Problem 12 - 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 A child is given a prescription for \(9 \mathrm{mg}\) of a drug. If she has to take 3 chewable tablets, what is the strength of each tablet?

Step-by-Step Solution

Answer

Each tablet is 3 mg.

Step 1: Understand the Problem

We need to find the strength of each tablet. The child takes 3 tablets to receive a total of 9 mg of the drug.

Step 2: Set Up the Proportion

The total dosage of 9 mg is received from 3 tablets. Therefore, the proportion is: \[ \frac{x}{1} = \frac{9 \text{ mg}}{3 \text{ tablets}} \] where \( x \) is the strength of each tablet.

Step 3: Solve the Proportion

Cross-multiply and solve for \( x \):\[ x \times 3 = 9 \] Divide both sides by 3:\[ x = \frac{9}{3} \] \[ x = 3 \text{ mg/tablet} \]

Step 4: Verify the Solution

Ensure that the calculated strength satisfies the conditions of the problem. If each tablet is 3 mg, then 3 tablets would be \( 3 \times 3 = 9 \) mg, matching the prescription.

Key Concepts

Understanding Proportions

Proportions are mathematical expressions that represent the equality of two ratios. When facing word problems, using proportions becomes incredibly useful to find unknown values. For example, if you need to know how much of one item compares to another, a proportion can help solve that problem.
One practical application is determining the strength of a medication, as seen in the exercise above. A proportion equation typically looks like this:

• \( \frac{a}{b} = \frac{c}{d} \)

In this context, \( a \) and \( b \) represent one ratio, and \( c \) and \( d \) represent another. When solving these, you can cross-multiply to find the unknown variable, ensuring that the two sides of the equation remain balanced. For example, if the equation is \( \frac{x}{1} = \frac{9}{3} \), solving for \( x \) will help you find the strength of one tablet in the provided word problem.

Drug Dosage Calculation Made Simple

Calculating drug dosage can seem daunting, but it's all about clarity and precision. Dosage calculations can often be solved using proportions, where the known dosage amount is compared against an unknown, which you're solving for.
In the provided scenario, the child's full dosage of 9 mg is spread over 3 tablets. To find how many milligrams each tablet should contain, you set up a proportion. This involves noting:

• Total drug required: 9 mg
• Number of tablets: 3 tablets

You set the proportion as \( \frac{x}{1} = \frac{9}{3} \), where \( x \) represents the strength of one tablet. Solving through cross-multiplication, \( x \times 3 = 9 \), you find \( x = 3 \). Thus, each tablet should be 3 mg. Dosage calculations often involve understanding these units and operations to ensure that patients receive the right amount of medication.

Solving Word Problems in Prealgebra

Word problems in prealgebra can be tricky, yet they play a crucial role in developing problem-solving skills. A common strategy is to translate story elements into mathematical expressions, making them more manageable.
Many prealgebra problems revolve around proportions, rates, and ratios. For example, understanding the relationship between quantities like tablets and milligrams allows students to use arithmetic methods to find unknown values.
To solve such problems, you'll typically:

• Identify what you're solving for
• Set up an equation or proportion
• Simplify and solve using basic operations like multiplication or division

These steps not only solve the present problem but also build a foundation for more complex algebraic concepts, making them indispensable in a student's mathematical journey.