Problem 49 - Prealgebra

Question

Change each percent to a fraction in lowest terms. $$6 \frac{1}{4} \%$$

Step-by-Step Solution

Answer

The fraction in lowest terms is \(\frac{1}{16}\).

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number percentage to an improper fraction. The percentage given is \(6 \frac{1}{4} \%\), which can be expressed as \(\frac{25}{4}\).

Step 2: Convert Percent to Fraction

Next, convert the improper fraction percent to a fraction by dividing by 100. This means you multiply by \(\frac{1}{100}\). Thus, \(\frac{25}{4} \times \frac{1}{100} = \frac{25}{400}\).

Step 3: Simplify the Fraction

Finally, simplify the resulting fraction. The greatest common divisor of 25 and 400 is 25. Divide both the numerator and the denominator by 25 to get \(\frac{1}{16}\).

Key Concepts

Mixed Numbers

Mixed numbers combine whole numbers with fractions. When you're looking at something like \(6 \frac{1}{4}\), it has two parts: the whole number 6 and the fraction \(\frac{1}{4}\). To do calculations, especially conversions, it's often helpful to transform the mixed number into an improper fraction. This just means turning it into a fraction where the numerator is greater than the denominator.

• Multiply the whole number by the denominator of the fraction.
• Add the result to the numerator of the fractional part.
• Place this sum over the original denominator.

So, for \(6 \frac{1}{4}\), you multiply 6 by 4 (which gives you 24) and add 1, resulting in 25. This gives the improper fraction \(\frac{25}{4}\).

Improper Fractions

Improper fractions might sound complicated, but they're just fractions where the numerator is greater than the denominator. These are easy to work with in mathematical operations, like division and multiplication.
Here's the quick rundown with our example \(\frac{25}{4}\):

• The numerator is 25.
• The denominator is 4.
• This means we have 25 divided by 4.

Improper fractions simplify converting percentages or when needing to divide by a number, as seen when dividing by 100 to convert a percent. To convert our \(\frac{25}{4}\) when dealing with percentages, multiply by \(\frac{1}{100}\), producing the fraction \(\frac{25}{400}\).

Simplifying Fractions

Once you've arrived at a fraction, it's time to simplify it. Simplifying means breaking it down to its smallest terms, which often makes further calculations much easier. Simplification involves finding the greatest common divisor (GCD) and using it to divide both the numerator and the denominator.
Here's how simplification works with \(\frac{25}{400}\):

• Determine the GCD of 25 and 400. In this case, it's 25.
• Divide the numerator (25) by 25 to get 1.
• Divide the denominator (400) by 25 to get 16.

After simplification, the fraction reduces to \(\frac{1}{16}\). Simplifying fractions ensures that you express results in the simplest form, making interpretations and further math easier.