Problem 70 - Prealgebra

Question

Find each of the following squares, and write your answers as mixed numbers. $$\left(2 \frac{3}{4}\right)^{2}$$

Step-by-Step Solution

Answer

The square of \(2 \frac{3}{4}\) is \(7 \frac{9}{16}\).

Step 1: Convert the Mixed Number to an Improper Fraction

To solve \( \left(2 \frac{3}{4}\right)^{2} \), first convert the mixed number, \(2 \frac{3}{4}\), into an improper fraction. Multiply the whole number part (2) by the denominator (4) and add the numerator (3): \[ 2 \times 4 + 3 = 8 + 3 = 11 \] This gives \( \frac{11}{4} \).

Step 2: Square the Improper Fraction

Now square the improper fraction \( \frac{11}{4} \). This means you square both the numerator and the denominator: \[ \left(\frac{11}{4}\right)^2 = \frac{11^2}{4^2} = \frac{121}{16} \].

Step 3: Convert the Result to a Mixed Number

Convert the improper fraction \(\frac{121}{16}\) to a mixed number. Divide the numerator by the denominator: \(121 \div 16\) gives 7 with a remainder of 9. So, the mixed number is \(7 \frac{9}{16}\).

Key Concepts

Mixed Numbers

Mixed numbers are numbers made up of a whole number and a proper fraction combined. Imagine you have 2 full pizzas and three-quarters of another pizza. In this case, you would have a mixed number: \(2 \, \frac{3}{4}\).
Mixed numbers are a great way to represent numbers that are not whole but include both whole and fractional parts.

• The whole number (like 2 in \(2 \frac{3}{4}\)) is the easy bit to understand; it's simply 2 full parts.
• The fraction (like \(\frac{3}{4}\) in \(2 \frac{3}{4}\)) means you have a part of something whole—in this case, three of the four equal parts (or quarters).

Using mixed numbers can make real-world understanding easier, like if you're cooking or dividing things up. Converting them into improper fractions, however, often becomes necessary for mathematical calculations like addition, subtraction, multiplication, or squaring of fractions.

Improper Fractions

An improper fraction is a type of fraction where the numerator (the top number) is larger than or equal to the denominator (the bottom number).
Improper fractions, while not always intuitive in day-to-day contexts, are very useful in various mathematical operations. Converting a mixed number into an improper fraction makes calculations simpler because they allow us to work with just one fraction instead of a whole and a part.
To convert a mixed number into an improper fraction:

• Multiply the whole number by the denominator.
• Add the numerator to that product.
• Place the sum over the original denominator.

For example, to convert \(2 \frac{3}{4}\), multiply 2 by 4 to get 8, then add 3 to get 11, resulting in an improper fraction of \(\frac{11}{4}\). These steps might take a little practice, but once you are familiar, it becomes an easy walk in the park.

Fraction Conversion

Fraction conversion may involve changing a mixed number to an improper fraction, or vice versa. Knowing how to switch between these two forms will greatly broaden your skills in managing and solving fraction problems. To convert an improper fraction back to a mixed number, you do the reverse:

• Divide the numerator by the denominator to find the whole number part.
• Use the remainder as the new numerator for the fractional part.
• Keep the same denominator.

In our example of squaring \(2 \frac{3}{4}\), we ended up with the improper fraction \(\frac{121}{16}\). Dividing 121 by 16 gives 7, with a remainder of 9, so it converts to the mixed number \(7 \frac{9}{16}\). Understanding these conversions is essential, as they help simplify solutions and allow easier arithmetic computations.