Problem 82 - Elementary Algebra

Question

For exercises \(67-82\), use the five steps and a proportion. Cyclosporine is an anti-rejection drug given to organ transplant patients. A bottle contains \(50 \mathrm{~mL}\) of liquid. Each milliliter of liquid contains \(100 \mathrm{mg}\) of cyclosporine. A kidney transplant patient needs to take \(850 \mathrm{mg}\) of cyclosporine each day. Find the amount of solution that the patient should take each day.

Step-by-Step Solution

Answer

The patient should take 8.5 mL of the solution each day.

Step 1 - Determine the known values

Identify the values given in the problem. The bottle contains 50 mL of liquid. Each mL of liquid contains 100 mg of cyclosporine. The patient needs 850 mg of cyclosporine each day.

Step 2 - Set up the proportion

We need to determine how many mL the patient should take to get 850 mg. Set up the proportion using the known values: \[\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\]

Step 3 - Solve for x

Cross-multiply to solve for x: \[100x = 850\]

Step 4 - Divide both sides

Divide both sides by 100 to isolate x: \[x = \frac{850}{100} = 8.5 \text{ mL}\]

Step 5 - Verify the solution

Check the solution by calculating the amount of cyclosporine in 8.5 mL of liquid: \[8.5 \text{ mL} \times 100 \text{ mg/mL} = 850 \text{ mg}\]The solution is correct.

Key Concepts

Proportions

Understanding proportions is crucial when solving many algebra problems, including this one about medication dosage. A proportion is a statement where two ratios or fractions are equal. In this exercise, we use the relationship between milliliters (mL) and milligrams (mg) of cyclosporine to set up a proportion. This helps us find the amount of solution the patient needs. One key point is to ensure that both sides of the proportion are comparing the same type of units. Ratios, like \(\frac{100 \text{ mg}}{1 \text{ mL}}\), show the amount of cyclosporine per mL.

Cross-Multiplication

Cross-multiplication is a helpful technique to solve proportions. It involves multiplying across the equals sign diagonally. When you have a proportion \(\frac{a}{b} = \frac{c}{d}\), you can cross-multiply to find \(\text{a} \times \text{d} = \text{b} \times \text{c}\). In our exercise, we begin with the proportion \(\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\). By cross-multiplying, we get \(\text{100} \times \text{x} = 850 \text{ mg} \times 1 \text{ mL}\), leading us to the next steps of solving for \(\text{x}\).

Algebraic Equations

Once we've set up our proportion and cross-multiplied, we have an algebraic equation to solve. In this problem, solving \(\text{100x} = 850\) requires isolating \(\text{x}\). We do this by dividing both sides of the equation by 100: \[x = \frac{850}{100} = 8.5 \text{ mL}\]. Solving algebraic equations often involves these steps: simplifying expressions, isolating the variable, and performing arithmetic operations. Remember, the goal is to isolate the variable to find its value.

Unit Conversion

Unit conversion is essential in the context of proportions and real-world problems like medication dosages. Always keep track of units when solving problems. In this example, we convert from mg to mL using known ratios. Each mL contains 100 mg of cyclosporine, and the patient needs 850 mg. Our task is to convert 850 mg into the equivalent mL. By using the proportion \(\frac{100 \text{ mg}}{1 \text{ mL}} = \frac{850 \text{ mg}}{x \text{ mL}}\), we performed a unit conversion to find the correct dosage per day.