Question

# Find the rational numbers having the following decimal expansions : (i) 0.¯3 (ii) 0.¯¯¯¯¯¯¯¯231 (iii) 3.¯¯¯¯¯¯52 (iv) 0.6¯¯¯8

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Solution

## (i) 0.¯¯¯3=0.3333 =0.3+0.03+0.003+...∞ =310+3102+3103+....∞ =310(1+110+1102+.....) =310(11−110) [∵Sum of G.P.=a1−r] =310×109 =39 Hence, 0.¯¯¯3=13 (ii) 0.¯¯¯¯¯¯¯¯231=0.231231231.... =0.231+0.000231+0.000000231+....∞ =231103=231106+231109+....∞ =231103(1+1103+1106+.....) =2311000(11−11000) Hence,0.¯¯¯¯¯¯¯¯231=231999 (iii) 3.¯¯¯¯¯¯52=3+0.52222..... =3+0.5+0.02+0.002+0.0002+....∞ =3.5+2102+2103+2104+....∞ =3.5+2102(1+110+1102+.....) =3510+2100(11−110) =3510+2100×109 =3510+290 =315+290 Hence, 3.¯¯¯¯¯¯52=31790 (iv) The rational number can be written as : 0.6¯¯¯8=0.6+0.08+0.008+0.0008+....∞ =35+8[0.01+0.001+0.0001+....∞] =35+8[1100+11000+....∞] This is an infinite GP with first term 1100 and common ratio 110 =35+8.1100.11−110 =35+445=3145Hence, 0.¯¯¯¯¯¯52=3145.

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