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Chapter 3: Problem 64

Write each fraction in lowest terms. $$ \frac{140}{315} $$

Short Answer

Expert verified
\( \frac{4}{9} \)

Step by step solution

01

- Identify the Greatest Common Divisor (GCD)

The first step in simplifying a fraction is to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder. For the fraction \( \frac{140}{315} \), list the factors of 140 and 315 to find the GCD.
02

- List the Factors

List the factors of 140: 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, 140. List the factors of 315: 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, 315.
03

- Determine the GCD

From the lists of factors in the previous step, identify the common factors: 1, 5, 7, 35. The greatest common factor is 35.
04

- Simplify the Fraction

Divide both the numerator and the denominator by the GCD. Simplify: \[ \frac{140 \div 35}{315 \div 35} = \frac{4}{9} \]

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Key Concepts

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

Greatest Common Divisor
When simplifying fractions, the Greatest Common Divisor (GCD) is a crucial concept. The GCD is the largest number that can perfectly divide both the numerator (top number) and the denominator (bottom number) of a fraction. It helps us find the simplest form of a fraction. For example, if we simplify the fraction \( \frac{140}{315} \), we need to find the GCD of 140 and 315 first. By listing out all the factors of each number, we can find the highest common one, which is our GCD. In this case, the factors of 140 and 315 include 35, which is the largest common factor, hence the GCD is 35.
factors
Finding factors is an important step in reducing fractions. Factors are the numbers you can multiply together to get another number. For instance, the factors of 140 are 1, 2, 4, 5, 7, 10, 14, 20, 28, 35, 70, and 140. Similarly, the factors of 315 are 1, 3, 5, 7, 9, 15, 21, 35, 45, 63, 105, and 315. Notice that the factors include both 1 and the number itself. In the fraction \( \frac{140}{315} \), we look for common factors between 140 and 315. The shared factors from both lists are 1, 5, 7, and 35. The largest one, 35, is the one we use to simplify the fraction.
lowest terms
Once you have the GCD, you can simplify your fraction to its lowest terms. This means you divide both the numerator and the denominator by their GCD. For \( \frac{140}{315} \), the GCD is 35. We divide the numerator (140) and the denominator (315) by 35. The division results are \( \frac{140 \div 35}{315 \div 35} = \frac{4}{9} \). So, \( \frac{4}{9} \) is the simplest form, or lowest terms, of the fraction. This process ensures that the fraction cannot be reduced any further while still maintaining the same value.

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In Investigation 1, you explored positive integer powers of 2 and of 4. $$\begin{array}{|c|c|c|c|c|c|c|c|}\hline n & {1} & {2} & {3} & {4} & {5} & {6} & {7} & {8} & {9} \\ \hline 2^{n} & {2} & {4} & {8} & {16} & {32} & {64} & {128} & {256} & {512} \\ \hline 4^{n} & {4} & {16} & {64} & {256} & {1,024} & {4,096} & {16,384} & {65,536} & {262,144} \\ \hline\end{array}$$ Now think about positive integer powers of 8. a. List the first five positive integer powers of 8. b. Name three numbers that are on all three lists鈥攖hat is, three numbers that are powers of 2, 4, and 8. c. List three numbers greater than 16 that are powers of 2 but are not powers of 8. d. List three numbers greater than 16 that are powers of 4 but are not powers of 8. e. Describe the powers of 2 that are also powers of 8. f. Describe the powers of 4 that are also powers of 8.
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