Problem 176 - Intermediate Algebra

Question

In the following exercises, add or subtract. $$ \frac{7}{16}-\frac{5}{12} $$

Step-by-Step Solution

Answer

\( \frac{1}{48} \)

Step 1: Find a common denominator

To add or subtract fractions, the first step is to find a common denominator. In this case, the denominators are 16 and 12. The least common multiple (LCM) of 16 and 12 is 48.

Step 2: Convert the fractions

Convert each fraction to have the common denominator of 48. For \(\frac{7}{16}\): \[ \frac{7}{16} \times \frac{3}{3} = \frac{21}{48} \] For \(\frac{5}{12}\): \[ \frac{5}{12} \times \frac{4}{4} = \frac{20}{48} \]

Step 3: Subtract the numerators

Now that the fractions have the same denominator, subtract the numerators: \[ \frac{21}{48} - \frac{20}{48} = \frac{1}{48} \]

Step 4: Simplify if possible

Check if the resulting fraction can be simplified. In this case, \(\frac{1}{48}\) is already in its simplest form.

Key Concepts

common denominator

When you're subtracting fractions, finding a common denominator is essential. This ensures that you can compare and combine the fractions easily. The common denominator is the smallest number that both denominators can divide into without leaving a remainder.
For instance, if you have fractions like \(\frac{7}{16}\) and \(\frac{5}{12}\), their denominators are 16 and 12. To find a common denominator, you look for their least common multiple (LCM), which is the smallest number that both 16 and 12 can divide into. For these denominators, the LCM is 48.
By converting each fraction so that they have this common denominator, it makes subtraction straightforward and ensures the fractions are compatible.

least common multiple

The least common multiple (LCM) is a key concept when working with fractions that have different denominators. It is the smallest number that is a multiple of both denominators.
To find the LCM of 16 and 12 in our example, list the multiples of each number until you find the smallest common one. Multiples of 16 are 16, 32, 48, 64..., while multiples of 12 are 12, 24, 36, 48.... The smallest common multiple here is 48.
Using the LCM helps to align the fractions under a common denominator, which simplifies the process of adding or subtracting them.

fraction subtraction

Subtracting fractions becomes simple once you have a common denominator. Here’s how you do it:
- **Step 1**: Find the least common multiple of the denominators
- **Step 2**: Convert each fraction to have the common denominator. For \(\frac{7}{16}\), multiply by \(\frac{3}{3}\) to get \(\frac{21}{48}\). For \(\frac{5}{12}\), multiply by \(\frac{4}{4}\) to get \(\frac{20}{48}\).
- **Step 3**: Subtract the numerators directly because the denominators are now the same: \(\frac{21}{48} - \frac{20}{48} = \frac{1}{48}\).
This method ensures that you are only working with the numerators once the fractions share a common base. It simplifies what could otherwise be a tricky operation.

simplifying fractions

In fraction arithmetic, simplifying the result is the final step. Simplifying a fraction means reducing it to its lowest terms, where the numerator and the denominator have no common factors besides 1.
In our example, \(\frac{1}{48}\) is already in its simplest form because 1 is a prime number and has no common factors with 48 other than 1. If the result hadn’t been in its simplest form, you would need to divide both the numerator and the denominator by their greatest common divisor (GCD).
Simplifying fractions not only makes the result cleaner and more elegant but also easier to understand and work with in subsequent calculations.