Problem 102 Given that \(0.3 \overline{3}=\f... [FREE SOLUTION]

Chapter 1: Problem 102

Given that $0.3 \overline{3}=\frac{1}{3}$ and $0.6 \overline{6}=\frac{2}{3},$ express each of the following as a ratio of two integers. $$7.7 \overline{7}$$

Short Answer

Expert verified

7.7\overline{7} = \frac{70}{9}

Step by step solution

- Represent the Number as a Sum

Express the number 7.7\overline{7} as the sum of a whole number, a terminating decimal, and an infinitely repeating decimal. So, $7.7\overline{7} = 7 + 0.7 + 0.0\overline{7}$.

- Write the repeating decimal separately

Next, rewrite the repeating decimal part separately: $0.7\overline{7}$ can be split into 0.7 and 0.0\overline{7}. Hence, we have $7 + 0.7 + 0.0\overline{7} $.

- Express Repeating Decimal as Fraction

We know that $0.0\overline{7}$ can be expressed as $\frac{7}{90}$. Hence, $0.7\overline{7} = 0.7 + \frac{7}{90}$.

- Convert 0.7 to a Fraction

Convert 0.7 to a fraction: $0.7 = \frac{7}{10}$. So, $0.7\overline{7} = \frac{7}{10} + \frac{7}{90}$.

- Combine the Fractions

To combine $\frac{7}{10} + \frac{7}{90} $, find a common denominator, which is 90. Hence, it becomes: $\frac{63}{90} + \frac{7}{90} = \frac{70}{90} = \frac{7}{9}$.

- Combine with the Whole Number

Combine this result with the whole number 7. Thus, $7.7\overline{7} = 7 + \frac{7}{9} = \frac{63}{9} + \frac{7}{9} = \frac{70}{9}$.

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Key Concepts

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

Repeating Decimals

Repeating decimals are decimal numbers in which a digit or group of digits repeats infinitely. These are commonly written with a line over the repeating part, called a vinculum. For example, 0.333... is usually written as 0.\bar{3}. To understand repeating decimals, let's break it down:

**Identifying:** Look at the decimal after the point. See if there's a repeated pattern.
**Notation:** Use the vinculum over the repeating digits, like 0.\bar{7} for 0.777...
**Conversion:** Convert to a fraction using algebra. For example, let $x = 0.\bar{7}$. Multiply both sides by 10 to shift the decimal and subtract the original from this result: $10x - x = 7.7... - 0.7...$, giving $9x=7$. Solving, $x = \frac{7}{9}$.

Repeating decimals reveal neat patterns in number theory. Converting them to fractions makes them easier to handle.

Fraction Conversion

Fraction conversion turns a decimal or repeating decimal into a fraction鈥攅ssentially, a ratio of two integers. Steps for conversion are:

**Identify the Decimal Type:** Determine if the decimal is terminating (ends) or repeating (continues indefinitely).
**Terminating Decimals:** To convert, simply place the digits over a power of 10. For instance, 0.7 becomes $\frac{7}{10}$. Simplify if possible.
**Repeating Decimals:** Use algebra to set up equations. For example, to convert 0.77..., let $x=0.\bar{7}$. Then, $10x=7.\bar{7}$. Subtract the original: $10x - x = 7.\bar{7} - 0.\bar{7}$, gives $9x = 7$. Thus, $x = \frac{7}{9}$.

By converting to fractions, you simplify computations and compare values more easily, reinforcing understanding of the number's exact value.

Simplification

Simplification reduces a fraction to its smallest form. This makes the fraction easier to understand and work with. Here's how:

**Greatest Common Divisor (GCD):** Find the GCD of the numerator and denominator. Divide both by this number.
**Example:** Consider $\frac{70}{90}$. The GCD of 70 and 90 is 10. Divide: $\frac{70 \times \frac{1}{10}}{90 \times \frac{1}{10}} = \frac{7}{9}$.
**Check for Simplification:** Ensure no further common factors exist.

Simplification ensures fractions are in their simplest form, aiding in solving equations and comparing fractions. This final step confirms the fraction accurately represents and maintains its value without extra bulk.

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91影视

Given that \(0.3 \overline{3}=\frac{1}{3}\) and \(0.6 \overline{6}=\frac{2}{3},\) express each of the following as a ratio of two integers. $$7.7 \overline{7}$$

Short Answer

Step by step solution

- Represent the Number as a Sum

- Write the repeating decimal separately

- Express Repeating Decimal as Fraction

- Convert 0.7 to a Fraction

- Combine the Fractions

- Combine with the Whole Number

Key Concepts

Repeating Decimals

Fraction Conversion

Simplification

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