Problem 2 - Prealgebra

Question

Convert the given fraction to a terminating decimal. \(\frac{19}{5}\)

Step-by-Step Solution

Answer

The fraction \( \frac{19}{5} \) as a decimal is 3.8.

Step 1: Understand the Problem

The task is to convert the fraction \( \frac{19}{5} \) into a decimal form. We can do this by performing division, as fractions represent division problems.

Step 2: Perform the Division

Divide the numerator by the denominator. Here, 19 is divided by 5. Start by seeing how many times 5 fits into 19. This can be calculated as follows: 1. 5 goes into 19 three whole times (as 5 times 3 is 15, which is less than 19). 2. Subtract 15 from 19, which gives a remainder of 4. 3. Bring a decimal point and add a zero to 4, making it 40. 4. Divide 40 by 5, resulting in 8 (as 5 times 8 is exactly 40). So the division yields 3.8.

Step 3: Write Down the Result

The result from the division is 3.8. Since the division came out evenly with no non-zero remainders left, we have a terminating decimal.

Key Concepts

Fraction to Decimal Conversion

Converting a fraction to a decimal is a fundamental skill in mathematics that often requires division. Every fraction essentially represents a division: the numerator is divided by the denominator. To convert \( \frac{19}{5} \) into a decimal, we divide 19 by 5. This process converts the fraction into a value more compactly represented in decimal form.

However, not all fractions convert into neat decimals. Sometimes they can result in repeating decimals, but in this exercise, since 19 divided by 5 results in 3.8, which is exact, the decimal terminates. Recognizing whether the decimal terminates or repeats is crucial.

• Terminating decimal: Ends after a few decimal places.
• Repeating decimal: Goes on indefinitely in a repeating pattern.

It's an essential concept to grasp, especially moving forward in mathematics, as decimals often make it easier to understand and compare numbers in real-world contexts.

Long Division in Mathematics

Long division is a method used to divide larger numbers in a systematic way, making it much simpler to handle challenging division problems.

In our exercise, dividing 19 by 5 using long division helped us discover the decimal form of \( \frac{19}{5} \). Here's a recap of the process using long division:

• Determine how many times the denominator (5) fits into the initial digits of the numerator (19), which is 3 times.
• Multiply 3 by 5, getting 15, and subtract this from 19, leaving a remainder of 4.
• Place a decimal point in the quotient and move the remainder over into the decimal place by adding a 0, forming 40.
• 5 goes into 40 exactly 8 times, resulting in a terminating decimal of 3.8.

Long division simplifies large number division by breaking it down step-by-step into manageable parts. Mastering this method equips you to handle various mathematical tasks and verify calculation accuracy.

Mathematics Problem-Solving

Problem-solving in mathematics involves understanding what is being asked and then applying suitable mathematical techniques. Let's consider the steps taken to convert the fraction \( \frac{19}{5} \) into a decimal as an example of good problem-solving practice.

First, recognize that the problem involves a division, as fractions indicate division operations. Next, decide on the method to use—in this case, long division ensures accuracy and clarity. Finally, follow through with the division step-by-step, just as we did, to arrive at the result of 3.8.

Problem-solving in math isn’t just about getting the right answer; it is about understanding the process and applying methods correctly:

• Understand the problem guidelines and requirements.
• Select appropriate mathematical tools or methods.
• Carry out calculations meticulously and check the work for errors.
• Interpret the results to ensure they make sense within the given context.

Building these problem-solving skills not only helps in mathematics but also prepares you to tackle various challenges in everyday life with confidence.