Verified Answer :

  • The expression "A is to B as C is to D" is a standard way to write the equality of two ratios \(\frac{A}{B} = \frac{C}{D}\).
  • "4 is to 12" translates to the ratio \(\frac{4}{12}\).
  • "1 is to 3" translates to the ratio \(\frac{1}{3}\).
  • The final proportion is \(\boxed{\frac{4}{12} = \frac{1}{3}}\).

Explanation

  1. Analysis of the Problem

The problem asks us to translate the English sentence "4 is to 12 as 1 is to 3" into a mathematical proportion. A proportion is a statement that two ratios are equal. The phrasing "A is to B as C is to D" identifies two ratios, \(\frac{A}{B}\) and \(\frac{C}{D}\), and equates them: \(\frac{A}{B} = \frac{C}{D}\).

  1. Translating the Sentence into a Proportion

The first part of the sentence, "4 is to 12", corresponds to the ratio \(\frac{4}{12}\). The second part of the sentence, "1 is to 3", corresponds to the ratio \(\frac{1}{3}\). The word "as" indicates that these two ratios are equal, so we write \(\frac{4}{12} = \frac{1}{3}\). This can also be represented using colon notation as \(4:12 = 1:3\). Both forms are correct ways to express the proportion.

  1. Conclusion

A proportion is the equality of two ratios. Since \(\frac{4}{12}\) simplifies to \(\frac{1}{3}\), the statement "4 is to 12 as 1 is to 3" is a true mathematical proportion. The standard mathematical way to represent this is \(\frac{4}{12} = \frac{1}{3}\) or \(4:12 = 1:3\), which is \(\boxed{\frac{4}{12} = \frac{1}{3}}\).

Examples

Proportions are used in many real-life situations, such as scaling recipes. If a recipe says that the ratio of sugar to flour is 1:3, and you need to use 4 cups of sugar, you must use 12 cups of flour to maintain the correct taste. This is represented by the proportion \(\frac{4}{12} = \frac{1}{3}\). Understanding proportions helps in many areas, from cooking to construction and map reading.

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