Problem 99 - Elementary Algebra

Question

Use the definition of percent to convert to fractions. $$ 0.7 \% $$

Step-by-Step Solution

Answer

0.7\% as a fraction is \(\frac{7}{1000}\).

Step 1: Write Percent as a Fraction

The percentage is given as 0.7\%. To solve this, first note that percentages are a way to express a number as a part of 100. Therefore, we can write 0.7\% as the fraction \( \frac{0.7}{100} \).

Step 2: Remove the Decimal from the Fraction

To remove the decimal from \( \frac{0.7}{100} \), multiply both the numerator and the denominator by 10 to eliminate the decimal point. This results in \( \frac{7}{1000} \).

Step 3: Simplify the Fraction

Check if the fraction \( \frac{7}{1000} \) can be simplified. Since 7 is a prime number and does not divide 1000, \( \frac{7}{1000} \) is already in its simplest form.

Key Concepts

Fraction Conversion

Converting a percentage into a fraction builds on the fundamental understanding that a percent represents a part of 100. To transform a percentage to a fraction, you simply express the number as a ratio of a fraction over 100. For example, when given a percentage like 0.7%, the initial step in fraction conversion is to write it as \( \frac{0.7}{100} \). This means that 0.7% is equivalent to 0.7 parts out of every 100 parts.

• Recognize the percentage sign (%) and understand it denotes division by 100.
• Place the percentage value over 100 to create the fraction.

After completing this step, you have successfully converted the percentage into a fraction. However, often your fraction might contain a decimal, which leads us to the next important concept: eliminating decimals from fractions.

Decimal Elimination

Once your percentage is expressed as a fraction, there could still be a decimal in the numerator, as seen in \( \frac{0.7}{100} \). This requires a crucial next step: decimal elimination. Removing the decimal ensures the fraction is in a more standard form.
To eliminate the decimal from a fraction like \( \frac{0.7}{100} \), multiply both the numerator and the denominator by 10. This transforms \( 0.7 \) into \( 7 \), resulting in the fraction \( \frac{7}{1000} \).

• Identify the position of the decimal and determine how many places you need to shift it to the right to make it an integer.
• Multiply by 10 for each decimal place.
• Apply this multiplication to both the numerator and the denominator.

Achieving a whole number in the numerator might make calculating and simplifying the fraction easier, which is the next part of our conversion journey.

Simplifying Fractions

The final step in converting percentages to fractions is simplifying the fraction where possible. Simplification involves reducing the fraction to its lowest terms, ensuring no common factors exist other than 1 between the numerator and the denominator.
Taking the example from before, \( \frac{7}{1000} \), notice that 7 is a prime number. Prime numbers have no divisors other than 1 and themselves. As 7 cannot divide 1000, the fraction is already in its simplest form.

• Check if the numerator and denominator share any common factors.
• Use divisibility rules if necessary to find common factors.
• Divide evenly by these common factors to simplify the fraction.

If a fraction is composed of a prime number in the numerator, like in our example, it's often an indication that further simplification may not be needed. This ensures the fraction remains as concise and simple as possible.