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Mathematics

Without actual division, find whether the following rational numbers are terminating decimals or recurring decimals :

(i)1345(ii)556(iii)7125(iv)2380(v)1566\begin{matrix} \text{(i)} & \dfrac{13}{45} \\[1.5em] \text{(ii)} & -\dfrac{5}{56} \\[1.5em] \text{(iii)} & \dfrac{7}{125} \\[1.5em] \text{(iv)} & -\dfrac{23}{80} \\[1.5em] \text{(v)} & -\dfrac{15}{66} \\[1.5em] \end{matrix}

In case of terminating decimals, write their decimal expansions.

Rational Irrational Nos

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Answer

(i) 1345\text{(i) } \dfrac{13}{45}

The given number 1345\dfrac{13}{45} is in its lowest form.

Prime factorization of denominator 45:

345315551\begin{array}{l|l} 3 & 45 \ \hline 3 & 15 \ \hline 5 & 5 \ \hline & 1 \end{array}

45 = 3 x 3 x 5 x 1
= 32 x 5 x 1

Denominator is not of the form 2m x 5n, where m, n are non-negative integers.

∴ The given number 1345\dfrac{13}{45} is recurring decimal.

(ii) 556\text{(ii) } -\dfrac{5}{56}

The given number 556-\dfrac{5}{56} is in its lowest form.

Prime factorization of denominator 56:

256228214771\begin{array}{l|l} 2 & 56 \ \hline 2 & 28 \ \hline 2 & 14 \ \hline 7 & 7 \ \hline & 1 \end{array}

56 = 2 x 2 x 2 x 7 x 1
= 23 x 7 x 1

Denominator is not of the form 2m x 5n, where m, n are non-negative integers.

∴ The given number 556-\dfrac{5}{56} is recurring decimal.

(iii) 7125\text{(iii) } \dfrac{7}{125}

The given number 7125\dfrac{7}{125} is in its lowest form.

Prime factorization of denominator 125:

5125525551\begin{array}{l|l} 5 & 125 \ \hline 5 & 25 \ \hline 5 & 5 \ \hline & 1 \end{array}

125= 5 x 5 x 5 x 1
= 53 x 1
= 53 x 20

Denominator is of the form 2m x 5n, where m, n are non-negative integers.

7125=720×53=7×2323×53=56(2×5)3=56(10)3=561000=0.056\dfrac{7}{125} = \dfrac{7}{2^0 × 5^3} \\[1.5em] = \dfrac{7 × 2^3}{2^3 × 5^3} \\[1.5em] = \dfrac{56}{(2 × 5)^3} \\[1.5em] = \dfrac{56}{(10)^3} \\[1.5em] = \dfrac{56}{1000} \\[1.5em] = 0.056

∴ The given number 7125\dfrac{7}{125} is a terminating decimal and its decimal expansion is 0.056.

(iv) 2380\text{(iv) } \dfrac{-23}{80}

The given number 2380-\dfrac{23}{80} is in its lowest form.

Prime factorization of denominator 80:

280240220210551\begin{array}{l|l} 2 & 80 \ \hline 2 & 40 \ \hline 2 & 20 \ \hline 2 & 10 \ \hline 5 & 5 \ \hline & 1 \end{array}

80 = 2 x 2 x 2 x 2 x 5 x 1
= 24 x 51

Denominator is of the form 2m x 5n, where m, n are non-negative integers.

2380=2324×51=23×5324×54=23×125(2×5)4=2875(10)4=287510000=0.2875-\dfrac{23}{80} = -\dfrac{23}{2^4 × 5^1} \\[1.5em] = -\dfrac{23 × 5^3} {2^4 × 5^4} \\[1.5em] = -\dfrac{23 × 125}{(2 × 5)^4} \\[1.5em] = -\dfrac{2875}{(10)^4} \\[1.5em] = -\dfrac{2875}{10000} = -0.2875

∴ The given number 2380-\dfrac{23}{80} is a terminating decimal and its decimal expansion is -0.2875.

(v) 1566\text{(v) } -\dfrac{15}{66}

The given number 1566-\dfrac{15}{66} is in its lowest form.

Prime factorization of denominator 66:

26633311111\begin{array}{l|l} 2 & 66 \ \hline 3 & 33 \ \hline 11 & 11 \ \hline & 1 \end{array}

66 = 2 x 3 x 11 x 1
= 2 x 3 x 11

Denominator is not of the form 2m x 5n, where m, n are non-negative integers.

∴ The given number 1566-\dfrac{15}{66} is recurring decimal.

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