Problem 63 - Introductory Algebra for College Students

Question

Add: \(\frac{1}{5}+\left(-\frac{3}{4}\right) .\) (Section 1.5, Example 4)

Step-by-Step Solution

Answer

The result of \(\frac{1}{5}+\left(-\frac{3}{4}\right)\) is \(\frac{19}{20}\).

Step 1: Identify the Least Common Denominator

The denominators of the fractions are 5 and 4. Thus, the least common denominator (LCD) for these two fractions is 20. We find this by identifying the least common multiple (LCM) of 5 and 4, which is 20.

Step 2: Convert each Fraction

Next, convert each fraction to have the LCD as the denominator. The fraction \(\frac{1}{5}\) becomes \(\frac{4}{20}\), by multiplying the numerator and denominator by 4. The fraction \(-\frac{3}{4}\) becomes \(-\frac{15}{20}\), by multiplying the numerator and denominator by 5.

Step 3: Perform the Subtraction

Now, subtract the two fractions: \(\frac{4}{20}-(-\frac{15}{20}) =\frac{4}{20}+\frac{15}{20} =\frac{19}{20}\) . This is equivalent to the original problem.

Key Concepts

Least Common Denominator

Understanding the least common denominator (LCD) is crucial when it comes to adding or subtracting fractions. The LCD is the smallest number that can be evenly divided by each of the denominators in question. Imagine the LCD as the common ground where we can combine fractions without losing their values.

To find the LCD, first list the multiples of each denominator and then identify the smallest multiple they have in common. In our exercise, we have denominators 5 and 4. The multiples of 5 are 5, 10, 15, 20, 25, and so on. For 4, they are 4, 8, 12, 16, 20, and onwards. The smallest common multiple is 20, which becomes our LCD. This step ensures that fractions can be added or subtracted on an equal footing, preparing them for seamless mathematical interaction.

Fraction Arithmetic

When performing fraction arithmetic, especially adding and subtracting, you must have a common denominator. Without it, fractions would be akin to apples and oranges—similar but not quite the same thing. After finding the LCD, the next move is converting the fractions so that both have the same denominator, hence 'speaking the same language'.

Let's apply this to our fractions, \frac{1}{5}\ and \(-\frac{3}{4}\). To convert \frac{1}{5}\, we multiply the top and bottom of the fraction by 4, obtaining \frac{4}{20}\. In the same vein, to convert \(-\frac{3}{4}\), we multiply by 5 to get \(-\frac{15}{20}\). Now we can easily add these converted fractions as we would with regular numbers. The negative sign in front of a fraction indicates it's a value to be subtracted, which in our case turns subtraction into addition when we combine the fractions that share the common denominator.

Improper Fractions

Improper fractions are those where the numerator (the top number) is larger than or equal to the denominator (the bottom number). They are simply another way to express quantities that are greater than one whole.
If during the addition our sum results in an improper fraction, it might be necessary to convert it into a mixed number to make it easier to understand. A mixed number has a whole number part and a fractional part, like 1\(\frac{1}{4}\).

In our exercise example, our sum is \(\frac{19}{20}\), which is not an improper fraction as the numerator is smaller than the denominator. However, if we had arrived at a fraction like \(\frac{21}{20}\), we would have one whole, since 20 fits into 21 once, and a remaining fraction of \(\frac{1}{20}\), so \(\frac{21}{20}\) would be equivalent to 1\(\frac{1}{20}\). This conversion is helpful for fully grasping the quantity being represented.