Question

Recently, the number of Americans under age 18 with student loans has increased from \(\frac{14}{100}\) to \(\frac{1}{5}\). Express the difference as a fraction reduced to its lowest terms. A. \(\frac{3}{50}\) B. \(\frac{6}{100}\) C. \(\frac{7}{50}\) D. \(\frac{13}{95}\)

Short answer

\(\frac{3}{50}\)

Step-by-step solution

Convert Fractions to Common Denominator

To find the difference between the fractions, they first need to have a common denominator. The fractions are \(\frac{14}{100}\) and \(\frac{1}{5}\). Convert \(\frac{1}{5}\) to a fraction with a denominator of 100.

Calculate Equivalent Fraction

Convert \(\frac{1}{5}\) to a denominator of 100: \(\frac{1}{5} = \frac{1 \times 20}{5 \times 20} = \frac{20}{100}\). Now, we have \(\frac{14}{100}\) and \(\frac{20}{100}\).

Subtract the Fractions

Subtract \(\frac{14}{100}\) from \(\frac{20}{100}\): \(\frac{20}{100} - \frac{14}{100} = \frac{6}{100}\).

Reduce the Fraction to Lowest Terms

Simplify \(\frac{6}{100}\) by finding the greatest common divisor (GCD) of 6 and 100, which is 2. Divide the numerator and the denominator by their GCD: \(\frac{6 ÷ 2}{100 ÷ 2} = \frac{3}{50}\).

Key concepts

Common Denominator

When you're dealing with fraction subtraction or addition, you need to make sure both fractions share the same denominator. This common denominator makes it easy to directly compare and perform arithmetic on fractions since the parts of the whole are the same size.

To find a common denominator, look at the original denominators of the fractions. The least common denominator is essentially the smallest multiple shared by the denominators. For instance, in our exercise, the fractions are \(\frac{14}{100}\) and \(\frac{1}{5}\). The first denominator is 100, and we need to convert \(\frac{1}{5}\) to have a denominator of 100. We do this by multiplying both the numerator and the denominator by the same number. Here, 5 multiplied by 20 gives 100, so we multiply the numerator, 1, by 20 too. This way, \(\frac{1}{5}\) becomes \(\frac{20}{100}\).

This conversion step means that now we can easily subtract one fraction from the other. Remember, getting a common denominator is crucial for any fraction operations.

Equivalent Fraction

Equivalent fractions represent the same quantity even though they might look different. For example, \(\frac{1}{5}\) and \(\frac{20}{100}\) are equivalent.

To convert a fraction to an equivalent fraction with a different denominator, you multiply both the numerator and the denominator by the same number. Here, \(\frac{1}{5}\) becomes \(\frac{20}{100}\) because 1 x 20 is 20, and 5 x 20 is 100.

It's like resizing a photo; the proportion remains the same, even though the size changes. This technique ensures you can compare and combine fractions correctly since they represent the same part of the whole.

Greatest Common Divisor

The greatest common divisor (GCD) is key to simplifying fractions. The GCD is the largest number that can evenly divide both the numerator and the denominator.

For our exercise, we found that \(\frac{20}{100} - \frac{14}{100} = \frac{6}{100}\) but this fraction can be reduced. To simplify \(\frac{6}{100}\), we need the GCD of 6 and 100. Here, the GCD is 2.

We divide both the numerator and the denominator by their GCD: \(\frac{6}{2}\) is 3 and \(\frac{100}{2}\) is 50. Thus, \(\frac{6}{100}\) simplifies to \(\frac{3}{50}\).

The GCD method ensures you get the simplest, most straightforward form of a fraction. This makes it easier to read, compare, and further calculate with other fractions.

Simplifying Fractions

Simplifying fractions makes them easier to work with. Once you have the GCD, you divide the numerator and the denominator by this number.

For example in our exercise, we simplified \(\frac{6}{100}\) to \(\frac{3}{50}\). Using the GCD, found to be 2, we divided 6 by 2 to get 3 and 100 by 2 to get 50.

Much like reducing a recipe, simplifying fractions helps present the most manageable form of the number. No matter the initial fraction, you always seek the simplest equivalent form so that calculations stay clean and straightforward. This helps in reducing errors and understanding the core numerical relationships at play.