Question

The nurse gave a client \(3 / 4\) tablespoons (tbs) of medication with breakfast, \(1 / 2\) tbs at lunch, \(1 / 2\) tbs at dinner, and \(11 / 4\) tbs at bedtime. How much medication did the nurse administer? ______

Short answer

4.5 tablespoons

Step-by-step solution

Convert Fractions to Like Denominators

To add fractions, the denominators must be the same. In this problem, the fractions have different denominators (4 and 2). Let's convert them all to have the denominator of 4.\(\frac{3}{4}\) stays the same, while \(\frac{1}{2} = \frac{2}{4}\).

Sum Up the Fractions

Now, add the fractions together using the common denominator. First, add the morning, lunch, and dinner doses: \(\frac{3}{4} + \frac{2}{4} + \frac{2}{4}\), which equals \(\frac{7}{4}\).

Add Bedtime Dosage

Finally, add the bedtime dose \(\frac{11}{4}\) to the previous total. Use the same denominator: \(\frac{7}{4} + \frac{11}{4} = \frac{18}{4}\).

Simplify the Result

Divide the numerator by the denominator to simplify the fraction: \(\frac{18}{4} = 4\frac{1}{2}\) or equivalently \(4.5\) tablespoons.

Key concepts

Fraction Addition

Adding fractions might initially seem daunting, but it's quite straightforward once you understand the steps. In simple terms, fraction addition involves combining the numerators (top numbers) of fractions with the same denominator (bottom number). However, if the denominators are different, you must first adjust them to be the same before you can proceed. This ensures that you are adding equivalent parts of a whole.To add fractions:

• Ensure the denominators are the same (we'll delve into this in the 'like denominators' section).
• Add the numerators together.
• Place the sum over the common denominator.

For example, in the case of the medication doses, after converting denominators, you can simply add: \[\frac{3}{4} + \frac{2}{4} + \frac{2}{4} = \frac{7}{4}\]Adding fractions is just about putting together the parts you have, so they make up a new, larger part, called a sum. And just like piecing together a puzzle, when the pieces fit well, the process becomes much simpler!

Like Denominators

Like denominators are key to performing fraction addition easily. Denominators being 'like' simply means they are the same, which allows you to combine fractions easily without any extra steps.To convert fractions to have like denominators, you might need to adjust the fractions so that their denominators match. This is because adding fractions with different denominators is like trying to add apples to oranges; you end up with mixed results instead of a straightforward sum.Here’s how you can convert them:

• Find a common denominator, which is typically the least common multiple (LCM) of the denominators you're working with.
• Adjust the numerators based on this new denominator. For instance, \(\frac{1}{2}\) can be converted to \(\frac{2}{4}\) to match the denominator of \(\frac{3}{4}\). This involves multiplying both the numerator and the denominator by the same number to achieve the new fraction.

Once your fractions have like denominators, you can add the numerators directly, making the addition seamless!

Simplifying Fractions

Once you've completed adding fractions, it's not uncommon to end up with a larger fraction than you started with. This is where simplifying fractions comes into play. Simplifying is the process of reducing a fraction to its most basic form, making it easier to understand and use.To simplify a fraction:

• Find the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCF.

For example, the solution to the nurse's medication dosage problem results in \(\frac{18}{4}\). By dividing both 18 and 4 by their GCF, which is 2, this fraction simplifies to \(\frac{9}{2}\). Expressing this as a mixed number gives \(4\frac{1}{2}\) or as a decimal, 4.5.Simplifying helps communicate your result with greater clarity and can sometimes make further calculations more manageable.