Question

A client consumed \(2 \frac{1}{2}\) pints of water in a day. How many cups of water is this equivalent to? ______ cups

Short answer

5 cups

Step-by-step solution

Understand the Problem

First, we need to determine the units involved in the problem. The client consumed his water in pints, and we need the equivalent in cups.

Find Conversion Factor

The conversion factor between pints and cups is 1 pint = 2 cups. This means we need to multiply the number of pints by 2 to find out how many cups there are.

Convert Mixed Number to Improper Fraction

The client consumed \(2 \frac{1}{2}\) pints. We need to convert this into an improper fraction first. \(2 \frac{1}{2}\) is equivalent to \(\frac{5}{2}\).

Multiply by Conversion Factor

Now, we multiply the number of pints by the conversion factor. \(\frac{5}{2} \times 2 = 5\).

Result Interpretation

The result, 5, means that \(2 \frac{1}{2}\) pints is equivalent to 5 cups.

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. In our case, the client consumed water measured in mixed numbers: \(2 \frac{1}{2}\) pints. To better understand and perform calculations, mixed numbers can often be converted into improper fractions. This allows us to easily operate on them mathematically.
For instance, \(2 \frac{1}{2}\) can be rewritten as an improper fraction \(\frac{5}{2}\). This is done by multiplying the whole number by the denominator of the fraction and then adding the numerator of the fraction. Specifically, \(2 \times 2 + 1 = 5\), hence \(\frac{5}{2}\).
Using improper fractions is especially helpful when performing multiplication or division, as you can see in our problem where the mixed number is transformed to facilitate conversion into cups.

Improper Fractions

Improper fractions are fractions where the numerator is larger than or equal to the denominator. These are essential in mathematical calculations because they simplify the process of performing algebraic operations.
In our exercise, converting the mixed number \(2 \frac{1}{2}\) into the improper fraction \(\frac{5}{2}\) allows us to perform the multiplication needed for converting the pints to cups.
By working with improper fractions, we avoid dealing with whole numbers and fractions separately, making calculations straightforward and less error-prone.
Converting back to a mixed number is often done after calculations to better interpret the result in realistic terms, although here, expressing \/ the quantity in cups suffices for understanding.

Conversion Factors

Conversion factors are numbers used to change one set of units to another, by multiplying or dividing. They are crucial for changing from one measurement unit to another seamlessly.
In the case of the exercise, the conversion factor between pints and cups is 2. This is because 1 pint is equivalent to 2 cups.
To find out how many cups are in \(2 \frac{1}{2}\) pints, we multiply \(\frac{5}{2}\) by this conversion factor, which gives us an easy way to convert measurements correctly and efficiently.
Always ensure the conversion factor is accurate for precision in conversions, and remember it might vary depending on the system of units being used.

Measurement Units

Measurement units are standards for expressing quantities or amounts. In daily life and science, they are vital for ensuring the standardization of measurements.
In our exercise, we are converting between two common liquid measurement units: pints and cups, which are both part of the U.S. customary system.
Understanding different measurement units and how to convert between them enhances problem-solving skills, especially in situations involving cooking, science experiments, or even daily tasks.
Learning to convert and interpret measurement units ensures you're speaking the same 'language' across different fields and applications, promoting accuracy and clarity in communication.