Answer :
Recurring decimal is decimal representation of a number whose digits are periodic and infinite. Proved algebraically that the recurring decimal 0.178 can be written as the fraction 59/330 below.
Given information;
Given number in the decimal form is [tex]0. 1 \overline 7 \overline 8[/tex]
Suppose the number is equal to the x,
[tex]x=0. 1 \overline 7 \overline 8[/tex]
Recurring decimal
Recurring decimal is decimal representation of a number whose digits are periodic and infinite.
As the number 78 is the recurring number. Thus the recurring number can be written as,
[tex]x=0.1787878.....[/tex] .......equation 1.
Suppose this is equation number 1.
Multiply the above equation with 100 both the sides,
[tex]100\times x=100\times0. 1 787878....[/tex]
[tex]100x=100\times0.1787878...[/tex]
[tex]100x=17.87878...[/tex]
Subtract the above equation from equation number 1. Thus,
[tex]\begin{aligned}\ 100x-x&=17.87878-0.1787878\\ 99x&=17.7\\ \end[/tex]
Solve for x ,
[tex]x=\dfrac{17.7}{99} [/tex]
Multiply with 10 in both numerator and denominator,
[tex]x=\dfrac{177}{990} \\ x=\dfrac{59}{330} \\[/tex]
Hence proved algebraically that the recurring decimal 0.178 can be written as the fraction 59/330
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