Problem 69 - Prealgebra

Question

Find each of the following squares, and write your answers as mixed numbers. $$\left(1 \frac{3}{4}\right)^{2}$$

Step-by-Step Solution

Answer

\((1 \frac{3}{4})^2 = 3 \frac{1}{16}\).

Step 1: Convert to Improper Fraction

First, we convert the mixed number \(1 \frac{3}{4}\) into an improper fraction. A mixed number is composed of a whole number and a fraction. We multiply the whole number (1) by the denominator of the fraction (4) and add the numerator of the fraction (3): \(1 \times 4 + 3 = 7\). Thus, the mixed number \(1 \frac{3}{4}\) becomes \(\frac{7}{4}\).

Step 2: Square the Improper Fraction

Next, we square the improper fraction \(\frac{7}{4}\). To do this, we square both the numerator and the denominator separately: \(\left(\frac{7}{4}\right)^2 = \frac{7^2}{4^2} = \frac{49}{16}\).

Step 3: Convert to Mixed Number

Now we convert the improper fraction \(\frac{49}{16}\) back into a mixed number. We divide the numerator (49) by the denominator (16) to determine the whole part. This division gives us 3 with a remainder of 1 (since \(49 \div 16 = 3 R1\)). Therefore, the mixed number is \(3 \frac{1}{16}\).

Key Concepts

Improper Fractions

Improper fractions are fractions where the numerator (top number) is larger than or equal to the denominator (bottom number). This means the value of the fraction is greater than or equal to 1. Understanding improper fractions is essential because they often appear when performing operations with mixed numbers or when the result of multiplying fractions results in a larger numerator.

• To convert a mixed number into an improper fraction, multiply the whole number by the denominator and then add the numerator. This total becomes the new numerator, while the denominator remains the same.
• For example, when converting the mixed number \(1 \frac{3}{4}\) to an improper fraction, multiply 1 (the whole number) by 4 (the denominator), then add 3 (the numerator): \(1 \times 4 + 3 = 7\). So, \(1 \frac{3}{4}\) is \(\frac{7}{4}\).

Learning to convert between mixed numbers and improper fractions is a handy skill that helps simplify complex math tasks smoothly.

Mixed Numbers

Mixed numbers consist of both a whole number and a proper fraction. They are often used in everyday measurements and calculations because they clearly separate the whole part from the fractional part.

• Understanding mixed numbers is crucial for operations like addition, subtraction, and multiplication involving fractions, especially when you need the result presented in a way that is easy to understand.
• In mathematics, after performing operations on fractions, you might end up with an improper fraction. Converting it to a mixed number can simplify the final answer, making it more intuitive.

For example, in the square of the mixed number \(1 \frac{3}{4}\), expressing the final result as \(3 \frac{1}{16}\) makes the outcome more understandable than in an improper fraction format like \(\frac{49}{16}\).

Fraction Conversion

Fraction conversion is the process of changing a fraction from one form to another, such as from a mixed number to an improper fraction, or vice versa. This is a fundamental operation necessary for many mathematical tasks.

• Starting with a mixed number, you convert it to an improper fraction for multiplication or division tasks because improper fractions are easier to manipulate mathematically.
• After performing calculations, convert the result back to a mixed number to present a clearer answer, particularly when dealing with square fractions or determining actual quantities.

Knowing how to convert fractions seamlessly ensures accuracy and fluency, reducing errors in both computation and presentation. This skill is especially useful when dealing with problems like squaring mixed numbers, where conversion is a key part of the solution process.