Problem 25 - Basic Engineering Mathematics

Question

Express as percentages: (a) \(\frac{5}{16}\) and (b) \(1 \frac{2}{5}\)

Step-by-Step Solution

Answer

(a) 31.25%, (b) 140%

Step 1: Understand the Concept

To convert a fraction or a mixed number to a percentage, you need to multiply the number by 100. This will give you its equivalent in percentage form.

Step 2: Convert Fraction to Percentage

For part (a), convert the fraction \(\frac{5}{16}\) to a percentage by multiplying it by 100:\[\left( \frac{5}{16} \right) \times 100 = \frac{500}{16} = 31.25\%\] So, \(\frac{5}{16}\) is 31.25\%.

Step 3: Convert Mixed Number to Improper Fraction

For part (b), first convert the mixed number \(1 \frac{2}{5}\) to an improper fraction. Multiply the whole number (1) by the denominator (5) and add the numerator (2):\[1 \frac{2}{5} = \frac{5 \times 1 + 2}{5} = \frac{7}{5}\]

Step 4: Convert Improper Fraction to Percentage

Now convert \(\frac{7}{5}\) to a percentage by multiplying by 100:\[\left( \frac{7}{5} \right) \times 100 = \frac{700}{5} = 140\%\] So, \(1 \frac{2}{5}\) is 140\%.

Key Concepts

Percentage Conversion

Percentage conversion is all about expressing fractions or mixed numbers as percentages. To convert a number to a percentage, you simply multiply it by 100. This is because 'percent' means per hundred. Essentially, you are finding out how many parts out of 100 the fraction or number represents.

Let's say you have the fraction \( \frac{5}{16} \). By multiplying \( \frac{5}{16} \) by 100, you get \( 31.25\% \). Here, you've translated it into a form that's easier to understand in relation to a whole or a total.

Similarly, for any fraction or mixed number, following the formula consistently provides you with the desired percentage form.

• Start by converting any given fraction.
• Don’t forget to change mixed numbers to improper fractions first.
• Finally, multiply by 100 to get the percentage.

This method allows for quick and accurate conversions, which is especially useful in calculations related to statistics, finance, and everyday measurements.

Fractions

Fractions represent a part of a whole. They are shown with two numbers: one on top (the numerator) and one on the bottom (the denominator). The fraction \( \frac{5}{16} \) tells us we have 5 parts out of 16 equal parts.

When converting fractions to percentages, it's key to understand what each part means:

• The numerator (5) tells you how many parts you have.
• The denominator (16) tells you how many parts in total make up a whole.

Multiplying the fraction by 100 scales it to a percentage format. This way, you can see the fraction in terms of parts per hundred, making it easier to compare with other percentages or to understand its magnitude in relation to a whole.

To find a fraction's percentage representation, always perform these steps: divide the numerator by the denominator to get a decimal, then multiply by 100. This consistent application of steps ensures clear and correct conversions.

Mixed Numbers

Mixed numbers, such as \( 1 \frac{2}{5} \), contain both a whole number and a fraction. To convert these to percentages, it's often easiest to first convert them to improper fractions.

Here's how you change a mixed number to an improper fraction:

• Multiply the whole number by the denominator.
• Add the result to the numerator.

For example, with \( 1 \frac{2}{5} \), multiply 1 by 5 to get 5, then add 2 (the numerator). You get \( \frac{7}{5} \) as the improper fraction. From here, multiplying by 100 converts it to a percentage, yielding \( 140\% \).

Converting mixed numbers to improper fractions simplifies calculations because fractions allow you to see both the parts greater than one whole and the partial portions together. Understanding this process helps when handling mixed numbers in equations, comparisons, and practical applications like cooking, budgeting, or understanding changes in data.