Problem 11 - 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 patient is given a prescription of 10 pills. The total prescription contains 355 milligrams. How many milligrams is contained in each pill?

Step-by-Step Solution

Answer

Each pill contains 35.5 milligrams.

Step 1: Understand the Problem

We are given that a prescription of 10 pills contains a total of 355 milligrams of medication. We need to determine how many milligrams are contained in each pill.

Step 2: Set Up the Proportion

Since we know the total amount of milligrams for 10 pills and need to find the milligrams per pill, we set up a proportion where the ratio of pills to milligrams for one pill is equal to the ratio for all 10 pills. This can be written as: \[ \frac{1\, \text{pill}}{x\, \text{mg}} = \frac{10\, \text{pills}}{355\, \text{mg}} \] where \(x\) is the unknown milligrams per pill.

Step 3: Solve the Proportion

Cross-multiply to eliminate the fractions: \[ 1 \times 355 = 10 \times x \] This simplifies to: \[ 355 = 10x \] Now, solve for \(x\) by dividing both sides by 10: \[ x = \frac{355}{10} \] which simplifies to \(x = 35.5\) milligrams per pill.

Step 4: Conclusion

The calculation shows that each pill contains 35.5 milligrams of medication. We used a proportion to divide the total milligrams by the number of pills to find the amount per individual pill.

Key Concepts

Solving Proportions

Proportions are equations that state that two ratios are equivalent. They are particularly useful in solving real-life problems, such as determining medication dosages. When solving proportions, the key is to set two ratios equal to each other.
For example, when given a problem involving a total dosage of medication spread across multiple pills, you can set up a proportion to find out how much medication each pill contains.
Here’s a step-by-step approach:

• Identify what you know and what you need to know.
• Set up a proportion where the first ratio compares units you know.
• Ensure that the units match across the ratio.
• Solve for the unknown variable using techniques like cross-multiplication.

Proportions provide a reliable way to break down and solve complex problems involving parts and wholes.

Translating Word Problems

Translating word problems into mathematical equations is an important skill. It consists of converting a verbal statement into mathematical expressions or equations. This requires understanding the problem by identifying keywords and comprehending the relationship between quantities.
First, underline or highlight numbers and phrases that indicate mathematical operations, such as 'total,' 'divided by,' or 'each.'
For instance, in the given problem, we recognize 'a prescription of 10 pills contains a total of 355 milligrams' and 'how many milligrams per pill' as a signal to formulate a ratio between pills and milligrams.
Next, set up a clear mathematical representation reflecting those relationships. Once the equation is set, use algebraic techniques, like cross-multiplication, to solve for unknowns.

Medication Dosage Calculation

Calculating medication dosages accurately is vital for ensuring patient safety. It involves determining the correct amount of medication a patient receives based on prescription requirements. In these calculations, proportions are crucial as they allow you to find unknown amounts based on known guidelines.
When calculating how much medication is contained in each pill from a given total dosage, think of it as dividing the total dose by the number of usages, which are the pills in this context.
So,

• Identify the total dose and the number of doses (pills).
• Set up a proportion to represent the known and unknown values.
• Use cross-multiplication if necessary to solve for the unknown.

This method ensures each administration, like a pill, provides the correct dosage.

Cross-Multiplication

Cross-multiplication is a fundamental algebraic tool used in solving proportion problems. It simplifies the process by eliminating the fractions and allowing direct manipulation of numbers to find the unknown quantity.
Here's how to apply cross-multiplication:

• Identify two ratios set equal in a proportion.
• Multiply the numerator of one ratio by the denominator of the other.
• Do the same for the other numerator and denominator.
• This yields an equation that is simpler to solve.
• Solve for the missing value by isolating the variable.

In the example, we used cross-multiplication to derive a clear equation: \[ 1 \times 355 = 10 \times x \]. This removed the fractions and allowed us to solve for \( x \), revealing the milligrams per pill, showcasing the power of cross-multiplication in simplifying complex problems.