Problem 27 - Elementary Algebra

Question

For the following problems, convert each fraction to a decimal fraction. If the decimal form is nonterminating,round to 3 decimal places. \(\frac{9}{20}\)

Step-by-Step Solution

Answer

Question: Convert the fraction 9/20 into a decimal fraction and round it to 3 decimal places if it is nonterminating. Answer: 0.45

Step 1: Convert the Fraction to a Decimal

To convert the fraction \(\frac{9}{20}\) to a decimal, we simply need to divide the numerator (9) by the denominator (20). So, we perform the division 9 ÷ 20.

Step 2: Perform the Division

After performing the division \(9 ÷ 20\), we get a decimal value of \(0.45\).

Step 3: Check for Decimal Nonterminating

The decimal value of \(0.45\) is not nonterminating, so we don't need to round it to 3 decimal places.

Step 4: Write the Final Answer

The decimal form of the given fraction \(\frac{9}{20}\) is \(0.45\).

Key Concepts

Fraction to Decimal Conversion

The process of transforming a fraction into a decimal is a fundamental skill in mathematics that's frequently utilized in various applications. A fraction essentially represents a division: the numerator (the top number) tells us how many parts we have, and the denominator (the bottom number) tells us into how many equal parts the whole is divided.

To convert a fraction to a decimal, you divide the numerator by the denominator. For the fraction \(\frac{9}{20}\), you would take 9 (the number of equal parts you have) and divide it by 20 (the total number of parts). Doing this calculation gives us \(9 \div 20 = 0.45\), a straightforward transformation showcasing that every fraction corresponds to a unique decimal representation.

Nonterminating Decimals

When dividing certain fractions, the result can be a nonterminating decimal. This type of decimal goes on indefinitely without repeating a pattern. Nonterminating decimals are a result of fractions where the denominator has prime factors other than 2 or 5, leading to an infinite string of digits after the decimal point. In practice, to manage these infinite decimals, we round them to a given number of decimal places to make them finite and easier to work with.

For example, the fraction \(\frac{1}{3}\) results in a nonterminating decimal of 0.333..., which can be rounded to 0.333 if we limit it to three decimal places. The key here is to define the degree of precision you need for your work and round the number accordingly.

Rounding Decimals

Rounding decimals is a technique used to reduce the number of digits after the decimal point, making the number simpler or 'neater' and more convenient for guesstimates, comparisons, or when a specific precision level is required. For rounding to 3 decimal places, you look at the fourth decimal place. If it is 5 or greater, you increase the third decimal place by one; otherwise, you leave it as is.

Let's say you have a nonterminating decimal 0.6666..., and you're asked to round to three decimal places. You would look at the fourth digit, which is a 6, so you'd round up the third digit resulting in 0.667. If the original decimal was 0.6664..., the third decimal place would stay 6, as the fourth digit is less than 5.