Problem 23 - Basic Engineering Mathematics

Question

Express as decimal fractions: (a) \(\frac{9}{16}\) and (b) \(5 \frac{7}{8}\)

Step-by-Step Solution

Answer

(a) 0.5625, (b) 5.875

Step 1: Understand the problem

We need to convert the expressions \(\frac{9}{16}\) and \(5 \frac{7}{8}\) from fractions into decimal format.

Step 2: Convert \(\frac{9}{16}\) to a Decimal

Divide the numerator by the denominator: \(9 \div 16 = 0.5625\). So, \(\frac{9}{16} = 0.5625\).

Step 3: Separate the Whole Number from the Fraction in \(5 \frac{7}{8}\)

The expression \(5 \frac{7}{8}\) is a mixed number, where 5 is the whole number and \(\frac{7}{8}\) is the fractional part. We need to convert the fractional part to a decimal and then add it to the whole number.

Step 4: Convert \(\frac{7}{8}\) to a Decimal

Divide the numerator by the denominator: \(7 \div 8 = 0.875\). So, \(\frac{7}{8} = 0.875\).

Step 5: Combine the Whole Number and Decimal

Add the decimal obtained from \(\frac{7}{8}\) to the whole number: \(5 + 0.875 = 5.875\). So, \(5 \frac{7}{8} = 5.875\).

Key Concepts

Fraction to Decimal Conversion

Converting fractions to decimals involves a simple but important process. The goal is to understand how the fractional part of a number can be expressed in decimal format. A fraction consists of a numerator (the number on top) and a denominator (the number on the bottom). To convert a fraction into a decimal:

• Divide the numerator by the denominator. This operation will give you a decimal representation of the fraction.
• This is done using simple division, just as you would with whole numbers.

For example, converting \(\frac{9}{16}\) involves dividing 9 (numerator) by 16 (denominator), resulting in a decimal of 0.5625. This means \(\frac{9}{16}\) is equivalent to 0.5625 in decimal form.
Understanding this method is crucial as it builds the foundation for working with decimals, making it easier to handle more complex number conversions in mathematics.

Mixed Number Conversion

A mixed number is a combination of a whole number and a fraction. To convert mixed numbers into decimals, you need to work on both the whole number and the fractional part separately.

• First, identify the whole number and the fractional part. In the example \(5 \frac{7}{8}\), '5' is the whole number and \(\frac{7}{8}\) is the fraction.
• Next, convert the fractional part into a decimal—this is done as mentioned in the fraction to decimal conversion process.
• After converting the fraction, simply add the decimal to the whole number to get the final result.

In our example, once you convert \(\frac{7}{8}\) to 0.875, add this decimal to the whole number 5. This gives us a final value of 5.875.
Mastering mixed number conversion is particularly useful in everyday applications, such as measurements or dealing with quantities, ensuring that calculations remain accurate and understandable.

Numerator and Denominator Division

The step of dividing the numerator by the denominator is foundational in converting fractions into decimals. This step essentially breaks down the fraction to understand how many parts of the denominator the numerator can account for.Here's how you approach it:

• Take the top number (numerator) and divide it by the bottom number (denominator) using standard division methods.
• The result of this division gives the precise decimal representation of the fraction.
• Ensure precision in division, especially in cases where the division may result in a repeating or long decimal. This ensures the decimal remains as accurate as possible.

Let's apply this to \(\frac{7}{8}\), where 7 divided by 8 results in 0.875. This division works similarly with other fractions. Understanding this will help you tackle various kinds of fraction-related problems with confidence.