Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(8 \times 1 \frac{3}{4}=\) ______

Short answer

14

Step-by-step solution

Convert Mixed Number to Improper Fraction

The first task is to convert the mixed number \(1 \frac{3}{4}\) into an improper fraction. A mixed number consists of a whole number and a fraction. To do this, multiply the whole number by the denominator, and add the numerator.\[1 \times 4 + 3 = 7\]So, \(1 \frac{3}{4}\) becomes \(\frac{7}{4}\).

Set Up the Multiplication of Fractions

Now that both numbers are in fraction form, we can set up the multiplication. The number 8 can be written as a fraction as well. Any whole number \(a\) can be written as \(\frac{a}{1}\). So we write 8 as \(\frac{8}{1}\).The equation now looks like this:\[\frac{8}{1} \times \frac{7}{4}\]

Multiply the Fractions

To multiply fractions, multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. Multiply:\[8 \times 7 = 56\]\[1 \times 4 = 4\]So, \(\frac{8}{1} \times \frac{7}{4} = \frac{56}{4}\).

Simplify the Fraction

The last step is to simplify the fraction \(\frac{56}{4}\). To do this, divide the numerator by the denominator:\[\frac{56}{4} = 14\]So, the answer is 14.

Key concepts

Improper Fractions

Improper fractions are fractions where the numerator (the top part) is equal to or larger than the denominator (the bottom part). This means there is more of the part in relation to the whole being represented. It is a key concept in fraction multiplication, especially when trying to combine different types of numbers.

• To understand improper fractions, think about a pizza. An improper fraction like \(\frac{7}{4}\) means you have 7 slices out of a whole that is divided into 4 slices. This translates into more than a whole pizza.
• Improper fractions are often used when converting mixed numbers to make multiplication simpler, as it turns everything into a straightforward fraction format.
• Starting with any fraction task, recognize if you need to convert to an improper fraction to make operations easier, especially for multiplication and addition.

Recognizing and converting into improper fractions is a crucial skill that helps in manipulating and solving equations efficiently.

Mixed Numbers Conversion

Converting mixed numbers into improper fractions is an essential skill when dealing with operations like multiplication. A mixed number consists of a whole number and a fraction, like in the example of \(1 \frac{3}{4}\). Here's how to convert:

• First, multiply the whole number by the denominator of the fraction. For our example, multiply 1 by 4 to get 4.
• Then, add the numerator of the fraction to this product. So, add 3 to 4 to get 7.
• This gives you the new improper fraction, which is \(\frac{7}{4}\), instead of the original mixed number \(1 \frac{3}{4}\).

This method allows for easier multiplication. Operating without mixed numbers avoids confusion and permits straightforward arithmetic where you simply work with the numerators and denominators.

Simplifying Fractions

Simplifying fractions, often referred to as reducing, is the process of making the fraction as simple as possible. Simplified fractions are the most reduced form where the numerator and the denominator have no common factors other than 1. This always results in the smallest possible whole numbers.

• To simplify a fraction such as \(\frac{56}{4}\), you will need to divide both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 56 and 4 is 4.
• Divide 56 by 4 to get 14, and 4 by 4 to get 1. This gives you \(\frac{14}{1}\), which simplifies to just 14. Simplification helps express the result in the simplest, most understandable format.
• Remember, a simplified fraction does not change the value of the fraction, it's just shorter and more straightforward. Always check to see if there are factors you can divide out to help make handling fractions less cumbersome.

By mastering simplifying, you'll make computation simpler and clearer, making your work results polished and correct.