Problem 55 - Algebra for College Students

Question

Simplify each numerical expression. $$7 \frac{1}{8}-\left(2 \frac{1}{4}-3 \frac{7}{8}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(8 \frac{3}{4}\).

Step 1: Convert Mixed Numbers to Improper Fractions

The expression contains mixed numbers, so we first need to convert them into improper fractions. The first number is \(7 \frac{1}{8}\) which converts to \(\frac{57}{8}\) and the second number is \(2 \frac{1}{4}\) which converts to \(\frac{9}{4}\). The third number is \(3 \frac{7}{8}\), converting it gives \(\frac{31}{8}\).

Step 2: Simplify the Innermost Parentheses

Our task is to simplify \(2 \frac{1}{4} - 3 \frac{7}{8}\). Using the improper fractions from the previous step, we have \(\frac{9}{4} - \frac{31}{8}\). To subtract these fractions, they must have the same denominator. We multiply \(\frac{9}{4}\) by \(\frac{2}{2}\) to get \(\frac{18}{8}\). Thus, \(\frac{18}{8} - \frac{31}{8} = \frac{18 - 31}{8} = \frac{-13}{8}\).

Step 3: Simplify the Outer Expression

Now, simplify the entire expression: \(7 \frac{1}{8} - (-\frac{13}{8})\). This turns into \(\frac{57}{8} + \frac{13}{8}\) because subtracting a negative is the same as adding. Thus, \(\frac{57}{8} + \frac{13}{8} = \frac{57 + 13}{8} = \frac{70}{8}\).

Step 4: Simplify the Improper Fraction

Finally, we simplify \(\frac{70}{8}\). Dividing both the numerator and the denominator by their greatest common divisor, which is 2, we get \(\frac{35}{4}\), and then convert it back to a mixed number: \(8 \frac{3}{4}\).

Key Concepts

Mixed Numbers to Improper Fractions

Mixed numbers are those that include both a whole number and a fraction, like \(7 \frac{1}{8}\), \(2 \frac{1}{4}\), and \(3 \frac{7}{8}\). To perform operations like addition or subtraction, we first need to convert these mixed numbers to improper fractions. This simplifies calculations and allows us to use regular fraction arithmetic.
To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction part.
• Add the result to the numerator of the fraction part.
• Write this sum over the original denominator.

For example, converting \(7 \frac{1}{8}\) involves multiplying 7 by 8, which gives 56, and then adding 1 to get 57. The improper fraction is therefore \(\frac{57}{8}\). Repeat similar steps for other mixed numbers in the expression to convert them to \(\frac{9}{4}\) and \(\frac{31}{8}\). By doing this, the arithmetic becomes much simpler.

Common Denominator

When subtracting fractions, they need to have the same denominator. This is called finding a common denominator. For instance, when simplifying \(\frac{9}{4} - \frac{31}{8}\), we notice \(9/4\) and \(31/8\) have different denominators.
To find a common denominator:

• Identify the denominators in the fractions.
• Find the least common multiple (LCM) of these numbers.
• Convert each fraction to an equivalent fraction with the LCM as the new denominator.

In our example, the LCM of 4 and 8 is 8. We multiply both the numerator and the denominator of \(\frac{9}{4}\) by 2 to get \(\frac{18}{8}\). Now both fractions have a common denominator of 8, allowing for straightforward subtraction. This process ensures the fractions can be easily compared or combined.

Simplifying Improper Fractions

Improper fractions have numerators larger than their denominators, which means they can be converted into a simpler form or back into a mixed number. In the previous operations, we got an improper fraction \(\frac{70}{8}\). Simplifying helps in organizing and simplifying expressions for clearer results.
To simplify an improper fraction:

• Find the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

For \(\frac{70}{8}\), the GCD is 2. Dividing both the numerator and the denominator by 2 results in \(\frac{35}{4}\). You can then convert \(\frac{35}{4}\) back into a mixed number by dividing 35 by 4, which results in \(8 \frac{3}{4}\). This final result is often easier to interpret and use in further calculations.

Adding and Subtracting Fractions

Adding and subtracting fractions is a common task in mathematics that requires attention to the denominators. In our problem, once you've reached the step \(\frac{57}{8} - (-\frac{13}{8})\), you've arranged everything nicely for operation.
The key points include:

• Ensure that fractions have a common denominator (as explained previously).
• When subtracting a negative, add instead (because \(a - (-b) = a + b\)).
• Add or subtract the numerators and keep the denominator unchanged.

So, in our case, we actually end up adding: \(\frac{57}{8} + \frac{13}{8} = \frac{70}{8}\). By practicing these steps consistently, you'll be able to tackle any combination of fractions with confidence. Remember, it's all about maintaining balance in the operations.