Find the quotient.
$\frac{-120}{-28}$
$\frac{-120}{-28}=\square$ (Type an integer or a simplified fraction.)
The problem asks to simplify the quotient of two negative numbers: \(\frac{-120}{-28}\). When dividing two negative numbers, the result is always positive because the negative signs cancel each other out, leaving us with the positive fraction \(\frac{120}{28}\).
To simplify the fraction \(\frac{120}{28}\), we need to find the greatest common divisor (GCD) of the numerator, 120, and the denominator, 28. We look for the largest number that divides both 120 and 28 evenly. The factors of 28 are 1, 2, 4, 7, 14, and 28. Testing these against 120, we find that 4 is the largest factor that divides both 120 and 28 (since \(120 = 4 \times 30\) and \(28 = 4 \times 7\)).
Now, we divide both the numerator and the denominator by their greatest common divisor, which is 4:
\[\frac{120 \div 4}{28 \div 4} = \frac{30}{7}.\]
Since 30 and 7 share no common factors other than 1, the fraction \(\frac{30}{7}\) is in its simplest form. Thus, \(\frac{-120}{-28} = \frac{30}{7}\).
Simplifying fractions is a fundamental skill in arithmetic used whenever you work with ratios or parts of a whole. For example, if you are measuring ingredients for a recipe and a conversion results in a complex ratio of measurements, simplifying it makes it much easier to measure accurately. Knowing how to reduce fractions helps maintain clarity and efficiency in everyday tasks like cooking, building, or managing finances.