity of B = 12% : 8%
To simplify the ratio 12 : 8, we find the greatest common divisor (GCD), which is 4. Dividing both parts of the ratio by 4:
12 ÷ 4 = 3
8 ÷ 4 = 2
So, the simplified ratio is 3 : 2.
Therefore, Solution A and Solution B should be mixed in the ratio 3 ∶ 2 to obtain a mixture with a sugar concentration of 73%.
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