Problem 15 - 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 tablet has a strength of 45 mg. If a patient is prescribed a dose of \(112.5 \mathrm{mg},\) how many tablets does he take?

Step-by-Step Solution

Answer

The patient needs to take 2.5 tablets.

Step 1: Identify known quantities

The known quantities are the strength of one tablet, which is 45 mg, and the prescribed dose, which is 112.5 mg.

Step 2: Set up the proportion

We know that 1 tablet corresponds to 45 mg. We need to find how many tablets (let's call it \( x \)) corresponds to 112.5 mg. We set up the proportion: \[ \frac{1 \text{ tablet}}{45 \text{ mg}} = \frac{x \text{ tablets}}{112.5 \text{ mg}} \]

Step 3: Solve the proportion

To solve for \( x \), we cross-multiply the proportion: \( 1 \times 112.5 = 45 \times x \). This simplifies to: \[ 112.5 = 45x \]

Step 4: Isolate x

Divide both sides by 45 to isolate \( x \): \[ x = \frac{112.5}{45} \] This simplifies to \( x = 2.5 \).

Step 5: Interpret the result

The computed \( x \) value means that the patient needs to take 2.5 tablets to achieve the prescribed dose of 112.5 mg.

Key Concepts

Word Problems

Word problems in mathematics provide a scenario where math concepts are applied to solve real-life situations. They require you to first understand the problem's scenario, then identify the quantities involved, and finally translate these into mathematical expressions or equations. In the context of the exercise at hand, we were given a healthcare-related situation where a patient's medication dosage needed calculation. Consider the example: A doctor prescribes 112.5 mg of a drug to a patient, and each tablet contains 45 mg.

• First, identify what you know: the dosage and the strength of each tablet.
• Next, formulate the question: How many tablets does the patient need?

Translating the situation into a math problem involves setting up a proportion, which we'll discuss more next. The ability to decipher such problems is a key outcome when learning to solve word problems. They help develop critical thinking and problem-solving skills.

Cross-Multiplication

Cross-multiplication is a core technique used in solving proportions in mathematics. When you have an equation in the form of two fractions set equal to each other, you can use cross-multiplication to solve for an unknown. For example, we set up the proportion for tablets: \[\frac{1 \text{ tablet}}{45 \text{ mg}} = \frac{x \text{ tablets}}{112.5 \text{ mg}}\]To utilize cross-multiplication, multiply the numerator of one fraction by the denominator of the other fraction, equating both products:- The calculation becomes, \(1 \times 112.5 = 45 \times x\).- This simplifies further to \(112.5 = 45x\).The advantage of this method is that it provides a straightforward way to eliminate the fractions and solve for unknown values. Cross-multiplication is especially useful in dealing with proportions, as it ensures accuracy and simplicity.

Prealgebra

Prealgebra forms the foundation for more advanced mathematical studies and is essential for solving a range of problems, including proportions. Concepts learned in prealgebra such as understanding numbers, fractions, and how to manipulate equations are crucial here. In the provided problem, we used prealgebra skills to isolate the variable \(x\) and solve for the number of tablets needed.

• Once we reached \(112.5 = 45x\), understanding basic algebra allowed us to solve this by dividing both sides by the known value, 45.
• The solution to this division, \(x = \frac{112.5}{45}\), becomes our answer, simplifying to \(x = 2.5\).

Overall, prealgebra equips students with the tools needed to approach problems methodically, build a solid mathematical foundation, and gain confidence in their ability to tackle real-world scenarios involving arithmetic and algebraic reasoning.