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Chapter 2: Problem 72

A bottle contains 24 ounces of a liquid pain medication. If a typical dose is 戮 ounce, how many doses are there in the bottle? ______________________________

Short Answer

Expert verified
There are 32 doses in the bottle.

Step by step solution

01

Identify the total volume

We know the bottle contains a total of 24 ounces of the liquid medication.
02

Determine the dose per serving

The question states that each dose of medication is 戮 of an ounce.
03

Calculate the number of doses

To find how many doses are in the bottle, we divide the total volume by the volume per dose. So, calculate \( \frac{24}{\frac{3}{4}} \).
04

Perform the division

Divide 24 by 戮. This is equivalent to multiplying by the reciprocal: \( 24 \times \frac{4}{3} = 32 \).
05

Verify the calculation

Check your work by multiplying the number of doses by the dose size: \( 32 \times \frac{3}{4} = 24 \), confirming that the calculation is correct.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Volume calculation
Calculating volume is essential in many real-life problems, such as determining how much liquid a container holds. In this exercise, we started by identifying the total volume of liquid available, which was 24 ounces. Volume calculation involves understanding two key elements:
  • Total volume: The complete amount of substance within a defined space鈥攆or example, the liquid in a bottle.
  • Measurement units: Common units for volume, like ounces, liters, or milliliters. These are crucial for accuracy in calculations.
Knowing the total volume helps us proceed with further calculations, such as determining how many doses the liquid can provide. A clear grasp of volume concepts allows us to connect practical information to mathematical solutions effectively.
Division
Division is a fundamental mathematical operation used to separate a total amount into equal parts. In our example, we used division to find out how many doses of medication are in the bottle. Here's a breakdown of the process:
  • Identify the total volume that needs to be divided. This was 24 ounces in the problem.
  • Determine the size of each part, which was the dose size of \( \frac{3}{4} \) ounce.
  • Perform the operation by dividing the total volume by the dose size: \( \frac{24}{\frac{3}{4}} \).
Division allows us to find out how many smaller units fit into a larger unit. It helps us distribute quantities evenly, which is a common requirement in everyday tasks like dividing medicine, food portions, or resources.
Multiplication
Multiplication and division are closely related, and sometimes one operation can simplify the other. In our problem, to divide a number by a fraction, we multiplied by the reciprocal of that fraction. Here's how it worked:
  • Understand the concept of reciprocals: The reciprocal of a fraction like \( \frac{3}{4} \) is \( \frac{4}{3} \), because \( \frac{3}{4} \times \frac{4}{3} = 1 \).
  • Instead of dividing by a fraction, multiply by its reciprocal, switching the operation: \( 24 \times \frac{4}{3} \).
  • Perform the multiplication to get the result: \( 24 \times \frac{4}{3} = 32 \).
Multiplication through reciprocals simplifies division problems, making calculations more straightforward. This technique is handy when dealing with fractions, enabling easier manipulation and accurate results.
Problem-solving steps
Problem-solving in mathematics involves a series of logical steps, making complex problems simpler and more approachable. Let's examine the steps we followed in this exercise:
  • Identify the problem: We needed to find out how many doses are in a 24-ounce bottle.
  • Break down the problem: By knowing the dose size was \( \frac{3}{4} \) ounce, we could calculate the number of doses.
  • Choose mathematical operations: Our task was made easier by using division with multiplication via reciprocals.
  • Perform and verify calculations: Always check the results鈥攎ultiplying back to ensure accuracy confirmed the solution: \( 32 \times \frac{3}{4} = 24 \).
Effectively solving problems involves clear identification, strategic calculation, and verification. These skills are crucial not only in academics but in everyday decision-making, boosting logical thinking and precision.

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