Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{29}{343}$<\/strong> 29 = 29 \u00d7 1 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{13}{125}$<\/strong><\/p>\n\n\n\n Sol : 13 = 13 \u00d7 1 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{27}{8}$<\/p>\n\n\n\n Sol : 27 = 3 \u00d7 3 \u00d7 3 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{7}{80}$ 7 = 7 \u00d7 1 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{64}{455}$<\/p>\n\n\n\n Sol : 64 = 26<\/sup> Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{6}{15}$<\/strong> \u21d22 and 5 have no common factor Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{35}{50}$<\/strong> 7 = 1 \u00d7 7 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{129}{2^{2} \\times 5^{7} \\times 7^{5}}$ 129 = 3 \u00d7 43 Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{2^{2} \\times 7}{5^{4}}$ 28 = 7 \u00d7 22<\/sup> Without actually performing the long division, state whether the following rational numbers have terminating or non-terminating repeating (recurring) decimal expansion.<\/strong><\/p>\n\n\n\n $\\dfrac{29}{243}$ 29 = 29 \u00d7 1 Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{17}{8}$<\/strong> Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{3}{8}$<\/strong> Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{29}{343}$ 29 = 29 \u00d7 1 Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{13}{125}$<\/strong> Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{27}{8}$<\/strong><\/p>\n\n\n\n Sol : Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{7}{80}$ Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{64}{455}$<\/strong> Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{6}{15}$<\/strong> Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{35}{50}$ Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\frac{129}{2^{2} 5^{7} 7^{5}}$<\/p>\n\n\n\n Sol : Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{2^{2} \\times 7}{5^{4}}$ Write down the decimal expansions of the following numbers which have terminating decimal expansions.<\/strong><\/p>\n\n\n\n $\\dfrac{29}{243}$ The following real numbers have decimal expansions as given below. In each case examine whether they are rational or not. If they are a rational number of the form p\/q, what can be said about q?<\/strong><\/p>\n\n\n\n (i) 7.2345<\/strong> (i)<\/strong> 7.2345 (ii)<\/strong> $5 . \\overline{234}$ (v)<\/strong> 0.120120012000120000\u2026
Sol :
Given rational number is $\\frac{29}{343}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
343 = 7 \u00d7 7 \u00d7 7
\u21d229 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
343 = 73<\/sup>
So, denominator is not of the form 2n<\/sup>5m<\/sup>
Thus, $\\frac{29}{343}$ is a non-terminating<\/strong> repeating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 D <\/h4>\n\n\n\n
Given rational number is $\\frac{13}{125}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
125 = 5 \u00d7 5 \u00d7 5
\u21d213 and 125 have no common factors
Therefore, 13 and 125 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
125 = 53<\/sup>
= 1 \u00d7 23<\/sup>
= 20<\/sup> \u00d7 53<\/sup>
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 0 and m = 3
Thus, $\\frac{13}{125}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 E <\/h4>\n\n\n\n
Given rational number is $\\frac{27}{8}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
8 = 2 \u00d7 2 \u00d7 2
\u21d227 and 8 have no common factors
Therefore, 27 and 8 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
8 = 23<\/sup>
= 1 \u00d7 23<\/sup>
= 50<\/sup> \u00d7 23<\/sup>
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 3 and m = 0
Thus, $\\frac{27}{8}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 F <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{7}{80}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
80 = 2 \u00d7 2 \u00d7 2 \u00d7 2 \u00d7 5
\u21d27 and 80 have no common factors
Therefore, 7 and 80 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
80 = 24<\/sup> \u00d7 5
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 4 and m = 1
