Problem 42 - Prealgebra

Question

Simplify each expression. $$\frac{7}{8}\left(5 \frac{3}{4}-2 \frac{1}{2}\right)$$

Step-by-Step Solution

Answer

The simplified expression is \(\frac{91}{32}\).

Step 1: Convert Mixed Numbers to Improper Fractions

Begin by converting the mixed numbers inside the parentheses into improper fractions. First, for \(5 \frac{3}{4}\), you multiply 5 by 4 and add 3, giving \(\frac{23}{4}\). Next, for \(2 \frac{1}{2}\), you multiply 2 by 2 and add 1, giving \(\frac{5}{2}\). The expression now becomes: \[\frac{7}{8}\left(\frac{23}{4} - \frac{5}{2}\right)\]

Step 2: Find Common Denominator

Identify the least common denominator for the fractions \(\frac{23}{4}\) and \(\frac{5}{2}\). The denominators are 4 and 2, so the least common denominator is 4. Change \(\frac{5}{2}\) to \(\frac{10}{4}\). The expression becomes: \[\frac{7}{8}\left(\frac{23}{4} - \frac{10}{4}\right)\]

Step 3: Subtract Fractions

Subtract the two fractions \(\frac{23}{4}\) and \(\frac{10}{4}\) since they now share a common denominator. This results in: \(\frac{13}{4}\). Therefore, the expression simplifies to: \[\frac{7}{8} \times \frac{13}{4}\]

Step 4: Multiply the Fractions

Multiply \(\frac{7}{8}\) by \(\frac{13}{4}\). Multiply the numerators together and the denominators together: \(7 \times 13 = 91\) and \(8 \times 4 = 32\). The product is \(\frac{91}{32}\).

Step 5: Simplify the Fraction

Check if \(\frac{91}{32}\) can be simplified. Since 91 and 32 have no common factors aside from 1, the fraction \(\frac{91}{32}\) is in its simplest form.

Key Concepts

Mixed Numbers

Understanding mixed numbers is the first step in many fraction exercises. A mixed number is a way of expressing numbers that are greater than one as a whole number and a proper fraction combined. For example, in the exercise, you see mixed numbers like \(5 \frac{3}{4}\) and \(2 \frac{1}{2}\). Here, \(5\) and \(2\) are whole numbers, and \(\frac{3}{4}\) and \(\frac{1}{2}\) are proper fractions.

When working with mixed numbers in arithmetic operations, it is often necessary to convert them to improper fractions. This is because improper fractions, where the numerator is larger than the denominator, make calculations such as addition, subtraction, and multiplication straightforward.

• Convert \(5 \frac{3}{4}\) by multiplying the whole number \(5\) by the denominator \(4\) and adding the numerator \(3\). This gives \(\frac{23}{4}\).
• Do the same for \(2 \frac{1}{2}\) to get \(\frac{5}{2}\).

Improper Fractions

Improper fractions are extremely useful for performing operations involving fractions. An improper fraction has a numerator larger than or equal to the denominator, like \(\frac{23}{4}\) and \(\frac{5}{2}\) from our example.

Once you have a mixed number converted into an improper fraction, you can easily perform various operations such as addition and subtraction. Why? Because improper fractions allow for easy manipulation of numbers without needing to convert back to mixed numbers until the final step, if desired.

• Improper fractions are helpful for multiplying fractions as they simplify the multiplication of the parts.
• They can be easily added or subtracted once a common denominator is found.

Common Denominator

Finding a common denominator is key to adding or subtracting fractions. Without a common denominator, fractions cannot be combined directly. In our example, the fractions inside the parentheses \(\frac{23}{4}\) and \(\frac{5}{2}\) must have the same denominator to perform subtraction.

To find the least common denominator (LCD), check the denominators of the fractions. Here \(4\) and \(2\) are the denominators. The LCD of these two numbers is \(4\). By converting \(\frac{5}{2}\) to \(\frac{10}{4}\), you now have identical denominators, making subtraction straightforward.

• Ensures fractions are compatible for addition or subtraction.
• Simplifies expressions into a form easy to manipulate with arithmetic operations.

Multiplying Fractions

Multiplying fractions involves multiplying the numerators together and multiplying the denominators together. This operation can often seem complex, but it's simplified by working with improper fractions. In our exercise, you carry out \( \frac{7}{8} \times \frac{13}{4}\).

To multiply these fractions:

• Multiply the numerators: \(7 \times 13 = 91\).
• Multiply the denominators: \(8 \times 4 = 32\).

The product of these fractions is \(\frac{91}{32}\).
After multiplying, it's always good practice to check if your result can be simplified. However, as in our example, \(\frac{91}{32}\) is already in its simplest form.

• Multiplication is straightforward: just multiply across for both numerators and denominators.
• Always check for simplification at the end to ensure the fraction is in its simplest state.