Question

Multiply the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{1}{2} \times \frac{3}{4} \times \frac{3}{5}=\) ______

Short answer

\(\frac{9}{40}\) is the final answer.

Step-by-step solution

Multiply the Numerators

First, multiply the numerators (the top numbers) of the fractions together. The numerators are 1, 3, and 3. Calculate the product as follows: \(1 \times 3 \times 3 = 9\).

Multiply the Denominators

Next, multiply the denominators (the bottom numbers) of the fractions together. The denominators are 2, 4, and 5. Calculate the product as follows: \(2 \times 4 \times 5 = 40\).

Form the New Fraction

Now, use the results from the previous steps to form a new fraction. The new fraction is \(\frac{9}{40}\) because the numerator is 9 and the denominator is 40.

Reduce to Lowest Terms

Check if \(\frac{9}{40}\) can be reduced further by finding the greatest common divisor (GCD) of 9 and 40. Since 9 and 40 have no common factors other than 1, \(\frac{9}{40}\) is already in its simplest form.

Key concepts

Numerators and Denominators

When working with fractions, two key elements must always be considered: numerators and denominators. But what exactly are they? In a fraction, the numerator is the top number. It tells you how many parts of the whole or set you're dealing with. For example, in the fraction \( \frac{1}{2} \), the numerator is 1, indicating one part out of a total of two.

The denominator is the bottom number, which gives you the total number of equal parts the whole is divided into. Returning to \( \frac{1}{2} \), the denominator is 2, showing that the whole is split into two equal sections. This basic understanding allows you to perform essential operations like addition, subtraction, and as we focus on here, multiplication of fractions.

When multiplying fractions, you take the numerators of each fraction and multiply them together. This calculated result becomes the new numerator for the resulting fraction. Similarly, the denominators are multiplied to determine the denominator of the resulting fraction.

• Example: For \( \frac{1}{2} \times \frac{3}{4} \times \frac{3}{5} \), multiply the numerators: \( 1 \times 3 \times 3 = 9 \).
• Multiply the denominators: \( 2 \times 4 \times 5 = 40 \).

Understanding numerators and denominators is foundational to mastering fractions.

Mixed Numbers

Fractions can also appear as mixed numbers. These are numbers that combine a whole number with a fraction, like 2\( \frac{1}{3} \). Understanding mixed numbers is crucial when you're faced with more complex multiplication problems.

Mixed numbers require conversion to improper fractions before performing multiplication. An improper fraction has a numerator larger than its denominator. To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fractional part.
• Add the numerator of the fractional part to this product.
• Use the result as your new numerator while keeping the original denominator.

Let's say you have 2\( \frac{1}{3} \). Convert it to an improper fraction:

• Multiply 2 by 3, which equals 6.
• Add 1, making it 7.
• Resulting in \( \frac{7}{3} \).

With this transformation, fractions, whether simple or mixed, can be multiplied efficiently.

Simplifying Fractions

After multiplying fractions, you'll often need to simplify the result, a process known as reducing to its lowest terms. Simplifying a fraction means making the numerator and denominator as small as possible without changing their ratio.

To simplify a fraction, find the greatest common divisor (GCD) of the numerator and denominator. Divide both by this number. For example, in the fraction \( \frac{18}{24} \):

• Find the GCD. Here, it's 6.
• Divide both 18 and 24 by 6 to get \( \frac{3}{4} \).

This gives you the simplest form.

For \( \frac{9}{40} \), the GCD is 1, meaning \( \frac{9}{40} \) cannot be simplified further. Always verify that no number apart from 1 can divide both your numerator and denominator, ensuring the fraction is in its simplest form.

Simplifying not only makes fractions easier to understand, but it is also often required in mathematical problems to achieve the correct answer in its most concise form.