Question

Consider fractions with denominator 11 a. Find decimal equivalents for \(\frac{1}{11}, \frac{2}{11}, \frac{3}{11}, \frac{4}{11},\) and \(\frac{5}{11} .\) What pattern do you see? b. Use the pattern you discovered to predict the decimal equivalents of \(\frac{7}{11}\) and \(\frac{9}{11}\)

Short answer

Using the pattern, \( \frac{7}{11} \) ≈ \( 0.\bar{63} \) and \( \frac{9}{11} \) ≈ \( 0.\bar{81} \)

Step-by-step solution

Find Decimal Equivalent for 1/11

Divide 1 by 11: \( \frac{1}{11} = 0.090909\text{...} \) which is a repeating decimal \( 0.\bar{09} \)

Find Decimal Equivalent for 2/11

Divide 2 by 11: \( \frac{2}{11} = 0.181818\text{...} \) which is a repeating decimal \( 0.\bar{18} \)

Find Decimal Equivalent for 3/11

Divide 3 by 11: \( \frac{3}{11} = 0.272727\text{...} \) which is a repeating decimal \( 0.\bar{27} \)

Find Decimal Equivalent for 4/11

Divide 4 by 11: \( \frac{4}{11} = 0.363636\text{...} \) which is a repeating decimal \( 0.\bar{36} \)

Find Decimal Equivalent for 5/11

Divide 5 by 11: \( \frac{5}{11} = 0.454545\text{...} \) which is a repeating decimal \( 0.\bar{45} \)

Identify the Pattern

Look at the repeating decimals found: \(0.\bar{09}, 0.\bar{18}, 0.\bar{27}, 0.\bar{36}, 0.\bar{45}\). Notice each decimal increases by a pattern of 0.090909...

Predict Decimal for 7/11

Following the observed pattern, skip the decimal equivalent for 6/11: \( 0.\bar{54} \) to find \( \frac{7}{11} = 0.636363\text{...} \) which is a repeating decimal \( 0.\bar{63} \)

Predict Decimal for 9/11

Following the observed pattern as above, skip the decimal equivalent for 8/11: \( 0.\bar{72} \) to find \( \frac{9}{11} = 0.818181\text{...} \) which is a repeating decimal \( 0.\bar{81} \)

Key concepts

fractions

Fractions represent a part of a whole. They consist of a numerator (the upper part) and a denominator (the lower part). For example, in the fraction \(\frac{1}{11}\), 1 is the numerator and 11 is the denominator. Simplifying fractions or converting them into decimals makes it easier to understand and compare different fractional values.
Splitting numbers into parts is a fundamental skill in mathematics. It helps us measure, compare, and perform operations like addition, subtraction, multiplication, and division on various quantities. Understanding fractions is crucial since they appear in many areas, including everyday life, science, and engineering.

repeating decimals

When a fraction is converted into a decimal, sometimes it results in a repeating decimal. For instance, \(\frac{1}{11}\) equals 0.090909... or \(0.\overline{09}\). This notation indicates that the sequence '09' repeats infinitely.
Repeating decimals are common when dealing with fractions that have denominators not easily divisible by powers of 10. Recognizing the repeating pattern can help you predict and simplify calculations. For example:

• \(\frac{2}{11} = 0.181818...\) or \(0.\overline{18}\)
• \(\frac{3}{11} = 0.272727...\) or \(0.\overline{27}\)

Spotting and understanding these repeating sequences can save time and simplify calculations involving fractions.

patterns in mathematics

Mathematics often reveals beautiful and consistent patterns. For instance, the repeating decimals of fractions with the denominator 11, like \(\frac{1}{11}, \frac{2}{11}, \frac{3}{11}\), and so on, show increasing patterns: 0.090909..., 0.181818..., 0.272727..., respectively.
Observing these sequences helps:

• Identify the behavior of numbers
• Predict future values without complex calculations
• Recognize mathematical relationships

Identifying patterns not only aids in solving specific problems but also builds foundational skills for more advanced mathematical concepts. For instance, in this exercise, recognizing the pattern allowed us to predict the decimals for \(\frac{7}{11}\) and \(\frac{9}{11}\) based on earlier results.

division

Division is the process of splitting a number into equal parts. When dividing a smaller number (numerator) by a larger number (denominator), the result often includes decimal places. For example, dividing 1 by 11 gives us 0.090909... (\(0.\overline{09}\)).
Here's a step-by-step breakdown of the division process using \(\frac{1}{11}\):

• 1 divided by 11 equals 0 with a remainder of 1
• Add a decimal point and bring down the next 0 to make it 10
• 10 divided by 11 still equals 0, with a remainder of 10
• Again, add a zero and repeat
• This process reveals the repeating pattern 0.090909...

Mastering division helps understand fractions, repeating decimals, and the behavior of numbers, paving the way for higher mathematical skills and problem-solving.