Thus, $\\frac{7}{80}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 G <\/h4>\n\n\n\n
Given rational number is $\\frac{64}{455}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
455 = 5 \u00d7 7 \u00d7 13
\u21d264 and 455 have no common factors
Therefore, 64 and 455 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
455 = 5 \u00d7 7 \u00d7 13
So, denominator is not of the form 2n<\/sup>5m<\/sup>
Thus, $\\frac{64}{455}$ is a non-terminating repeating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 H <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{6}{15}$
$\\frac{6}{15}=\\frac{2}{5}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
Therefore, 2 and 5 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
5 = 51<\/sup> \u00d7 1
= 51<\/sup> \u00d7 20<\/sup>
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 0 and m = 1
Thus, $\\frac{6}{15}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 I <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{35}{50}$
$\\frac{35}{50}=\\frac{7}{10}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
10 = 2 \u00d7 5
\u21d2 7 and 10 have no common factor
Therefore, 7 and 10 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
10 = 51<\/sup> \u00d7 21<\/sup>
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 1 and m = 1
Thus, $\\frac{35}{50}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 J <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{129}{2^{2} \\times 5^{7} \\times 7^{5}}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
Denominator = 22<\/sup> \u00d757<\/sup> \u00d775<\/sup>
\u21d2129 and 22<\/sup> \u00d757<\/sup> \u00d775<\/sup> have no common factors
Therefore, 129 and 22<\/sup> \u00d757<\/sup> \u00d775<\/sup> are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
Denominator = 22<\/sup> \u00d757<\/sup> \u00d775<\/sup>
So, denominator is not of the form 2n<\/sup>5m<\/sup>
Thus, $\\frac{129}{2^{2} \\times 5^{7} \\times 7^{5}}$ is a non-terminating repeating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 K <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{2^{2} \\times 7}{5^{4}}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
625 = 54<\/sup>
\u21d2 28 and 625 have no common factors
Therefore, 28 and 625 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
625 = 54<\/sup> \u00d7 1
= 54<\/sup> \u00d7 20<\/sup>
So, denominator is of the form 2n<\/sup>5m<\/sup> where n = 0 and m = 4
Thus,$\\frac{2^{2} \\times 7}{5^{4}}$ is a terminating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 1 L <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{29}{243}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly, we check co-prime<\/p>\n\n\n\n
243 = 35<\/sup>
\u21d229 and 243 have no common factors
Therefore, 29 and 243 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
243 = 35<\/sup>
So, the denominator is not of the form 2n<\/sup>5m<\/sup>
Thus, $\\frac{29}{243}$ is a non- terminating<\/strong> repeating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 2 A <\/h4>\n\n\n\n
Sol :
We know, $\\frac{17}{8}=\\frac{17}{2^{3} \\times 5^{0}}$
Multiplying and dividing by 53<\/sup>
$=\\frac{17 \\times 5^{3}}{2^{3} \\times 5^{0} \\times 5^{3}}$
$=\\frac{17 \\times 125}{2^{3} \\times 1 \\times 5^{3}}$
$=\\frac{2125}{(2 \\times 5)^{3}}$
$=\\frac{2125}{(10)^{3}}$
$=\\frac{2125}{1000}$
= 2.125<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 B <\/h4>\n\n\n\n
Sol :
We know, $\\frac{3}{8}=\\frac{3}{2^{3} \\times 5^{0}}$
Multiplying and dividing by 53<\/sup>
$\\frac{3 \\times 5^{3}}{2^{3} \\times 5^{0} \\times 5^{3}}$
$=\\frac{3 \\times 125}{2^{3} \\times 1 \\times 5^{3}}$
$=\\frac{375}{(2 \\times 5)^{3}}$
$=\\frac{375}{(10)^{3}}$
$=\\frac{375}{1000}$
= 0.375<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 C <\/h4>\n\n\n\n
Sol :
We know, $\\frac{29}{343}=\\frac{29}{7^{3}}$
Given rational number is $\\frac{29}{343}$
$\\frac{p}{q}$ is terminating if
a) p and q are co-prime &
b) q is of the form of 2n<\/sup> 5m<\/sup> where n and m are non-negative integers.Firstly we check co-prime<\/p>\n\n\n\n
343 = 7 \u00d7 7 \u00d7 7
\u21d229 and 343 have no common factors
