If Mark can mow \(\frac{2}{3}\) of a lawn in 1 hour, how many hours does it take him to mow the entire lawn? A. \(\frac{2}{3}\) B. \(1 \frac{1}{3}\) C. \(1 \frac{1}{2}\) D. 2

To grasp proportion problems, it's essential to know that a proportion is essentially an equation that states two ratios are equivalent. In the context of the given exercise, the ratio \(\frac{2}{3}\) of the lawn is mowed in 1 hour is a part-to-whole ratio comparing the part of the lawn that is mowed to the whole lawn. To solve proportion problems, you can set up two ratios that compare corresponding parts: the one given and the one you're trying to find.

One common method for solving proportions is 'cross-multiplication', where the numerator of one ratio is multiplied by the denominator of the other, and these products are set equal to each other. It's a powerful technique because it reduces the problem to a simple equation that can be solved typically with basic algebra. For students tackling proportion problems, remember it's like solving a puzzle where you find the missing piece by ensuring both sides of the equation balance.

Algebraic fractions are simply fractions that include variables, such as \(\frac{x}{y}\). When you're solving equations that have fractions, you often need to find a common denominator, cross-multiply, or sometimes multiply each side of the equation by the least common multiple (LCM) to eliminate the fractions.

In our exercise, we didn't have to find a common denominator because we set up a proportion involving a known fraction and a variable. By cross-multiplying, the problem simplifies to a linear equation that's solvable for the variable. It's crucial for students to practice various algebraic fraction problems to become comfortable with these techniques so that they can apply them confidently in different scenarios.

Improper fractions are fractions where the numerator (top number) is larger than the denominator (bottom number). Unlike proper fractions, they represent a quantity larger than one whole unit. Converting an improper fraction to a mixed number involves a division process where the numerator is divided by the denominator.

The quotient becomes the whole number part, and the remainder is the new numerator of the fractional part, with the denominator remaining the same. For instance, to convert \(\frac{3}{2}\) to a mixed number, you divide 3 by 2. The quotient is 1, and the remainder is 1. So, \(\frac{3}{2}\) becomes \(1\frac{1}{2}\).

Understanding how to convert between these forms is crucial for comparing and estimating sizes of fractions, particularly in real-world problems like the one provided with Mark and his lawn mowing task. Students should practice this skill to improve their versatility in handling fractions in various contexts.