Convert each of the following ratios into a percentage:
(i) $37 : 100$
(ii) $16 : 25$
(iii) $3 : 5$
(iv) $5 : 4$

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Solution

We have the following:
(i) $37 : 100 = \frac{37}{100} = \frac{37}{100} \times 100 % = 37 %$
(ii) $16 : 25 = \frac{16}{25} = \frac{16}{25} \times 100 % = 16 \times 4 % = 64 %$
(iii) $3 : 5 = \frac{3}{5} = \frac{3}{5} \times 100 % = 3 \times 20 % = 60 %$
(iv) $5 : 4 = \frac{5}{4} = \frac{5}{4} \times 100 % = 5 \times 25 % = 125 %$

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Express each one of the following with rational denominator :

(i) $\frac{1}{3 + \sqrt{2}}$

(ii) $\frac{1}{\sqrt{6} - \sqrt{5}}$

(iii) $\frac{16}{\sqrt{41} - 5}$

(iv) $\frac{30}{5 \sqrt{3} - 3 \sqrt{5}}$

(v) $\frac{1}{2 \sqrt{5} + \sqrt{3}}$

(vi) $\frac{3 \sqrt{3} + 1}{2 \sqrt{2} - \sqrt{3}}$

(vii) $\frac{6 - 4 \sqrt{2}}{6 + 4 \sqrt{2}}$

(viii) $\frac{3 \sqrt{2} + 1}{2 \sqrt{5} - 3}$

(ix) $\frac{b^{2}}{\sqrt{a^{2} + b^{2} + a}}$

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