Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{3}{5} \times 3 \frac{1}{8}=\) ______

Short answer

\(\frac{15}{8}\) or 1 \(\frac{7}{8}\) as a mixed number.

Step-by-step solution

Convert the Mixed Number to an Improper Fraction

To multiply the fractions, first convert the mixed number \(3 \frac{1}{8}\) to an improper fraction. Multiply the whole number \(3\) by the denominator \(8\) and add the numerator \(1\): \((3 \times 8) + 1 = 24 + 1 = 25\), so \(3 \frac{1}{8} = \frac{25}{8}\).

Multiply the Fractions

Multiply \(\frac{3}{5}\) by \(\frac{25}{8}\) by multiplying the numerators and denominators: \(\frac{3 \times 25}{5 \times 8} = \frac{75}{40}\).

Simplify the Fraction

To simplify \(\frac{75}{40}\), find the greatest common divisor (GCD) of 75 and 40. The GCD is 5. Divide both the numerator and the denominator by 5: \(\frac{75 \div 5}{40 \div 5} = \frac{15}{8}\).

Confirm the Fraction is in Lowest Terms

Ensure \(\frac{15}{8}\) cannot be simplified further. The numerator 15 and the denominator 8 share no common factors other than 1. Therefore, \(\frac{15}{8}\) is already in its lowest terms.

Key concepts

Mixed Numbers

Mixed numbers are fractions that include both a whole number and a proper fraction combined together. For example, in the number \(3 \frac{1}{8}\), \(3\) is the whole number, and \(\frac{1}{8}\) is the fractional part. This means you have three whole parts and one-eighth more.
To multiply mixed numbers with other fractions, it is usually easier to convert them into improper fractions first. This is because multiplication is simpler when you only have fractions to deal with, without the added complexity of a whole number.
Conversion involves multiplying the whole number by the denominator of the fraction, then adding the numerator. This gives us all the parts termed in the fractional format, making calculations straightforward.

Improper Fractions

An improper fraction has a numerator that is larger than, or equal to, its denominator. This occurs when a fraction is expressing a quantity greater than one.

• For example, \(\frac{25}{8}\) is an improper fraction because 25, the numerator, is greater than 8, the denominator.
• Improper fractions often arise when converting mixed numbers as they capture the whole number into fractional parts.

Improper fractions allow us to perform arithmetic operations, like multiplication, more fluidly.
During multiplication, we simply multiply across the numerators and the denominators. Converting back to mixed numbers might be needed afterwards, depending on the problem requirements.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form. This means you should have the numerator and the denominator as small as possible while maintaining the same value of the fraction.
It is essential to spot common factors between the numerator and the denominator. You divide them both by their greatest common divisor (GCD).
In our problem, \(\frac{75}{40}\) can be simplified by dividing both numbers by 5, resulting in \(\frac{15}{8}\). Simplifying makes fractions easier to work with and understand.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both without leaving a remainder. It is a crucial concept when simplifying fractions as it helps find the simplest form of a fraction.

• To find the GCD of a fraction's numerator and denominator, like 75 and 40, you list the factors of each number and find the largest common one.
• In our example, the GCD is 5, so dividing both the numerator and denominator by 5 simplified the fraction \(\frac{75}{40}\) to \(\frac{15}{8}\).

By utilizing the GCD, we ensure the fraction is in its simplest form, making it comprehensible and manageable.