Problem 110 - Algebra 1: Concepts and Skills

Question

Subtract. Write the answer as a fraction in simplest form. (Skills Review p. 768) $$ \frac{4}{5}-45 \% $$

Step-by-Step Solution

Answer

The subtraction \( \frac{4}{5} - 45 \% \) in simplest fraction form equals \( \frac{7}{20} \).

Step 1: Conversion

Convert 45% to a fraction. The conversion is done by dividing the amount by 100. So \( \frac{45}{100} \) or \( \frac{9}{20} \) when simplified.

Step 2: Subtraction

Now subtract the converted 45% (in fraction form) from the fraction \( \frac{4}{5} \). Therefore, \( \frac{4}{5} - \frac{9}{20} \). Since the denominators aren't the same, find a common denominator that both 5 and 20 can divide into. The least common multiple between 5 and 20 is 20, so the first fraction can be converted to have 20 as its denominator by multiplying both the numerator and denominator by 4. This results in \( \frac{16}{20} - \frac{9}{20} \).

Step 3: Final Simplification

Subtract the numerators, the denominator stays the same. Therefore, \( \frac{16 - 9}{20} = \frac{7}{20} \). The result cannot be simplified any further, hence \( \frac{7}{20} \) is your final answer in its simplest form.

Key Concepts

Percentage Conversion

Converting a percentage to a fraction is an essential skill. It's quite simple when broken down. A percentage is effectively a fraction out of 100. For example, if we look at 45%, this means 45 out of 100. Therefore, we initially write it as \( \frac{45}{100} \).

After writing it as a fraction, we need to simplify it by finding the greatest common divisor of the numerator and the denominator. In this case, the greatest common divisor of 45 and 100 is 5. So we divide both the numerator and the denominator by 5:

• 45 divided by 5 equals 9
• 100 divided by 5 equals 20

This gives us the simplified fraction \( \frac{9}{20} \). And voilà, we have converted 45% into a simplified fraction form.

Common Denominator

When subtracting fractions, having the same denominator is crucial. This shared denominator is called the 'common denominator'. In our example, we're subtracting fractions \( \frac{4}{5} \) and \( \frac{9}{20} \).

To find a common denominator, we look for the least common multiple (LCM) of the two denominators. For 5 and 20, the LCM is 20 because both 5 and 20 divide into 20 perfectly. So, we need to adjust \( \frac{4}{5} \) to have a denominator of 20. Multiply both the numerator and the denominator by 4:

• 4 times 4 equals 16
• 5 times 4 equals 20

Thus, \( \frac{4}{5} \) becomes \( \frac{16}{20} \). Now both fractions have a common denominator, making subtraction straightforward: \( \frac{16}{20} - \frac{9}{20} \).

Fraction Simplification

Simplifying fractions is like tidy up work—you make fractions as simple as possible. Start by subtracting the numerators, while keeping the common denominator. In this problem:

\( 16 - 9 = 7 \)

So, we have \( \frac{7}{20} \). To confirm it’s in its simplest form, check if there is a number greater than 1 that divides both the numerator (7) and the denominator (20) evenly.

Here:

• 7 is a prime number, which means it only divides by 1 and itself.
• 20 divides by 1, 2, 4, 5, 10, 20.

Since 7 doesn’t divide any of 20's divisors, \( \frac{7}{20} \) is already simplified. Therefore, \( \frac{7}{20} \) is the simplest form of the answer.