Question

Evaluate each expression. $$ \frac{6}{25} \cdot \frac{10}{27} $$

Short answer

\( \frac{4}{45} \)

Step-by-step solution

- Multiply the Numerators

First, multiply the numerators of the fractions together. The numerators are 6 and 10.\[ 6 \times 10 = 60 \]

- Multiply the Denominators

Next, multiply the denominators of the fractions together. The denominators are 25 and 27.\[ 25 \times 27 = 675 \]

- Form the New Fraction

Combine the results from Step 1 and Step 2 to form a new fraction.\[ \frac{60}{675} \]

- Simplify the Fraction

Simplify the fraction \( \frac{60}{675} \). Find the greatest common divisor (GCD) of 60 and 675. The GCD of 60 and 675 is 15.Divide the numerator and denominator by their GCD:\[ \frac{60 \div 15}{675 \div 15} = \frac{4}{45} \]

Key concepts

simplifying fractions

Simplifying fractions means making the fraction as simple as possible. This process involves reducing the fraction to its smallest form without changing its value.
For instance, if you start with the fraction \(\frac{60}{675}\), you need to find a way to simplify it.
The process of simplification usually involves dividing both the numerator and the denominator by the same number known as the greatest common divisor (GCD).
After simplification, the fraction should be in its most reduced form.

greatest common divisor

The greatest common divisor, or GCD, is the largest number that can divide both the numerator and the denominator without leaving a remainder.
To simplify a fraction, you first need to find the GCD of the two numbers involved.
In our example, the GCD of 60 and 675 is 15.
We find this by listing the factors of each number and then identifying the largest common factor, or by using the Euclidean algorithm.

numerators and denominators

A fraction is made up of two parts: the numerator and the denominator.
The numerator is the top part of the fraction, representing how many parts we have.
The denominator is the bottom part, representing the total number of equal parts the whole is divided into.
For example, in the fraction \(\frac{6}{25}\), 6 is the numerator and 25 is the denominator.
In fraction multiplication, we multiply the numerators together and the denominators together to get a new fraction.

fraction reduction

Fraction reduction is the process of simplifying a fraction into its simplest form.
To reduce a fraction, you divide both the numerator and the denominator by their greatest common divisor (GCD).
For instance, if you have \(\frac{60}{675}\), you find the GCD, which is 15, and divide both numbers by 15.
This gives you \(\frac{4}{45}\), which is the reduced form of the original fraction.