Discussion
A mixture of two food items has a salt to sugar ratio of 7 : 32. The ratio of salt to sugar is 2 : 11 and 5 : 21, in the two individual types of food items. In what proportion are the 2 food items mixed?
Answer: B
The alligation rule gives the ratio in which two ingredients must be mixed:
\[ \frac{\text{cheaper quantity}}{\text{dearer quantity}} = \frac{\text{dearer} - \text{mean}}{\text{mean} - \text{cheaper}} \]
Work with the fraction of salt in each item.
\[ \text{Item 1} = \frac{2}{13}, \qquad \text{Item 2} = \frac{5}{26}, \qquad \text{mixture} = \frac{7}{39} \]
Over the common denominator 78 these are \( \tfrac{12}{78} \), \( \tfrac{15}{78} \) and \( \tfrac{14}{78} \):
\[ \frac{\text{Item 1}}{\text{Item 2}} = \frac{15 - 14}{14 - 12} = \frac{1}{2} \]
Converting each ratio into a single fraction first is what makes the alligation rule applicable.
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