Question

Find the fractions of denominator which are power's of 10 getting closer and closer to each of the fractions below and then write their decimal form 1) 5/6, 2) 3/11, 3) 23/11, 4) 1/13

Answer 1

10is terminating decimal fraction

Answer 2

Answer:

1. $\frac{5}{6}$ is closer to the fraction whose denominator 100 is $\frac{83}{100}$
2. $\frac{3}{11} is closer and closer to the fraction whose denominator 11000 is \frac{3}{11000}$
3. $\frac{23}{11} is closer and closer to the fraction whose denominator 11000 is \frac{10}{11000}$
4. $\frac{1}{3} is closer and closer to the fraction whose denominator 130000 is \frac{81}{130000}$

Step-by-step explanation:

• In mathematics, decimal fractions are fractions whose denominators should be the powers of 10, such as 101, 102, 103, and so on.
• Examples of decimal fractions are 124/1000, 55/100, 24/10, etc.
• Convert the given fractions into decimal fractions if the prime factorization of the denominator should contain either 2 or 5 or both.
• For example, 7/4 could be converted into a decimal fraction, as the prime factorisation of 4 is 2 × 2.
• Like integers, we can also perform different operations on decimal fractions such as addition, subtraction, multiplication and division.

1) $\frac{5}{6}=\frac{5}{10} \times \frac{10}{6}\\\\=\frac{1}{10}\left(\frac{50}{6}\right) \\=\frac{1}{10}\left(8+\frac{4}{12}\right)\\=\frac{8}{10}+\frac{2}{60} \\=\frac{8}{10}+\frac{4}{120}$

$&\frac{5}{6}-\frac{8}{10}=\frac{1}{30} ; \frac{5}{6} \text { is closer to } \frac{8}{10} \\&=\frac{1}{100} \times \frac{100}{30}=\frac{1}{100}\left(3+\frac{10}{30}\right) \\&=\frac{3}{100}+\frac{10}{3000}=\frac{3}{100}+\frac{1}{300}$

$&\frac{5}{6}-\frac{8}{10}-\frac{3}{100}=\frac{1}{300} \\&\frac{5}{6}-\frac{83}{100}=\frac{1}{300} \therefore \frac{5}{6}=0.83 \\&\frac{5}{6}=\frac{5}{100} \times \frac{100}{6}=\frac{1}{100}\left(\frac{500}{6}\right) \\&=\frac{1}{100}\left(\frac{8}{83}+\frac{2}{6}\right)$

$\frac{5}{6}=\frac{83}{100}+\frac{2}{600} ; \frac{5}{6}-\frac{83}{100}=\frac{2}{600}=\frac{1}{300}$

$\frac{5}{6}$ is closer to the fraction whose denominator 100 is $\frac{83}{100}$

$\frac{5}{6}-\frac{8}{10}=\frac{1}{30} ;\\ \frac{5}{6}-\frac{83}{100}=\frac{1}{300} \\ \frac{5}{6}-\frac{833}{1000}=\frac{1}{3000} \\ \frac{5}{6} in decimal form 0.8333...$

2. $\frac{3}{11}=\frac{3}{100} \times \frac{100}{11}=\frac{3}{100}\left(9+\frac{1}{11}\right)$

$=\frac{27}{100}+\frac{3}{1100}\\\frac{3}{11}-\frac{27}{100}=\frac{3}{1100}\\=\frac{3}{10000} \times \frac{10000}{1100}$

$=\frac{3}{10000} \times \frac{100}{11}=\frac{3}{1000 d}\left(9+\frac{1}{11}\right)=\frac{27}{10000}+\frac{3}{11000}$

$=\frac{3}{11}-\frac{27}{100}-\frac{27}{10000}=\frac{3}{11000}$

3. $\frac{23}{11}=\frac{23}{100} \times \frac{100}{11}$

$\begin{aligned}\frac{23}{11} &=2+\frac{1}{11} ; \frac{1}{11}=\frac{1}{100} \times \frac{100}{11} \\&=\frac{1}{100}\left(9+\frac{1}{11}\right)=\frac{9}{100}+\frac{1}{1100} \\&=\frac{1}{100}-\frac{9}{100}=\frac{1}{1100}\end{aligned}$

$\frac{1}{11} is closer to the fraction whose denominator 100 is \frac{9}{100}$

$\begin{aligned}\frac{1}{11} &=\frac{1}{1000} \times \frac{1000}{11} \\&=\frac{1}{1000}\left(90+\frac{10}{11}\right)=\frac{90}{1000}+\frac{10}{11000} \\\frac{1}{11} &-\frac{90}{1000}=\frac{1}{1100}\end{aligned}$

$\frac{1}{11} is closer and closer to the fraction whose denominator 100 is \frac{9}{1000}$

$\begin{aligned}&\frac{1}{11}-\frac{9}{100}=\frac{1}{1100} ; \frac{1}{11}-\frac{90}{1000}=\frac{10}{11000} \\&\frac{1}{11}-\frac{909}{10000}=\frac{1}{10000}\end{aligned}$

$\begin{aligned}&\frac{1}{11}=0.090909 \\&\therefore \frac{23}{11}=2.0907\end{aligned}$

4. $\frac{1}{13}=\frac{1}{100} \times \frac{100}{13}$

$=\frac{1}{100}\left[7+\frac{9}{13}\right]=\frac{7}{100}+\frac{9}{1300}$

$\frac{1}{13}-\frac{7}{100}=\frac{9}{1300}=\frac{9}{1300} \times \frac{10000}{1300}$

$=\frac{9}{10000} \times \frac{100}{13}=\frac{9}{10000}\left[7+\frac{9}{13}\right]$

$\frac{63}{10000}+\frac{81}{130000}$

$\begin{aligned}&=\frac{1}{13}-\frac{7}{100}-\frac{63}{10000}=\frac{81}{130000} \\&=\frac{1}{13}-\frac{763}{10000}=\frac{81}{130000}\end{aligned}$

$\therefore \frac{1}{13}=0.076$

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