Question

example:
observe the pattern
$$\frac { 1 } { 11 } = 0. overline { 09 }, \frac { 2 } { 11 } = 0. overline { 18 }, \frac { 3 } { 11 } = 0. overline { 27 }$$
a) determine the decimals for $$\frac { 5 } { 11 }$$ and $$\frac { 9 } { 11 }$$
b) what fraction will have $$0.636363...$$ as a decimal?
fractions with denominators of 10, 100, 1000
a fraction with a denominator of 10, 100 or 1000 can be easily converted into a decimal.
the number of zeros indicate the number of places the decimal needs to shift to the left in the numerator.
for example, consider $$\frac { 3 } { 10 }$$. in this case the numerator is 3 or 30.
since there is one zero in the denominator, move the decimal point one place to the left.
$$\frac { 3 } { 10 }$$
this means, $$3.0 = 0.3$$
lets consider $$\frac { 57 } { 100 }$$
$$\frac { 57 } { 100 }$$
this means, $$57.0 = 0.57$$

Explanation:

Step1: Analyze the pattern

Given \(\frac{1}{11}=0.\overline{09}\), \(\frac{2}{11}=0.\overline{18}\), \(\frac{3}{11}=0.\overline{27}\). The decimal part is \(09\times1\), \(09\times2\), \(09\times3\) respectively.

Step2: Calculate \(\frac{5}{11}\)

For \(\frac{5}{11}\), since the pattern is \(09\times n\) (where \(n\) is the numerator), \(09\times5 = 45\), so \(\frac{5}{11}=0.\overline{45}\)

Step3: Calculate \(\frac{9}{11}\)

For \(\frac{9}{11}\), \(09\times9=81\), so \(\frac{9}{11}=0.\overline{81}\)

Step4: Find the fraction for \(0.636363\cdots\)

Let \(x = 0.636363\cdots\). Then \(100x=63.636363\cdots\). Subtract \(x\) from \(100x\): \(100x - x=63.6363\cdots - 0.6363\cdots\). So \(99x = 63\), and \(x=\frac{63}{99}=\frac{7}{11}\)

Answer:

A) \(\frac{5}{11}=0.\overline{45}\), \(\frac{9}{11}=0.\overline{81}\)
B) \(\frac{7}{11}\)