Problem 18 - Calculate With Confidence

Question

Change each of the following percentages to a ratio, and reduce to lowest terms. 16% _____________

Step-by-Step Solution

Answer

The ratio is 4:25.

Step 1: Understand the Percentage

A percentage is a fraction out of 100. Thus, 16% can be written as a fraction: \( \frac{16}{100} \).

Step 2: Simplify the Fraction

To simplify the fraction \( \frac{16}{100} \), we find the greatest common divisor (GCD) of 16 and 100, which is 4. Then divide both the numerator and the denominator by this number: \( \frac{16 \div 4}{100 \div 4} = \frac{4}{25} \).

Step 3: Convert Simplified Fraction to Ratio

A ratio can be written from a fraction by using the colon ':' symbol. The simplified fraction \( \frac{4}{25} \) is equivalent to the ratio 4:25.

Key Concepts

Percentage Conversion

When you're working with percentages, it's essential to understand that they represent parts out of 100. This is why we can easily transform a percentage into a fraction by using 100 as the denominator. For example, the percentage 16% becomes the fraction \( \frac{16}{100} \). This transformation is crucial, as it sets the stage for further operations that require a fractional form, such as simplification or calculation.Keep in mind:

• Percentages are always out of 100.
• Converting them to fractions is a straightforward process.
• This conversion is the first step in finding other forms like ratios.

After this initial conversion, you're ready to move on to simplifying the fraction, making it more manageable for various applications.

Fraction Simplification

Once you have your fraction form of the percentage, the next step is to simplify it, which involves reducing it to its simplest terms. Simplifying a fraction means dividing both the numerator and the denominator by their greatest common divisor (GCD). For the fraction \( \frac{16}{100} \), the GCD is 4. This means we divide both 16 and 100 by 4, which results in \( \frac{4}{25} \).Here's a simple guide to fraction simplification:

• Find the greatest common divisor of the numerator and the denominator.
• Divide both the numerator and the denominator by this number.
• Ensure the resulting fraction is in its lowest terms, where no number other than 1 can divide both the numerator and denominator evenly.

Through simplification, the fractions become easier to work with in further mathematical operations.

Numerator and Denominator Reduction

The process of reducing the numerator and the denominator involves dividing them by their greatest common divisor, a critical step to simplify fractions. In our example, once we have the fraction \( \frac{16}{100} \), we seek the largest number that can evenly divide both 16 and 100, which is 4. We divide the numerator, 16, and the denominator, 100, by this number:1. Calculate \( 16 \div 4 = 4 \).2. Calculate \( 100 \div 4 = 25 \).Thus, the fraction simplifies to \( \frac{4}{25} \).Key pointers:

• Reduction ensures the fraction is in its simplest form.
• Always use the greatest common divisor for precise simplification.
• This step is vital for converting fractions to simple ratios without changing their value.

Reduction not only simplifies the appearance of the fraction but also makes it easier to understand and use in practical situations.