Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(2 \times 4 \frac{3}{8}=\) ______

Short answer

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

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, convert the mixed number \(4 \frac{3}{8}\) into an improper fraction. Multiply the whole number \(4\) by the denominator \(8\) and add the numerator \(3\). This results in \((4 \times 8) + 3 = 32 + 3 = 35\). So, \(4 \frac{3}{8}\) becomes \(\frac{35}{8}\).

Convert Whole Number to Fraction

The whole number \(2\) can be rewritten as a fraction by placing it over \(1\). This gives us \(\frac{2}{1}\).

Multiply the Fractions

To multiply \(\frac{2}{1}\) by \(\frac{35}{8}\), multiply the numerators together and the denominators together. This gives \(2 \times 35 = 70\) for the numerator and \(1 \times 8 = 8\) for the denominator. Thus, the product is \(\frac{70}{8}\).

Simplify the Fraction

To simplify \(\frac{70}{8}\), find the greatest common divisor (GCD) of \(70\) and \(8\). The GCD is \(2\). Divide both the numerator and the denominator by \(2\) to get \(\frac{35}{4}\).

Convert to Mixed Number

Convert the improper fraction \(\frac{35}{4}\) to a mixed number by performing the division: \(35 \div 4\) is \(8\) with a remainder of \(3\). Thus, \(\frac{35}{4}\) is equivalent to \(8 \frac{3}{4}\).

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. It is a way to express amounts greater than a whole but not entirely fractional. For example, in the mixed number \(4 \frac{3}{8}\), "4" is the whole number, and "\(\frac{3}{8}\)" is the fraction. This means there are 4 complete items, and an additional part of an item, described by the fraction.

Mixed numbers can be very useful in situations where you want to express a quantity that consists of both wholes and parts. However, when performing operations like multiplication, they are often converted to improper fractions for simplicity. This conversion allows for easier mathematical manipulations and calculations.

Improper Fractions

Improper fractions are fractions where the numerator is greater than or equal to the denominator. This means they represent values equal to or greater than one whole. For instance, converting the mixed number \(4 \frac{3}{8}\) into an improper fraction gives you \(\frac{35}{8}\).

To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. This brings the entire value into fractional form, making operations such as addition, subtraction, or multiplication more straightforward. This is because fractions are more directly compatible with mathematical operations than mixed numbers.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its smallest form, where the numerator and the denominator have no common divisors other than one. This simplifies calculations and makes numbers easier to work with.

To simplify \(\frac{70}{8}\), we look for the greatest common divisor (GCD) of 70 and 8, which is 2. We then divide both the numerator and the denominator by the GCD: \(\frac{70 \div 2}{8 \div 2} = \frac{35}{4}\). This gives you a fraction that is simple to read and understand.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that can divide both without leaving a remainder. It plays a crucial role in simplifying fractions, making them easier to use in calculations and to understand intuitively.

When simplifying the fraction \(\frac{70}{8}\), identifying the GCD helps to bring it down to its simplest form, \(\frac{35}{4}\). In this case, the GCD was 2, the largest number that divides both 70 and 8 evenly.

• Finding the GCD can often involve listing the factors of each number and finding the largest common one.
• There are also algorithmic methods, such as the Euclidean algorithm, to efficiently find the GCD for larger numbers.