Problem 36 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$-\frac{1}{30}+\frac{9}{40}$$

Step-by-Step Solution

Answer

The LCD is 120, and the result is \(\frac{23}{120}\).

Step 1: Identify the Denominators

The fractions are \(-\frac{1}{30}\) and \(\frac{9}{40}\). Identify the denominators, which are 30 and 40.

Step 2: Find the Least Common Denominator (LCD)

The least common denominator is the smallest number that both 30 and 40 divide into evenly. To find the LCD, compute the least common multiple (LCM) of the denominators. The prime factors are 30 = 2 × 3 × 5 and 40 = 2³ × 5. The LCM is 2³ × 3 × 5 = 120. So the LCD is 120.

Step 3: Convert to Equivalent Fractions

Convert \(-\frac{1}{30}\) into an equivalent fraction with the LCD, 120. Multiply the numerator and denominator by 4 to get \(-\frac{1 \times 4}{30 \times 4} = -\frac{4}{120}\). Next, convert \(\frac{9}{40}\) into an equivalent fraction with denominator 120 by multiplying the numerator and denominator by 3 to get \(\frac{9 \times 3}{40 \times 3} = \frac{27}{120}\).

Step 4: Add the Fractions

Now that both fractions have the same denominator of 120, add them together. Combine the numerators: \(-4 + 27 = 23\). This results in \(\frac{23}{120}\).

Step 5: Simplify the Result

Check if the fraction \(\frac{23}{120}\) can be simplified. Since 23 is a prime number and does not divide 120, the fraction is already in its simplest form.

Key Concepts

Least Common Denominator (LCD)

When adding or subtracting fractions, the first step is to find the Least Common Denominator (LCD). The LCD is the smallest number that both denominators can divide evenly into, enabling you to easily add or subtract the fractions. To find the LCD, you need to determine the least common multiple (LCM) of the denominators.

Here’s a helpful strategy to determine the LCM:

• Identify the prime factors of each denominator.
• For 30, the prime factors are 2, 3, and 5, making it 2 × 3 × 5.
• For 40, the prime factors are 2³ and 5, forming 2³ × 5.
• The LCM is found by taking the highest power of each prime factor available, which in this case is 2³ × 3 × 5 = 120.

Thus, the LCD of 30 and 40 is 120, allowing you to convert the fractions to have a common denominator, simplifying the addition or subtraction process.

Equivalent Fractions

Once you have identified the LCD, you need to convert each fraction to have this common denominator. This involves finding equivalent fractions. Equivalent fractions have the same overall value, even though they may look different.

To find these equivalent fractions, follow these steps:

• For the fraction \(-\frac{1}{30}\), multiply both its numerator and denominator by 4 to achieve the common denominator: \(-\frac{1\times4}{30\times4} = -\frac{4}{120}\).
• Similarly, for \(\frac{9}{40}\), multiply both components by 3: \(\frac{9\times3}{40\times3} = \frac{27}{120}\).

These new fractions, \(-\frac{4}{120}\) and \(\frac{27}{120}\), are equivalent to their original forms but now share a common denominator, making them ready for addition or subtraction.

Simplifying Fractions

After adding or subtracting fractions, the result needs to be simplified, if possible. Simplifying a fraction means reducing it to its smallest form where the numerator and denominator have no common factors other than 1. This step ensures clarity and precision in your answers.

To simplify the resultant fraction \(\frac{23}{120}\):

• Check if the numerator, 23, and the denominator, 120, share any common factors.
• Since 23 is a prime number and does not divide into 120 evenly, the fraction is already in its simplest form.

A keen eye is helpful here as it ensures you don't miss any further reduction opportunities. However, in this case, the fraction \(\frac{23}{120}\) cannot be simplified further.