Problem 38 - Prealgebra

Question

Find the LCD for each of the following; then use the methods developed in this section to add or subtract as indicated. $$\frac{a}{100}+\frac{7}{10}$$

Step-by-Step Solution

Answer

The sum is \( \frac{a + 70}{100} \).

Step 1: Identify the Denominators

The fractions are \( \frac{a}{100} \) and \( \frac{7}{10} \). The denominators are 100 and 10.

Step 2: Find the Least Common Denominator (LCD)

The LCD of 100 and 10 is the smallest number divisible by both. Since 100 is the larger and already divisible by 10, the LCD is 100.

Step 3: Adjust the Second Fraction

We need to adjust \( \frac{7}{10} \) so it has the denominator of 100. Multiply both the numerator and denominator by 10: \( \frac{7 \times 10}{10 \times 10} = \frac{70}{100} \).

Step 4: Add the Fractions

Now, both fractions have the same denominator: \( \frac{a}{100} + \frac{70}{100} \). Add the numerators: \( \frac{a + 70}{100} \).

Key Concepts

Adding Fractions

Adding fractions becomes simple once you have a common denominator for both fractions involved. A fraction has two parts: a numerator and a denominator. When you want to add two fractions, the denominators must be the same. This is because fractions can only be added if they refer to the same whole.

• The denominators tell us how many equal parts the whole is divided into.
• To add fractions, you first align these parts by finding a common denominator.
• Once the fractions have the same denominator, you simply add the numerators.

In our exercise, after adjusting the fraction \( \frac{7}{10} \) to have a denominator of 100, we were able to add it to \( \frac{a}{100} \) easily. It’s like adding apples to apples instead of oranges and apples! Overall, the process involves a little bit of adjusting to make sure the fractions are on the same level.

Denominators

Denominators play a crucial role when dealing with fractions. They represent the total number of equal parts in one whole. In fraction notation, the denominator sits at the bottom of the fraction. How Denominators Work:

• In \( \frac{a}{100} \), the denominator is 100, meaning the whole is divided into 100 equal parts.
• In \( \frac{7}{10} \), ten equal parts make up the whole.

Since different denominators mean different ways of dividing the whole, comparing or adding fractions with different denominators without adjustments is like comparing apples and oranges. Finding a common denominator neutralizes this issue and provides a common basis, making fraction operations possible. Comprehending denominators not only helps in understanding the size of each fractional piece but also in performing operations like addition or subtraction accurately.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. It is a shared multiple of the denominators involved. The least common denominator (LCD) helps simplify this task. Steps to Find the LCD:

• Look at both denominators and identify a common multiple.
• The smallest of these common multiples is the Least Common Denominator.
• For the fractions \( \frac{a}{100} \) and \( \frac{7}{10} \), since 100 is divisible by 10, the LCD is 100.

This makes it easier to convert fractions so both have the same denominator. Adjusting \( \frac{7}{10} \) by multiplying both its numerator and denominator gives us \( \frac{70}{100} \). This adjustment is crucial, as it allows the fractions to "speak the same language," enabling straightforward addition or subtraction. Solving fraction problems becomes much easier once this step is clear.