Therefore, 29 and 343 are co-prime.
Now, we have to check that q is in the form of 2n<\/sup>5m<\/sup>
343 = 73<\/sup>
So, the denominator is not of the form 2n<\/sup>5m<\/sup>
Thus, $\\frac{29}{343}$ is a non-terminating<\/strong> repeating<\/strong> decimal.<\/p>\n\n\n\n
\n\n\n\nQuestion 2 D <\/h4>\n\n\n\n
Sol :
We know, $\\frac{13}{125}=\\frac{13}{2^{0} \\times 5^{3}}$
Multiplying and dividing by 23<\/sup>
$=\\frac{13 \\times 2^{3}}{2^{0} \\times 5^{3} \\times 2^{3}}$
$=\\frac{13 \\times 8}{1 \\times 2^{3} \\times 5^{3}}$
$=\\frac{104}{(2 \\times 5)^{3}}$
$\\frac{104}{(10)^{3}}$
$=\\frac{104}{1000}$
= 0.104<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 E <\/h4>\n\n\n\n
We know, $\\frac{27}{8}=\\frac{3^{3}}{2^{3} \\times 5^{0}}$
Multiplying and dividing by 53<\/sup>
$\\frac{27 \\times 5^{3}}{2^{0} \\times 2^{3} \\times 5^{3}}$
$=\\frac{27 \\times 125}{1 \\times 2^{3} \\times 5^{3}}$
$=\\frac{3375}{(2 \\times 5)^{3}}$
$\\frac{3375}{(10)^{3}}$
$=\\frac{3375}{1000}$
= 3.375<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 F <\/h4>\n\n\n\n
Sol :
We know, $\\frac{7}{80}=\\frac{7}{2^{4} \\times 5^{1}}$
Multiplying and dividing by 53<\/sup>
$=\\frac{7 \\times 5^{3}}{2^{4} \\times 5^{1} \\times 5^{3}}$
$=\\frac{7 \\times 125}{2^{4} \\times 5^{4}}$
$=\\frac{875}{(2 \\times 5)^{4}}$
$=\\frac{875}{(10)^{4}}$
$=\\frac{875}{10000}$
= 0.0875<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 G <\/h4>\n\n\n\n
Sol :
We know, $\\frac{64}{455}=\\frac{2^{6}}{5 \\times 7 \\times 13}$
Since the denominator is not of the form 2n<\/sup>5m<\/sup>
$\\frac{64}{455}$ has a non-terminating repeating decimal expansion.<\/p>\n\n\n\n
\n\n\n\nQuestion 2 H <\/h4>\n\n\n\n
Sol :
We know, $\\frac{6}{15}=\\frac{2}{5}=\\frac{2}{2^{0} \\times 5}$
Multiplying and dividing by 21<\/sup>
$=\\frac{2 \\times 2}{2^{0} \\times 5^{1} \\times 2}$
$=\\frac{4}{1 \\times 10}$
$=\\frac{4}{10}$
= 0.4<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 I <\/h4>\n\n\n\n
Sol :
We know, $\\frac{35}{50}$
$=\\frac{7}{10}$
$=\\frac{7}{2 \\times 5}$
$=\\frac{7}{10}$
= 0.7<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 J <\/h4>\n\n\n\n
Given rational number is $\\frac{129}{2^{2} \\times 5^{7} \\times 7^{5}}$
Since the denominator is not of the form 2n<\/sup>5m<\/sup>
$\\frac{129}{2^{2} \\times 5^{7} \\times 7^{5}}$has a non-terminating repeating decimal expansion.<\/p>\n\n\n\n
\n\n\n\nQuestion 2 K <\/h4>\n\n\n\n
Sol :
We know, $\\frac{2^{2} \\times 7}{5^{4}}=\\frac{2^{2} \\times 7}{2^{0} \\times 5^{4}}$
Multiplying and dividing by 26<\/sup>
$\\frac{2^{2} \\times 7 \\times 2^{6}}{2^{0} \\times 5^{4} \\times 2^{6}}$
$=\\frac{2^{2} \\times 7 \\times 2^{6}}{1 \\times 5^{4} \\times 2^{6}}$
$=\\frac{7 \\times 2^{6}}{(2 \\times 5)^{4}}$
$=\\frac{448}{(10)^{4}}$
$=\\frac{448}{10000}$
= 0.0448<\/strong><\/p>\n\n\n\n
\n\n\n\nQuestion 2 L <\/h4>\n\n\n\n
Sol :
Given rational number is $\\frac{29}{243}$
$\\frac{29}{243}=\\frac{29}{3^{5}}$
Since the denominator is not of the form 2n<\/sup>5m<\/sup>.
$\\frac{29}{243}$ has a non-terminating repeating decimal expansion.<\/p>\n\n\n\n
\n\n\n\nQuestion 3 <\/h4>\n\n\n\n
(ii) <\/strong>$5 . \\overline{234}$<\/strong>
(iii) 23.245789<\/strong>
(iv) <\/strong>$7 . \\overline{3427}$<\/strong>
(v) 0.120120012000120000\u2026<\/strong>
(vi) 23.142857<\/strong>
(vii) 2.313313313331\u2026<\/strong>
(viii) 0.02002000220002\u2026<\/strong>
(ix) 3.300030000300003\u2026<\/strong>
(x) 1.7320508\u2026<\/strong>
(xi) 2.645713<\/strong>
(xii) 2.8284271\u2026<\/strong>
Sol :<\/p>\n\n\n\n
Here, 7.2345 has terminating decimal expansion.
So, it represents a rational number.
i.e. 7.2345 $=\\frac{7.2345}{10000}=\\frac{p}{q}$
Thus, q = 104<\/sup>, those factors are 23<\/sup> \u00d7 53<\/sup>
<\/p>\n\n\n\n
$5 . \\overline{234}$ is non-terminating but repeating.
So, it would be a rational number.
In a non-terminating repeating expansion of $\\frac{p}{q}$ ,
q will have factors other than 2 or 5.<\/p>\n\n\n\n
0.120120012000120000\u2026 is non-terminating and non-repeating.
So, it is not a rational number as we see in the chart.<\/p>\n\n\n\n