Problem 80 - Algebra 1

Question

Write the fraction as a decimal. Round to the nearest hundredth if necessary. $$\frac{2}{7}$$

Step-by-Step Solution

Answer

The decimal equivalent of the fraction \( \frac{2}{7} \) rounded to the nearest hundredth is 0.29.

Step 1: Identify the Fraction

The fraction given is \( \frac{2}{7} \).

Step 2: Division

Divide the numerator (2) by the denominator (7). Please note that in case the division results in a repeating decimal, only round to the nearest hundredth. The result of this division \( \frac{2}{7} \) is approximately 0.2857142857142857.

Step 3: Rounding

The result obtained from the division process is a repeating decimal, therefore it needs to be rounded. According to the question, it has to be rounded to the nearest hundredth. Rounding 0.2857142857142857 to the nearest hundredth gives 0.29.

Key Concepts

Division of Fractions

Understanding how to divide fractions is critical in mathematics, often requiring us to convert fractions into more manageable forms, such as decimals. Division of fractions is essentially a process of determining how many times the denominator can fit into the numerator. When you divide the numerator by the denominator, you are finding the value of one fraction in terms of another.

For instance, let's consider the fraction \( \frac{2}{7} \). To convert it to a decimal, you divide 2 (the numerator) by 7 (the denominator). This doesn't mean how many whole times 7 fits into 2, but rather what fraction of 7, 2 is. Using a calculator or long division, you perform the division and express 2 as a fraction of 7 in decimal form.

Why is this useful? Converting fractions to decimals can simplify the process of working with numbers, especially when dealing with measurements, probability, and certain algebraic equations. It helps students and mathematicians alike to compare and connect different numerical values.

Rounding Decimals

Once you've divided fractions and obtained a decimal result, you may need to round the number to make it more understandable or manageable. Rounding decimals is about converting a longer decimal to a shorter one by keeping the most significant digits and trimming the rest. This is based on the position of the numbers following the decimal point.

In the context of the exercise, we were asked to round the decimal to the nearest hundredth. To do this, you look at the third decimal place. If this number is 5 or greater, you round up; if it's less than 5, you keep the second decimal place as is. So for our given answer, 0.2857142857142857, we look at the third decimal place (5), and because it is 5, we round the second decimal place (8) up to 9, resulting in 0.29 as the rounded number.

This method of rounding is used extensively in financial transactions, scientific data, and when a high degree of precision is not required. It's a fundamental arithmetic skill that helps in presenting data that's easy to read and use.

Repeating Decimals

Sometimes when converting fractions to decimals, you come across repeating decimals. These are decimals in which a digit or group of digits repeats indefinitely. For the fraction \( \frac{2}{7} \), the decimal form is 0.2857142857142857, where the sequence '285714' repeats over and over again.

Repeating decimals are denoted using a bar above the repeating sequence, such as \( 0.\overline{285714} \). Recognizing a repeating pattern is crucial because it tells you that although the decimal representation appears to be lengthy, it's actually a simple pattern that repeats and can be noted in a more concise form. In this context, the concept is particularly essential because it lets us know when and why we would need to round a decimal—because writing out the entire repeating sequence is impractical and adds no value to clarity or precision.