Problem 3 - Algebra for College Students

Question

For Problems 1-12, perform the indicated operations involving rational numbers. Be sure to express your answers in reduced form. \(\frac{7}{8}-\frac{3}{5}\)

Step-by-Step Solution

Answer

The result is \(\frac{11}{40}\) in reduced form.

Step 1: Determine the Least Common Denominator (LCD)

To subtract fractions, we need a common denominator. Here, the denominators are 8 and 5. The least common multiple of 8 and 5 is 40, which becomes our least common denominator (LCD).

Step 2: Convert Fractions to Equivalent Fractions

We need to convert each fraction to have the LCD of 40. For \(\frac{7}{8}\), multiply both the numerator and denominator by 5 to get \(\frac{35}{40}\). For \(\frac{3}{5}\), multiply both the numerator and denominator by 8 to get \(\frac{24}{40}\).

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, subtract the numerators while keeping the denominator the same: \[\frac{35}{40} - \frac{24}{40} = \frac{35 - 24}{40} = \frac{11}{40} \].

Step 4: Simplify the Fraction (if possible)

Check if the resulting fraction \(\frac{11}{40}\) can be simplified. Since 11 and 40 have no common factors other than 1, this fraction is already in its simplest form.

Key Concepts

Subtracting Fractions

Subtracting fractions might seem tricky at first, but with a few simple steps, it becomes straightforward. When you need to subtract fractions, the key is to ensure they have the same denominator first. This means we take the numbers below the fraction line, called denominators, and make them equal for both fractions involved. Once this is done, the subtraction becomes a simple matter of subtracting the numerators, which are the numbers above the fraction line.
Here's a step-by-step approach:

• Identify the denominators of the fractions you want to subtract.
• Find a common denominator, ideally the least common denominator, which we'll explain in the next section.
• Convert the fractions so that they are expressed with the newly found common denominator.
• Subtract the numerators while keeping the common denominator the same.

By following these steps, subtracting fractions doesn't have to be a daunting task. It's all about making the rules work for you.

Least Common Denominator

The least common denominator (LCD) is crucial when subtracting fractions because it allows us to effectively compare and subtract the fractions. To find the LCD, we look for the smallest number that both denominators can divide evenly into, which helps align the fractions to make subtraction possible.
In our example of \(\frac{7}{8} - \frac{3}{5}\), the denominators are 8 and 5. We find the least common multiple of these two numbers. The smallest number that both 8 and 5 divide into without leaving a remainder is 40. This number, 40, becomes our least common denominator.
Using the LCD, we convert each fraction to an equivalent fraction with this common denominator, which we'll discuss more in the following section.
By using the LCD, the process of subtracting fractions becomes much more manageable, as it reduces the fractions to a common ground for simple arithmetical operations.

Equivalent Fractions

Once we've determined our least common denominator, the next step is to convert the original fractions into equivalent fractions with this common denominator. This process might seem complex, but it's quite straightforward.
To make an equivalent fraction, we multiply both the numerator and denominator of a fraction by the same number. This keeps the value of the fraction the same, even though it looks different.
For example, to convert \(\frac{7}{8}\) into an equivalent fraction with 40 as the denominator, we multiply the numerator (7) and the denominator (8) by 5, yielding \(\frac{35}{40}\). Similarly, for \(\frac{3}{5}\), we multiply both the numerator and the denominator by 8, resulting in \(\frac{24}{40}\).

• This makes both fractions ready for easy subtraction, sharing a common, least denominator of 40.

Remember, equivalent fractions are your friend when working towards a common goal, like subtraction or addition.

Simplifying Fractions

After subtracting fractions and obtaining your result, the final step is always to simplify the fraction, if possible. Simplifying fractions means reducing them to their smallest possible form where the numerator and denominator are as low as they can go while still representing the same value.
To simplify a fraction, check if there are any common factors in the numerator and denominator. Divide both by their greatest common divisor (GCD) to achieve simplification. In some cases, like our result of \(\frac{11}{40}\), the fraction is already in its simplest form because the numerator 11 and the denominator 40 share no common factors other than 1.
Simplifying is a good habit as it not only makes fractions easier to understand but also ensures that you always express your answers in the most accurate and standardized form.

• Identify common divisors of the numerator and denominator.
• Divide them by the highest common factor.
• Check to ensure the fraction cannot be simplified further.

This process will ensure your final answer is clear and properly reduced.