Question

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

Short answer

\(\frac{1}{2}\)

Step-by-step solution

Convert Mixed Numbers to Fractions

There are no mixed numbers in the given problem, so we can skip this step in this case.

Multiply the Numerators

To multiply two fractions, you first multiply the numerators (the top numbers of the fractions). So, calculate the product of the numerators: \(2 \times 5 = 10\).

Multiply the Denominators

Now, multiply the denominators (the bottom numbers of the fractions):\(5 \times 4 = 20\).

Form the Product Fraction

Combine the results of the previous two steps to form a new fraction:\(\frac{10}{20}\). This fraction combines the multiplied numerators and denominators.

Simplify the Fraction

To simplify, find the greatest common divisor (GCD) of 10 and 20. The GCD is 10. Divide both the numerator and the denominator by 10:\(\frac{10}{20} = \frac{1}{2}\).

Key concepts

Numerators

The numerator is a key component in any fraction, sitting above the fraction's line. In a fraction such as \(\frac{2}{5}\), the number 2 is the numerator. It tells us how many parts we have out of a whole that is divided by the denominator. When multiplying fractions, you focus first on the numerators. Just multiply them straight across. No fancy tricks here—simply multiply the top parts of each fraction. For the fractions \(\frac{2}{5}\) and \(\frac{5}{4}\), multiply the numerators: \(2 \times 5\) equals 10. That's it! The product of the numerators becomes the numerator of the resulting fraction.

Denominators

In a fraction, the denominator is beneath the line and shows how many equal parts the whole is divided into. If you look at \(\frac{2}{5}\), the 5 is the denominator. Like kicking a soccer ball through an open goalpost, in fraction multiplication, just multiply the denominators together. For our exercise with \(\frac{5}{4}\), you multiply: \(5 \times 4\), which is 20. This result is the denominator of your new fraction. Remember, ultimately this number serves as the partition of equal segments in the fraction.

Simplifying Fractions

Simplifying fractions is like tidying up a room; removing unnecessary clutter gives a clearer view. Once you've multiplied both numerators and denominators, you might have a fraction that's not in its simplest form. Simplifying means making the fraction as small as possible while keeping its value unchanged. After forming \(\frac{10}{20}\), you notice it's not fully reduced. To make things neat and simple, divide both the numerator and the denominator by their greatest common divisor.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest whole number that divides both the numerator and the denominator without leaving a remainder. It's like finding the common thread in two different yet related fabrics. For \(\frac{10}{20}\), the GCD is 10. Identify it by listing factors of both the numerator and denominator and picking the biggest one they share.

• Factors of 10: 1, 2, 5, 10
• Factors of 20: 1, 2, 4, 5, 10, 20

The GCD is the highest number they have in common, here being 10. Once we've identified this, reduce the fraction. Divide both the top and bottom by 10 to get \(\frac{1}{2}\), the simplest form of your product after multiplying.