Problem 50 - Algebra 1

Question

Evaluate the expression. $$ \frac{7}{12}-\left(-\frac{3}{4}\right)+\left(-\frac{1}{8}\right) $$

Step-by-Step Solution

Answer

The expression simplifies to \( \frac{29}{24} \)

Step 1: Break down the expression into parts

Express the expression as: \( \frac{7}{12} + \left(\frac{3}{4}\right) - \frac{1}{8} \). This helps to separate the parts of the expression, looking at each fraction separately, and better understanding the influence of the negative signs.

Step 2: Convert all fractions to equivalent fractions with same denominator

All fractions should have the same denominator for easy addition and subtraction. The least common multiple of 12, 4, and 8 is 24. Convert each fraction to an equivalent fraction with denominator 24: \( \frac{14}{24} + \frac{18}{24} - \frac{3}{24} \).

Step 3: Add and subtract the fractions

Now that all fractions have the same denominator, they can be added and subtracted. Perform the addition and subtraction to get the answer: \( \frac{14 + 18 - 3}{24} = \frac{29}{24} \).

Key Concepts

Addition of Fractions

When you add fractions, it's important to have a common denominator. The denominator is the bottom number of a fraction and it represents how many equal parts the whole is divided into.
To add fractions:

• Find a common denominator.
• Convert each fraction so that they have this common denominator.
• Add the numerators (the top numbers) together.

For example, to add \( \frac{7}{12} \) and \( \frac{3}{4} \), first express them with a common denominator, which is 24. Convert both fractions: \( \frac{7}{12} \) becomes \( \frac{14}{24} \), and \( \frac{3}{4} \) becomes \( \frac{18}{24} \). Once they have the same denominator, simply add the numerators:
\( \frac{14}{24} + \frac{18}{24} = \frac{32}{24} \). With this method you'll always ensure the fractions are combined correctly! Don't forget to simplify the result, if possible.

Subtraction of Fractions

Subtraction of fractions is quite similar to addition in terms of needing a common denominator.
Here's how to subtract fractions effectively:

• Ensure both fractions have the same denominator.
• Convert fractions if their denominators do not match.
• Subtract the second fraction's numerator from the first fraction's numerator.

Using the expression \( \frac{14}{24} - \frac{3}{24} \) from our example, you can see that because they have the same denominator, it's straightforward. Take 14 minus 3 in the numerators:
\( \frac{14 - 3}{24} = \frac{11}{24} \). Ensure you understand the difference between operations: always subtract the second fraction's numerator from the first's.

Least Common Multiple

The least common multiple (LCM) is crucial when working with fractions with different denominators. It helps in converting these fractions into equivalent fractions with a common denominator.
Here's the step-by-step guide to finding the LCM of numbers:

• List the multiples of each denominator.
• Identify the smallest multiple that appears in each list.

In our problem, we had denominators 12, 4, and 8, with an LCM of 24:

• Multiples of 12: 12, 24, 36, ...
• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 8: 8, 16, 24, ...

The smallest number common to all these is 24. By using the LCM, fractions are easily converted to have this common denominator, making calculations simpler and more efficient.

Negative Numbers

Negative numbers can sometimes be a bit tricky. Still, understanding how they work in operations can make your calculations much easier.
Here are a few key points about negative numbers in fraction calculations:

• A negative sign outside the fraction affects the whole fraction.
• When subtracting a negative fraction, it's the same as adding its positive.\( a - (-b) = a + b \)
• Adding a negative fraction is the same as subtracting its positive counterpart.

Look at our example with fractions: \( \frac{7}{12} - (-\frac{3}{4}) + (-\frac{1}{8}) \). When we see \( -(-\frac{3}{4}) \), this becomes \(+\frac{3}{4}\), and \(+(-\frac{1}{8})\) becomes \(-\frac{1}{8}\). Understanding these rules helps in tackling problems with ease.