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In mathematics, a ratio is a way to compare two quantities by using division. Ratios tell us how many times one quantity is contained within another. For example, a ratio of 5 to 3 can be expressed in several ways:

• 5:3
• 5/3
• For every 5 units of the first item, there are 3 units of the second item.

Ratios are used in many real-life situations like recipes, maps, and, as seen in this exercise, with nutritional balance. When working with ratios, it's important to keep the same units for both quantities to ensure accuracy. In the problem, the ratio of calcium to phosphorus is 5:3. This indicates that for every 5 parts of calcium, there should be 3 parts of phosphorus to maintain the healthy balance for bones. Understanding ratios is key to solving problems where two quantities are interdependent.

When we equate two ratios, we have a proportion. Proportions are statements that indicate two ratios are equal. They are incredibly useful for solving problems where one quantity is unknown, but the relationship between quantities is known. For instance, to find the amount of phosphorus in Mario's body, we use his calcium weight and set up the proportion \[\frac{2.5}{x} = \frac{5}{3}\]Here, 2.5 is the known amount of calcium, and \(x\) is the unknown amount of phosphorus. Solving this proportion helps us figure out that Mario needs 1.5 pounds of phosphorus. By cross-multiplying and solving the equation, we can find unknown quantities effectively. Proportions are powerful tools in word problems because they provide a straightforward way to resolve questions involving comparisons between quantities.

Word problems are an essential aspect of learning mathematics, as they allow us to apply mathematical concepts to real-world situations. When approaching word problems, it is important to read the problem carefully and identify the information given and what needs to be found.

• First, understand the problem and determine the quantities involved.
• Next, find the relationship between the quantities, often expressed as a ratio or proportion.
• Finally, set up an equation that represents the problem and solve for the unknown.

In the given exercise, we have two word problems. The first involves finding the amount of phosphorus needed in proportion to the given amount of calcium in Mario's body. The second problem requires understanding percentages to find how much of Kyla's weight is calcium, then using that information to calculate the phosphorus amount using a similar ratio. Word problems help in seeing the practical application of algebra, making abstract numbers more tangible and understandable.