Question

Add the following decimals. You are to give a client one tablet labeled \(0.15 \mathrm{mg}\) and one labeled \(0.025 \mathrm{mg}\). What is the total dosage of these two tablets?

Short answer

The total dosage is 0.175 mg.

Step-by-step solution

Understand the Problem

We need to find the total dosage by adding the milligram values of the two tablets: one with a dosage of \(0.15\) mg and the other with \(0.025\) mg.

Align the Decimals

Write the numbers \(0.15\) and \(0.025\) vertically, aligning their decimal points. It looks like this:\[\begin{array}{c} 0.15 \+0.025\end{array}\]

Add the Columns Starting from the Right

Add the numbers from right to left. Starting with the thousandths column, add \(5\) (from \(0.015\)) to \(5\) (from \(0.025\)), which gives you \(0\) carry \(1\).

Add the Tenths Column

Next, add the tenths column, \(1\) (from the carry), to \(1\) from \(0.15\) and \(2\) from \(0.025\). The sum is \(4\) (\(0\), carry \(0\) again).Add this with the hundredths column \(2+3=5\)

Verify Your Result

Double check each step to ensure no mistakes were made, aligning decimal points and ensuring carryovers are properly handled. The sum is \(0.175\) mg.

Key concepts

Dosage Calculation

When performing dosage calculations, especially in a medical context, accuracy is of utmost importance. Dosage calculation requires clear understanding and careful mathematical skills to ensure a patient receives the correct amount of medication.
In the problem provided, we need to determine the total dosage of two tablets. Each tablet has a specified dosage, given in decimal form: one at \(0.15 \text{ mg}\) and another at \(0.025 \text{ mg}\). By adding these values, you ensure that the patient receives the correct combined dosage, which is crucial for effectiveness and safety.
Incorrect calculations can lead to underdosing or overdosing, both of which could have serious consequences. Therefore, it’s important to double-check the calculations as seen in the solution's steps. This not only confirms the correctness of the calculation but also ensures patient safety.

Place Value Alignment

When adding decimals, aligning the decimals based on their place values is crucial for accurate results. It ensures that each column represents the same place value, such as tenths or hundredths. This alignment is similar to how you would add whole numbers, but it emphasizes the importance of the decimal point.
For instance, in the problem where you're calculating dosages, align \(0.15\) and \(0.025\) under each other so that each decimal point lines up neatly. Essentially, it looks like this:

• 0.15
• +0.025

Place value alignment makes it easier to see which numbers should be added together, thus preventing errors in calculations. It also ensures that when you add each column from right to left, you're adding values representing similar magnitudes, which keeps the entire operation organized and straightforward.

Carryover in Addition

Understanding the concept of carryover is vital when you are adding numbers column by column. Sometimes, when numbers are added, their sum exceeds the value of their place value, requiring a shift or 'carrying' of a number to the next column.
In the exercise involving adding \(0.15\) and \(0.025\), consider the thousandths column. If you add \(5\) from \(0.15\) to \(5\) from \(0.025\), the result is \(10\), which means you need to carry over a \(1\) to the hundredths column.
Next, continue with the addition: the added carryover needs to be included in the tenths place as well, ensuring every column is accurately calculated. Carryover ensures exactness, and incorrect handling can lead to errors that propagate through the problem. Double-checking your carryovers at each step can prevent mistakes.

Basic Arithmetic Steps

Performing the addition of decimals involves following the basic arithmetic steps carefully and in sequence. Although this might seem simple, each step has critical importance to achieving the correct outcome.
Start by positioning the numbers so their decimal points are aligned. This preliminary step sets the stage for accuracy in following steps. From there, add each column starting from the rightmost side.

• Add the smallest place value first, which is often the hundredths or thousandths in decimal numbers.
• Ensure you carry over numbers correctly if the column’s sum exceeds its maximum value (like \(9\) for tenths or units).
• Proceed to the next column repeating the same addition method, making sure to account for any carryover from the previous column.
• Always verify each step as you proceed to confirm the accuracy of your results.

These steps form the backbone of decimal addition and are essential for any calculation task, big or small.