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Chapter 6: Problem 84

Solve. $$ \frac{67}{9} $$

Short Answer

Expert verified
The answer is 7.\bar{4} or 7 \frac{4}{9}.

Step by step solution

01

- Perform the Division

Divide the numerator by the denominator: \[ \frac{67}{9} = 67 \text{ divided by } 9. \]Perform the division to get: \[ 67 \text{ divided by } 9 = 7.4444...\]So, the quotient is 7 with a remainder.
02

- Simplify to Decimal

To express the exact decimal value, continue the division until a repeating pattern or a decimal point is established.Performing the division: \[ 67 \text{ divided by } 9 = 7.4444\text{(repeating)} = 7.\bar{4} \]
03

- Express the Fraction as a Mixed Number

Express this result as a mixed number. The quotient is 7 and the remainder is 4. So:\[ \frac{67}{9} = 7 \frac{4}{9} \]
04

- Express as Repeating Decimal

If required, express the fraction as a repeating decimal: \[ 7.\bar{4} \]

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Key Concepts

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

Fraction Division
When you see a fraction like \(\frac{67}{9}\), it represents the division of 67 by 9. To solve for this, you simply divide the numerator (67) by the denominator (9).

In this exercise, dividing 67 by 9 gives us 7 with a remainder. It is important to know that fractions show us parts of a whole and division is the way to find out how many whole parts we have and what is left over.

Understanding fraction division is fundamental since it translates fractions into more familiar forms, such as decimals or mixed numbers. When you do the division step-by-step, you get the decimal form which can be either a terminating or repeating decimal.
Repeating Decimal
Not all fractions convert into neat, terminating decimals. When dividing 67 by 9, we find the quotient is 7.4444..., where the number 4 repeats infinitely. This type of decimal is called a repeating decimal.

Mathematically, you can write this repeating part as \(7.\bar{4}\). The bar over 4 indicates it repeats forever. Recognizing repeating decimals is important because it shows the continuing pattern without having to write the number out infinitely.
Mixed Number Conversion
A mixed number combines a whole number with a fraction. In this exercise, \(\frac{67}{9}\) can also be expressed as a mixed number.

The whole number part of the mixed number is obtained from the quotient, which is 7. For the fractional part, we take the remainder of the division (67 - 63 = 4) and place it over the original denominator (9).

Hence, \(7 \frac{4}{9}\) is the mixed number form of \(\frac{67}{9}\). This conversion is helpful in understanding the size of the fraction in a more intuitive and simplified form.

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