Find the quotient.
$\frac{30}{42}$
$\frac{30}{42}=\square$ (Type an integer or a simplified fraction.)
To simplify the fraction \(\frac{30}{42}\), we must find a common factor for both the numerator (30) and the denominator (42). A fraction is in its simplest form when its numerator and denominator share no common factors other than 1. Dividing both by their greatest common divisor (GCD) achieves this.
First, we identify the factors of the numerator, 30:
1, 2, 3, 5, 6, 10, 15, 30.
Next, we identify the factors of the denominator, 42:
1, 2, 3, 6, 7, 14, 21, 42.
The common factors are 1, 2, 3, and 6. The greatest common divisor (GCD) is 6.
Now, divide both the numerator and the denominator by their GCD, which is 6:
Numerator: \(\frac{30}{6} = 5\)
Denominator: \(\frac{42}{6} = 7\)
Thus, the simplified fraction is \(\frac{5}{7}\).
Simplifying fractions is a fundamental skill used in everyday life, such as when following a recipe that needs to be scaled down or when managing proportions in a project. For instance, if a recipe calls for 42 minutes of baking time, and you are only making a portion that requires \(\frac{30}{42}\) of that time, simplifying this fraction to \(\frac{5}{7}\) helps you understand the ratio of the time needed more clearly. Mastering this helps in making quick and accurate decisions when dealing with parts of a whole.