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Chapter 3: Problem 66

Simplify. $$ \frac{63}{75} $$

Short Answer

Expert verified
\frac{63}{75} = \frac{21}{25}

Step by step solution

01

- Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{63}{75}\), we first need to find the greatest common divisor (GCD) of the numerator and the denominator. Find the prime factorization of both numbers. \(\begin{aligned} 63 &= 3^2 \times 7 \ 75 &= 3 \times 5^2 \ \text{The GCD is } 3. \end{aligned}\)
02

- Divide Numerator and Denominator by the GCD

Next, divide both the numerator and the denominator by the GCD, which is 3. \[ \frac{63 \div 3}{75 \div 3} = \frac{21}{25} \]
03

- Simplified Fraction

The fraction is now simplified. \[ \frac{63}{75} = \frac{21}{25} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
When you're simplifying a fraction, the greatest common divisor (GCD) is super important. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder. To find the GCD, you can use prime factorization. This method involves breaking down both numbers into their prime factors.

For instance, let's look at simplifying \(\frac{63}{75}\). The prime factorization of 63 is \(\begin{aligned} 63 &= 3^2 \times 7 \ \text{and for } 75 &= 3 \times 5^2 \ \text{Thus, the GCD is } 3 \end{aligned} \). By dividing both the numerator and denominator by the GCD, we get \(\frac{63 \div 3}{75 \div 3} = \frac{21}{25}\).
Prime Factorization
Prime factorization means breaking a number down into its most basic building blocks. These are numbers that can only be divided by 1 and themselves, like 2, 3, 5, 7, 11, and so on. For example, to find the prime factors of 63, you keep dividing by prime numbers: \(\begin{aligned} 63 &= 3 \times 21 \ 21 &= 3 \times 7 \ \text{So, } 63 &= 3^2 \times 7 \end{aligned} \).

For 75, the steps are similar: \(\begin{aligned} 75 &= 3 \times 25 \ 25 &= 5 \times 5 \ \text{Thus, } 75 &= 3 \times 5^2\ \end{aligned} \). This breakdown helps to identify the GCD and simplify fractions.
Numerator and Denominator
In any fraction, the top number is called the numerator, while the bottom number is the denominator. For example, in the fraction \(\frac{63}{75}\), 63 is the numerator and 75 is the denominator.

The numerator tells you how many parts you have, and the denominator tells you how many equal parts the whole is divided into. Both terms are essential in simplifying fractions because simplifying involves finding common factors that can divide both the numerator and the denominator.

After finding the GCD (3 in this case), you divide both the numerator and denominator by it and get the simplified fraction \(\frac{21}{25}\).

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Most popular questions from this chapter

Consider only the numbers \(3,4,5,\) and 6 each can be placed in a blank in the following. $$ \square+\frac{\square}{\square} \cdot \square=? $$ What placement of the numbers in the blanks yields the largest sum?

Use a calculator to arrange the following in order from smallest to largest. $$ \frac{3}{4}, \frac{17}{21}, \frac{13}{15}, \frac{7}{9}, \frac{15}{17}, \frac{13}{12}, \frac{19}{22} $$

Estimate each of the following as \(0, \frac{1}{2},\) or 1 . $$ \frac{7}{15} $$

Subtract and simplify. $$ \frac{11}{12}-\frac{2}{12} $$

Subtract and simplify. $$ \frac{1}{8}-\frac{1}{12} $$
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