Problem 53 - Fundamentals of Mathematics

Question

Find the value of \(\frac{21}{40}+\frac{17}{30}\).

Step-by-Step Solution

Answer

The value is \(\frac{131}{120}\).

Step 1: Find the Least Common Denominator (LCD)

To add the fractions \(\frac{21}{40}\) and \(\frac{17}{30}\), we need a common denominator. The denominators are 40 and 30. The least common multiple of 40 and 30 is 120. Thus, the least common denominator (LCD) is 120.

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with a denominator of 120. For \(\frac{21}{40}\): - Multiply the numerator and denominator by \(\frac{3}{3}\) to get \(\frac{63}{120}\).For \(\frac{17}{30}\): - Multiply the numerator and denominator by \(\frac{4}{4}\) to get \(\frac{68}{120}\).

Step 3: Add the Fractions

Now, with a common denominator, add the numerators: - \(\frac{63}{120} + \frac{68}{120} = \frac{63 + 68}{120} = \frac{131}{120}\).

Step 4: Simplify the Result

Check if the fraction \(\frac{131}{120}\) can be simplified. Since 131 is a prime number and does not evenly divide into 120, the fraction is already in its simplest form.

Key Concepts

Least Common Denominator

When you want to add fractions like \( \frac{21}{40} + \frac{17}{30} \), you need to find the least common denominator (LCD). The LCD is the smallest number that both denominators can divide into without leaving a remainder.
This step is crucial because fractions need a common denominator for addition. Think of it like finding a common ground or language to seamlessly combine two different values.
Here’s how you find the LCD for the denominators 40 and 30:

• List the multiples of 40 (e.g., 40, 80, 120, 160) and 30 (e.g., 30, 60, 90, 120).
• Look for the smallest number that appears in both lists. That’s your LCD.

In this case, 120 is the smallest common number, making it the LCD. Developing this skill will help tremendously in quickly solving fraction problems.

Equivalent Fractions

Once you have the least common denominator, it's time to create equivalent fractions. This means adjusting the fractions so that they have the same denominator without changing their actual value.
Equivalent fractions are like resizing images without distorting their content. The original value and meaning stay intact, just in a different form.
Following our example, for \( \frac{21}{40} \), multiply both the numerator and the denominator by 3, giving \( \frac{63}{120} \). For \( \frac{17}{30} \), multiply both by 4, resulting in \( \frac{68}{120} \).
This transformation keeps the fractions equal in value and prepares them for easy addition. Remember:

• The multiplier for a fraction is found by dividing the LCD by the current denominator.
• The fraction’s value does not change; only its appearance does.

Understanding equivalent fractions helps in operations beyond just addition.

Simplifying Fractions

After performing the addition, you might want to simplify the fraction. Simplifying a fraction reduces it to its simplest form, where the numerator and the denominator are as small as possible without changing the fraction's value.
In our example, we ended up with \( \frac{131}{120} \) after adding \( \frac{63}{120} \) and \( \frac{68}{120} \). To check if this can be simplified:

• Determine if both 131 and 120 have common factors other than 1.
• Since 131 is a prime number and does not divide evenly into 120, it retains its simplest form.

Simplifying is important for clarity and for ensuring you present the result in the most concise way possible. It makes handling and interpreting fractions much simpler in both calculations and real-life scenarios.