If \(3 \frac{3}{5}=x+2 \frac{2}{3},\) then \(x=?\) F. \(\frac{4}{5}\) G. \(\frac{14}{15}\) H. 1\(\frac{1}{2}\) J. 1\(\frac{6}{15}\) K. 6\(\frac{4}{15}\)

Improper fractions are a key part of understanding how to work with different types of numerical expressions, especially when dealing with mixed numbers. An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). This means that the value of the fraction is greater than or equal to one.

For instance, let's take the number \(3 \frac{3}{5}\) from our example problem. To convert this mixed number into an improper fraction, you multiply the whole number by the denominator and then add the numerator. Here's the math: \(3 \times 5 + 3 = 15 + 3 = 18\), so the improper fraction becomes \(\frac{18}{5}\).

This conversion is essential for performing various operations such as addition, subtraction, multiplication, and division with mixed numbers.

When subtracting fractions, it is crucial to use a least common denominator (LCD), which is the smallest number that can be divided by each of the denominators of the fractions without leaving a remainder. The LCD is essentially the least common multiple (LCM) of the fractions' denominators.

In our ACT math problem, to subtract \(\frac{18}{5}\) from \(\frac{8}{3}\), we first identify the LCM of 5 and 3, which is 15. Having a common denominator of 15, we can then equate the value of both fractions to fifteenths. This step simplifies the problem to a simple subtraction of two fractions with the same denominator. Remembering to find the LCD is a crucial step to solve any problem involving the subtraction (or addition) of fractions.

Subtracting fractions can be challenging for many students, but it becomes much easier once you remember to use the least common denominator. Once both fractions are expressed with the same denominator, you simply subtract the numerators and keep the denominator the same.

Following the steps from our exercise, once \(\frac{18}{5}\) and \(\frac{8}{3}\) are both expressed with a denominator of 15, we get \(\frac{54}{15}\) and \(\frac{40}{15}\). Subtracting these gives us \(x = \frac{54}{15} - \frac{40}{15} = \frac{14}{15}\). This is possible because with a common denominator, fractions represent parts of the same size whole, so you can directly compare and subtract them.