Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(15 \times \frac{2}{3}=\) ______

Short answer

The product is 10.

Step-by-step solution

Convert the Whole Number

To multiply, we first need to convert the whole number 15 into a fraction. This can be done by writing it as 15/1. This is because any whole number, like 15, can be represented as 15/1.

Set Up the Multiplication

Now, set up the multiplication by writing both fractions next to each other: \( \frac{15}{1} \times \frac{2}{3} \).

Multiply the Numerators

Multiply the numerators: 15 and 2. \( 15 \times 2 = 30 \). So, the new numerator is 30.

Multiply the Denominators

Multiply the denominators: 1 and 3. \( 1 \times 3 = 3 \). So, the new denominator is 3.

Write the Resulting Fraction

The result of the multiplication before simplification is \( \frac{30}{3} \).

Simplify the Fraction

To simplify, divide the numerator and the denominator by their greatest common divisor, which is 3 in this case. So, \( \frac{30}{3} = 10 \). Thus, the simplified fraction is \( 10 \).

Key concepts

Convert Whole Numbers to Fractions

When multiplying fractions, it's essential to convert whole numbers into fractions to make the process straightforward.
Whole numbers can be very easily expressed as fractions. You simply put the whole number over 1.
For instance, with the number 15, you would write it as \( \frac{15}{1} \). This transformation doesn't change the value of the number; instead, it allows us to multiply it by other fractions with ease.
This step is vital because fractions have two parts: a numerator and a denominator, which enables us to perform operations like multiplication.

Simplifying Fractions

Simplifying fractions means making them easier to use and understand by reducing them to their smallest form.
The process involves dividing the numerator and the denominator by their greatest common divisor (GCD). The GCD is the largest number that can divide both the numerator and the denominator evenly.

• For example, with the fraction \( \frac{30}{3} \), the GCD is 3.
• Divide both 30 and 3 by their GCD (3), which simplifies the fraction to \( \frac{10}{1} \).

After simplifying, you might find that the fraction becomes a whole number, as seen in this example, where \( \frac{10}{1} \) simplifies to 10.
Simplifying not only makes fractions easier to read but often helps in further calculations or applications.

Mixed Numbers Multiplication

Mixed numbers, such as 3 1/2, contain both a whole number and a fraction.
To multiply mixed numbers, first convert them to improper fractions. This involves multiplying the whole number by the fraction's denominator, then adding the numerator.
For instance, to convert 3 1/2 to an improper fraction, multiply 3 (whole number) by 2 (denominator), yielding 6. Add 1 (numerator) to get 7. Thus, 3 1/2 becomes \( \frac{7}{2} \).
Once converted, multiply the fractions using standard fraction multiplication: multiply the numerators together and the denominators together.
After solving, always check if the resulting fraction can be simplified. If needed, convert back to a mixed number for final presentation.