Question

Set up the following problems as a proportion and solve. Include labels in the set up and on the answer. If one glass of milk contains 280 milligrams (mg) of calcium, how many mg of calcium is in \(21 / 2\) glasses of milk?

Short answer

2940 mg of calcium in \(\frac{21}{2}\) glasses.

Step-by-step solution

Understanding the Problem

We need to find out how many milligrams of calcium are in \(\frac{21}{2}\) glasses of milk, knowing that one glass contains 280 mg. We will use a proportion to solve this.

Setting Up the Proportion

We set up a proportion comparing the amount of calcium in one glass to the amount in \(\frac{21}{2}\) glasses. Let \(x\) be the amount of calcium in \(\frac{21}{2}\) glasses: \[ \frac{1 \text{ glass}}{280 \text{ mg}} = \frac{\frac{21}{2} \text{ glasses}}{x \text{ mg}} \]

Solving the Proportion

Cross-multiply to solve for \(x\): \[ 1 \cdot x = 280 \cdot \frac{21}{2} \] Calculate the right side: \[ x = 280 \times 10.5 = 2940 \]

Conclusion and Labeling the Answer

The result from solving the equation gives \(x = 2940\), meaning there are 2940 mg of calcium in \(\frac{21}{2}\) glasses of milk.

Key concepts

Understanding Calcium Content

Calcium is an essential mineral that is critical for maintaining healthy bones and teeth. It is commonly found in dairy products like milk and cheese. To understand the calcium content in a food item means knowing how much calcium is present per serving. In this exercise, we're looking specifically at milk, where one glass contains 280 mg of calcium. This specific measure helps us understand and control our mineral intake.

If you have more than one serving, like multiple glasses of milk, you can calculate the total calcium intake by multiplying the amount of calcium per glass by the number of glasses. For example, the problem presents \( \frac{21}{2} \) glasses, which is equivalent to 10.5 glasses. Calculating this total requires accurate proportioning, and it is important for dietary planning and nutritional studies.

The Role of Mathematical Problem-Solving

Mathematical problem-solving is a systematic process that helps in breaking down complex problems into simpler, more manageable tasks. In this exercise, solving the total calcium intake involves understanding and applying mathematical operations such as multiplication and setting up proportions.

Here are helpful strategies used in this problem:

• Understanding the problem: Identify the known and unknown quantities. You know the calcium content of one glass and need to find the calcium content of \( \frac{21}{2} \) glasses.
• Setting up a proportion: Represent the relationship between known and unknown quantities mathematically.
• Cross-multiply to find the unknown: This helps to isolate and solve for x, the unknown.

By following these steps, solving the problem becomes a matter of logical progression, honing your critical thinking and analytical skills.

Solving with Cross Multiplication

Cross multiplication is a mathematical technique used to solve equations that are structured as proportions. A proportion is an equation where two ratios are equal, which in this case helps us find out how much calcium is in several glasses of milk. The equation is originally set up as:
\[ \frac{1 \text{ glass}}{280 \text{ mg}} = \frac{\frac{21}{2} \text{ glasses}}{x \text{ mg}} \]
To solve it, you "cross-multiply," meaning you multiply the numerator of each ratio by the denominator of the opposite ratio.

Process:

• Multiply 1 glass by x mg.
• Multiply 280 mg by 10.5 glasses (\(\frac{21}{2}\)).

This gives you:
\[ 1 \cdot x = 280 \times 10.5 \]
Calculating the right side gives us \(2940\), and therefore:
\[ x = 2940 \]
Cross multiplication simplifies the process of solving proportions, making it straightforward to find the unknown variable when dealing with ratios and relative quantities.