Problem 13 - Prealgebra

Question

Solve each of the following word problems by translating the statement into a proportion. Be sure to show the proportion used in each case. [Examples \(1-4]\) Nursing An oral medication has a dosage strength of \(275 \mathrm{mg} / 5 \mathrm{mL} .\) If a patient takes a dosage of \(300 \mathrm{mg}\) how many milliliters does he take? Round to the nearest tenth

Step-by-Step Solution

Answer

The patient takes 5.5 mL of medication.

Step 1: Understand the Given Information

We are given that the medication has a concentration of 275 mg per 5 mL.

Step 2: Identify What Needs to Be Found

We need to find the number of milliliters (mL) for a dosage of 300 mg.

Step 3: Set Up the Proportion

We will set up a proportion comparing the known dosage to the unknown dosage: \(\frac{275 \text{ mg}}{5 \text{ mL}} = \frac{300 \text{ mg}}{x \text{ mL}}\), where \(x\) is the number of mL we need to find.

Step 4: Cross-Multiply to Solve for X

Cross-multiplication will help us solve for \(x\): \(275 \times x = 300 \times 5\).

Step 5: Simplify and Solve

Simplify the equation: \(275x = 1500\). Solve for \(x\) by dividing both sides by 275: \(x = \frac{1500}{275}\).

Step 6: Calculate and Round the Answer

Calculate \(x\): \(x \approx 5.4545\). Round this to the nearest tenth to get \(x = 5.5 \text{ mL}\).

Key Concepts

Dosage Calculations

Dosage calculations play a crucial role in healthcare, especially for administering the correct medication amount to patients. When we talk about dosage strength, it’s about how much of a drug is in a certain volume or weight. In the exercise, we had a concentration of 275 mg per 5 mL of medication. This means for every 5 mL, there’s 275 mg of the drug. Simple enough, right? But what if the patient needs a different dose, like 300 mg? That's where calculations come in!

Understanding how to properly calculate dosages ensures patient safety and treatment effectiveness. A wrong calculation can lead to underdosing or overdosing, both of which can be harmful. The key is setting up a proportion, a mathematical shortcut that lets us easily find out how much volume a certain amount of drug corresponds to. This way, health professionals can ensure they're delivering just the right amount of medicine every time.

• Assess the known dosage and volume.
• Know how much dosage the patient requires.
• Use proportions to find the volume of medication required.

By practicing dosage calculations in different scenarios, you become adept at ensuring patients receive the appropriate amount of medication.

Cross-Multiplication

Cross-multiplication is a key technique used in handling proportions. Whenever you have two ratios set equal to each other, like in the exercise, this method comes in handy. You just multiply across the equals sign diagonally. Think of it as forming an "X" with the numbers to bring them all together.

In our example, we have the ratio of 275 mg to 5 mL set equal to 300 mg to an unknown volume, represented by \( x \). With cross-multiplication:

\[ 275 imes x = 300 imes 5 \]

We solve for \( x \) by first multiplying the known numbers on one side — 300 mg and 5 mL. Then, you solve for \( x \) by dividing the result by 275. This step transforms what could be a complex algebraic task into a straightforward arithmetic solution.

Cross-multiplication effectively handles proportional problems, and once you master this method, you'll be able to solve a wide range of mathematical tasks quickly.

Solving Equations

Solving equations is all about finding that unknown number which completes the equation properly. With proportions like we have, the unknown is often straightforwardly represented as \( x \.\) In the context of the dosage word problem, our goal was to determine \( x \,\) representing how many milliliters were equivalent to 300 mg of medication.

After using cross-multiplication to set up our equation — \( 275x = 1500 \) — the next step was to isolate the variable \( x \). This means manipulating the equation to get \( x \) by itself on one side of the equals sign. We achieve this by dividing both sides of the equation by the number attached to \( x \), in this instance 275:

\[ x = \frac{1500}{275} \]

This solution gives us a decimal answer, which we then round appropriately — in this case, to the nearest tenth, yielding 5.5 mL. Mastering this approach enables you to tackle equations of varying complexity confidently, whether they're in dosage calculations or other real-world problems.