Problem 9 - Prealgebra

Question

Write your answers as proper fractions or mixed numbers, not as improper fractions. Find the following products. (Multiply.) $$2 \cdot 4 \frac{7}{8}$$

Step-by-Step Solution

Answer

The product is \( 9 \frac{3}{4} \).

Step 1: Convert Mixed Number to Improper Fraction

First we need to convert the mixed number \( 4 \frac{7}{8} \) into an improper fraction. The whole number is 4 and the fraction part is \( \frac{7}{8} \). To convert, multiply the whole number by the denominator and add the numerator: \( 4 \times 8 + 7 = 32 + 7 = 39 \). Thus, \( 4 \frac{7}{8} \) is equivalent to \( \frac{39}{8} \).

Step 2: Multiply Fractions

Now we'll multiply \( 2 \) by \( \frac{39}{8} \). Since \( 2 \) can be written as \( \frac{2}{1} \), we multiply the numerators and denominators: \( \frac{2}{1} \times \frac{39}{8} = \frac{2 \times 39}{1 \times 8} = \frac{78}{8} \).

Step 3: Simplify the Improper Fraction

The fraction \( \frac{78}{8} \) can be simplified by dividing both the numerator and the denominator by the greatest common divisor. The GCD of 78 and 8 is 2. Thus, \( \frac{78}{8} \) simplifies to \( \frac{78 \div 2}{8 \div 2} = \frac{39}{4} \).

Step 4: Convert Improper Fraction to Mixed Number

Convert the improper fraction \( \frac{39}{4} \) to a mixed number. Divide 39 by 4: 39 divided by 4 is 9 with a remainder of 3. Therefore, \( \frac{39}{4} \) as a mixed number is \( 9 \frac{3}{4} \).

Key Concepts

Converting Mixed Numbers

When you encounter mixed numbers, like \( 4 \frac{7}{8} \), it's often necessary to convert them to improper fractions to simplify calculations, especially multiplication. To do this, you *multiply* the whole number part by the fraction's denominator and then *add* the numerator.
So for \( 4 \frac{7}{8} \), multiply 4 by 8 (the denominator) resulting in 32, then add the 7 from the numerator, giving you 39.
Thus, \( 4 \times 8 + 7 = 39 \). So, \( 4 \frac{7}{8} \) becomes \( \frac{39}{8} \).
By converting mixed numbers this way, their true value can be used directly in multiplication.

Improper Fractions

Improper fractions can seem a bit daunting at first. They are fractions where the numerator (top number) is larger than the denominator (bottom number).
These types of fractions are particularly useful in calculations because they are often easier to manipulate mathematically.
Take our example with \( \frac{39}{8} \). Here, the numerator, 39, is larger than 8, the denominator.
When performing operations like multiplication, improper fractions are preferable because they eliminate the intermediate step of dealing with whole numbers separately.

• A key point is that they can be easily converted back to mixed numbers if needed.
• They can also be part of a larger solution involving further arithmetic operations.

Simplifying Fractions

Once you have multiplied or calculated using fractions, it's a good idea to simplify your result for clarity. Simplifying means reducing the fraction to its smallest form.
For example, using \( \frac{78}{8} \), find the greatest common divisor (GCD) of 78 and 8 to simplify.
The GCD is the largest number that divides both the numerator and the denominator without a remainder.
In this case, 2 is the GCD. So, divide both 78 and 8 by 2:

• \( 78 \div 2 = 39 \)
• \( 8 \div 2 = 4 \)

This simplifies \( \frac{78}{8} \) to \( \frac{39}{4} \). Simplifying ensures our answers are as neat and easy to interpret as possible.

Fraction Multiplication

Fraction multiplication involves two straightforward steps: multiply the numerators together and then multiply the denominators together.
In our case, we had \( 2 \times \frac{39}{8} \), which can be seen as multiplying two fractions: \( \frac{2}{1} \times \frac{39}{8} \).
Here are the key steps:

• Multiply the numerators: \( 2 \times 39 = 78 \)
• Multiply the denominators: \( 1 \times 8 = 8 \)
• Resulting in the fraction \( \frac{78}{8} \)

It's a simple process but requires accuracy. Ensure all numbers are correct, and remember to simplify afterward if possible. With practice, fraction multiplication becomes a quick and useful tool for many math problems.