Question

A bottle of medicine contains 30 doses. How many doses are in \(21 / 2\) bottles. ______

Short answer

There are 315 doses in \(\frac{21}{2}\) bottles.

Step-by-step solution

Understand the Problem

The exercise asks us to find how many doses are in \(\frac{21}{2}\) bottles, given that each bottle contains 30 doses of medicine.

Calculate the Number of Bottles

Convert the fractional bottle into a numerical value. Since \(\frac{21}{2}\) is equivalent to dividing 21 by 2, which results in 10.5. The medicine is in 10.5 bottles.

Multiply Bottles by Doses per Bottle

Calculate the total number of doses by multiplying the number of bottles (10.5) by the number of doses per bottle (30). Multiply 10.5 by 30 to get the total doses.

Perform the Calculation

Perform the multiplication: \(10.5 \times 30 = 315\). The total number of doses in \(\frac{21}{2}\) bottles is 315.

Key concepts

Fractions in Mathematics

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator and a denominator. The numerator, found at the top, signifies how many parts you have, and the denominator, at the bottom, shows how many parts the whole is divided into. For instance, in the fraction \(\frac{21}{2}\), the numerator is 21, and the denominator is 2. When you encounter a fraction, you can perform operations such as addition, subtraction, and multiplication on it, just like whole numbers. In this context, fractions can represent quantities less than a whole or even mixed numbers, which includes a whole number and a fractional part. Fraction conversion is often needed, especially in calculations involving proportions or division.To convert a simple fraction to a decimal, you divide the numerator by the denominator. So, \(\frac{21}{2}\) becomes 10.5. This method is useful when solving real-world problems, such as calculating doses.

Multiplication

Multiplication is a fundamental operation in mathematics that signifies repeated addition. It involves two or more numbers known as multiplicands and factors. The result of multiplication is termed a product.When you're dealing with whole numbers, fractions, or decimals, the process is essentially the same. You multiply the numbers to find the total quantity of units. For example, in our problem, calculating the total doses requires multiplying the number of bottles by the number of doses per bottle.Here's a step-by-step way to multiply when one or more of your numbers is a decimal, like 10.5:

• Ignore the decimal initially and multiply as if it's a whole number.
• Calculate \(105 \times 30\), which equals 3150.
• Adjust for the decimal place. Since 10.5 has one decimal place, the final answer is 315.0, equivalent to 315.

Understanding multiplication in this way helps simplify complex problems by breaking them into manageable parts.

Medicine Dosage

Medicine dosage calculations often involve understanding and manipulating numbers to ensure correct administration to patients. Accurate dosage is vital, as too little may be ineffective, while too much could be harmful. In the context of medicine, the term "dose" refers to the specific quantity of medicine to be administered at one time. Dosage calculations help determine the proper dose for different amounts of medication or varying patient needs. For our exercise, we have a known amount of doses per bottle and need to scale this quantity based on more than one bottle. By using fractions and multiplication, as shown in the exercise, we accurately determine how many doses are available in a given number of bottles—here, multiplying 10.5 bottles by 30 doses per bottle, resulting in 315 doses in total. Such operations are crucial in the healthcare setting to ensure safe and effective patient care. This illustrates why a deep understanding of basic mathematical concepts is essential not only in math classes but in everyday practical scenarios, including medication administration.