Problem 50 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$5 \frac{1}{4} \%$$

Step-by-Step Solution

Answer

The percentage \(5 \frac{1}{4}\%\) as a fraction in lowest terms is \(\frac{21}{400}\).

Step 1: Convert the mixed percentage to an improper fraction

Start by converting the mixed percentage to an improper fraction. The mixed number is 5 \( \frac{1}{4} \), which is equivalent to \(\frac{21}{4}\) as a fraction, since \(5 \times 4 + 1 = 21\).

Step 2: Convert the percentage to a fraction

To convert the percentage to a fraction, divide the fraction by 100. So, \(\frac{21}{4}\) percent is equal to \(\frac{21}{4} \div 100\). This is the same as \(\frac{21}{4} \times \frac{1}{100} = \frac{21}{400}\).

Step 3: Simplify the fraction

Check if the fraction \(\frac{21}{400}\) can be simplified. Since 21 and 400 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator, the top part, is larger than or equal to the denominator, the bottom part. This might sound strange at first, but it represents a value larger than or equal to one whole item. For example, if you have \( \frac{21}{4} \), you have 21 parts out of 4, which means you have more than 5 wholes. **Why Use Improper Fractions?**

• Improper fractions are useful for calculations because they do not mix whole numbers with parts.
• They help in keeping a consistent approach when performing arithmetic calculations like addition, subtraction, multiplication, and division with fractions.

To convert a mixed number to an improper fraction, first multiply the whole number by the denominator. Add this result to the numerator of the fractional part. For \(5 \frac{1}{4}\), the improper fraction becomes \( \frac{21}{4} \) because \(5 \times 4 + 1 = 21\). This technique is crucial for simplifying further into percentages or other fraction operations.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible — finding the smallest numerator and denominator that represent the same value. It's like tidying your room, but for numbers!**How to Simplify Fractions?**

• The main method to simplify a fraction is to divide both the numerator and the denominator by their greatest common divisor (GCD).
• If you can't find a common divisor besides 1, then the fraction is in its simplest form.
• For instance, \( \frac{21}{400} \) cannot be further simplified because 21 and 400 share no common divisor other than 1, meaning it is already as simplified as it can get.

Simplifying is essential in math because it makes fractions easier to understand and work with. You'll often see simplified fractions in solutions because they are generally more straightforward and tidier.

Mixed Numbers

A mixed number is a whole number combined with a fraction, such as 5 \( \frac{1}{4} \). It's a neat way to express numbers that are more than whole.**Benefits of Using Mixed Numbers**

• Mixed numbers give a clear idea of size or quantity in an intuitive way, which is visually and mentally easier for many to grasp than improper fractions.
• They are often used in everyday scenarios such as measurements, cooking, or any practical context involving parts of wholes.

To convert a mixed number to an improper fraction, first ensure you are comfortable breaking down the whole and part, and reverse for the opposite. This understanding bridges the gap between understanding a percentage and comprehending a fraction, and is vital when manipulating and converting numbers between forms in math exercises.