Question

Express the following in \[\dfrac{p}{q}\] form, where \[p\] and \[q\] are integers and \[q \ne 0\].
(a) \[0.\overline 6 \]
(b) \[0.4\overline 7 \]
(c) \[0.\overline {001} \]

Answer 1

Hint: Here, we will rewrite the given numbers using the bar sign. We will assume the number to be \[x\]. Then, we need to multiply the expression by a power of 10 depending on the number of digits repeating, and then subtract the two equations such that the numbers in the decimal places will be 0. Then, we will simplify the equation to find the value of \[x\], and hence, get the value of the numbers in \[\dfrac{p}{q}\] form.

Complete step-by-step answer:
(a)
We will solve this question by forming two equations and subtracting them. The purpose of this is to remove the numbers in the decimal places.
The bar sign above 6 indicates that the numbers 6 repeats in the decimal expansion.
Let \[x = 0.666666\].
Here, we can observe that 6 repeats in the decimal expansion.
Since 6 is one digit, we will multiply the expression by 10 to find the value of \[10x\].
Multiplying both sides of the expression by 10, we get
\[10x = 6.66666\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[10x\] is the same.
The number of decimal places in \[x = 0.666666\] is 6.
Thus, extending the decimal places in \[10x = 6.66666\] from 5 to 6, we get
\[10x = 6.666666\]
Next, we need to subtract the two equations to eliminate the repeating number in the expansion, that is 6.
Subtracting the equation \[x = 0.666666\] from the equation \[10x = 6.666666\], we get
\[10x - x = 6.666666 - 0.666666\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 9x = 6.000000\\ \Rightarrow 9x = 6\end{array}\]
The decimal places are removed from the expression.
Now, we will divide both sides by 9 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 9, we get
\[\begin{array}{l} \Rightarrow \dfrac{{9x}}{9} = \dfrac{6}{9}\\ \Rightarrow x = \dfrac{6}{9}\end{array}\]
Simplifying the expression, we get
\[\therefore x=\dfrac{2}{3}\]
Therefore, the number \[0.\overline 6 \] is converted into the \[\dfrac{p}{q}\] form, \[\dfrac{2}{3}\].
(b)
We will solve this question by forming two equations and subtracting them. The purpose of this is to remove the numbers in the decimal places.
The bar sign above 7 indicates that the numbers 7 repeats in the decimal expansion.
Let \[x = 0.477777\].
Here, we can observe that 7 repeats in the decimal expansion.
Since 7 is one digit, we will multiply the expression by 10 to find the value of \[10x\].
Multiplying both sides of the expression by 10, we get
\[10x = 4.77777\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[10x\] is the same.
The number of decimal places in \[x = 0.477777\] is 6.
Thus, extending the decimal places in \[10x = 4.77777\] from 5 to 6, we get
\[10x = 4.777777\]
Next, we need to subtract the two equations to eliminate the repeating number in the expansion, that is 7.
Subtracting the equation \[x = 0.477777\] from the equation \[10x = 4.777777\], we get
\[10x - x = 4.777777 - 0.477777\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 9x = 4.300000\\ \Rightarrow 9x = 4.3\end{array}\]
The repeating number 7 is removed from the decimal expansion.
Now, we will rewrite the decimal remaining in the expression.
Rewriting the decimal \[4.3\] as fraction, we get
\[ \Rightarrow 9x = \dfrac{{43}}{{10}}\]
Now, we will divide both sides by 9 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 9, we get
\[\begin{array}{l} \Rightarrow \dfrac{{9x}}{9} = \dfrac{{\dfrac{{43}}{{10}}}}{9}\\ \Rightarrow x = \dfrac{{43}}{{10 \times 9}}\end{array}\]
Simplifying the expression, we get
\[\therefore x=\dfrac{43}{90}\]
Therefore, the number \[0.4\overline 7 \] is converted into the \[\dfrac{p}{q}\] form, \[\dfrac{{43}}{{90}}\].
(c)
We will solve this question by forming two equations and subtracting them. The purpose of this is to remove the numbers in the decimal places.
The bar sign above 001 indicates that the numbers 001 repeat in the decimal expansion.
Let \[x = 0.001001001001\].
Here, we can observe that 001 repeats in the decimal expansion.
Since 001 is three digits, we will multiply the expression by 1000 to find the value of \[1000x\].
Multiplying both sides of the expression by 1000, we get
\[1000x = 1.001001001\]
We will extend the decimal places such that the number of decimal places in the value of \[x\] and \[1000x\] is the same.
The number of decimal places in \[x = 0.001001001001\] is 12.
Thus, extending the decimal places in \[1000x = 1.001001001\] from 9 to 12, we get
\[1000x = 1.001001001001\]
Next, we need to subtract the two equations to eliminate the repeating numbers in the expansion, that is 001.
Subtracting the equation \[x = 0.001001001001\] from the equation \[1000x = 1.001001001001\], we get
\[1000x - x = 1.001001001001 - 0.001001001001\]
Subtracting the terms in the equation, we get
\[\begin{array}{l} \Rightarrow 999x = 1.000000000000\\ \Rightarrow 999x = 1\end{array}\]
The decimal places are removed from the expression.
Now, we will divide both sides by 999 to get the value of \[x\] in the \[\dfrac{p}{q}\] form.
Dividing both sides by 999, we get
\[\begin{array}{l} \Rightarrow \dfrac{{999x}}{{999}} = \dfrac{1}{{999}}\\ \Rightarrow x = \dfrac{1}{{999}}\end{array}\]
Therefore, the number \[0.\overline {001} \] is converted into the \[\dfrac{p}{q}\] form, \[\dfrac{1}{{999}}\].


Note: A common mistake we can make in this question is to convert the decimal \[0.\overline 6 \] into the fraction \[\dfrac{6}{{10}}\] and leave the answer at that.
We multiplied \[0.666666\] by 10 in the solution. The product of a decimal number by a power of 10 (10, 100, 1000, etc) can be calculated using a simple method. The number of zeroes in 10 is 1. When a decimal number is multiplied by 10, the result can be obtained by shifting the decimal point 1 digit to the right. For example: When \[0.666666\] is multiplied by 10, the result is \[6.66666\], which is simply obtained by shifting the decimal point one place to the right from after 0 to after 6.
Similarly, we have multiplied \[0.477777\] by 10, and \[0.001001001001\] by 1000.