Problem 66 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{13}{24}-\frac{3}{40} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{7}{15} \).

Step 1: Find a Common Denominator

To subtract the fractions \( \frac{13}{24} \) and \( \frac{3}{40} \), we need to find a common denominator. The least common multiple (LCM) of 24 and 40 is 120. We will use 120 as the common denominator.

Step 2: Convert Fractions to Equivalent Fractions

Convert each fraction to an equivalent fraction with the denominator of 120. For \( \frac{13}{24} \):Multiply both numerator and denominator by 5 to get:\[ \frac{13}{24} \times \frac{5}{5} = \frac{65}{120} \]For \( \frac{3}{40} \):Multiply both numerator and denominator by 3 to get:\[ \frac{3}{40} \times \frac{3}{3} = \frac{9}{120} \]

Step 3: Subtract the Fractions

Now, subtract the fractions with the common denominator:\[ \frac{65}{120} - \frac{9}{120} = \frac{56}{120} \]

Step 4: Simplify the Result

Simplify \( \frac{56}{120} \) by finding the greatest common divisor (GCD) of 56 and 120, which is 8.Divide both numerator and denominator by 8:\[ \frac{56 \div 8}{120 \div 8} = \frac{7}{15} \]

Step 5: Final Answer

So, the simplified result of the operation is \( \frac{7}{15} \).

Key Concepts

Understanding Common Denominators

When it comes to fraction subtraction, finding a common denominator is a crucial step. A common denominator, especially the least common denominator (LCD), helps align the fractions so that their denominators are the same. This is essential because you can only directly subtract or add fractions when their denominators are equal.
To find a common denominator, look for the Least Common Multiple (LCM) of the existing denominators. In our example, the denominators are 24 and 40. Calculating the LCM, we find that 120 is the smallest number that both 24 and 40 evenly divide into. This makes 120 our common denominator.
Here's how to find it:

• List out the multiples of each denominator:
  
  • Multiples of 24: 24, 48, 72, 96, 120, ...
  • Multiples of 40: 40, 80, 120, ...
• Identify the smallest common multiple from these lists, which is 120.
• Use this common multiple to create equivalent fractions for subtraction.

Creating Equivalent Fractions

Equivalent fractions are fractions that denote the same value or proportion, even though they may have different numerators and denominators. To subtract fractions with different denominators, we first need to convert them into equivalent fractions with a common denominator.
For \(\frac{13}{24}\) and \(\frac{3}{40}\), we aim to make their denominators 120. We do this by scaling the fractions appropriately:

• For \(\frac{13}{24}\), multiply both numerator and denominator by 5 to get \(\frac{65}{120}\).
• For \(\frac{3}{40}\), multiply both by 3, resulting in \(\frac{9}{120}\).

By converting them to equivalent fractions with the same denominator, you make it possible to subtract the fractions directly, which will be handled in the next steps.

Simplifying Fractions

Once the subtraction is completed, it is often necessary to simplify the resulting fraction to its simplest form. Simplifying reduces a fraction to its smallest possible numerator and denominator so that it remains equivalent to the original result.
In our subtraction example, after finding \(\frac{65}{120} - \frac{9}{120} = \frac{56}{120}\), we simplify \(\frac{56}{120}\):

• First, find the Greatest Common Divisor (GCD) of 56 and 120. In this case, the GCD is 8.
• Next, divide both the numerator and denominator by this GCD:
  
  • Numerator: \(56 \div 8 = 7\)
  • Denominator: \(120 \div 8 = 15\)
• Thus, \(\frac{56}{120}\) simplifies to \(\frac{7}{15}\).

Simplifying not only makes the fraction easier to understand and work with but often helps in better comprehending the size of the fraction, providing clarity in mathematical communication.