Question

Which ratio is larger in the following pairs?
(i) 3 : 4 or 9 : 16
(ii) 15 : 16 or 24 : 25
(iii) 4 : 7 or 5 : 8
(iv) 9 : 20 or 8 : 13
(v) 1 : 2 or 13 : 27

Answer 1

(i) Writing the ratios as fractions, we have
3 : 4 = $\frac{3}{4}$ and 9 : 16 = $\frac{9}{16}$
Now, LCM of 4 and 16 = 16.
Making the denominator of each fraction = 16, we have
$\frac{3}{4}$ = $\frac{3\times 4}{4\times 4}$ = $\frac{12}{16}$ and the other fraction = $\frac{9}{16}$
Of $\frac{12}{16}$ and $\frac{9}{16}$, clearly $\frac{12}{16}$>$\frac{9}{16}$.
Therefore, $\frac{3}{4}$>$\frac{9}{16}$.
(ii) Writing the ratios as fractions, we have
15 : 16 = $\frac{15}{16}$ and 24 : 25 = $\frac{24}{25}$
Now, LCM of 16 and 25 = 400.
Making the denominator of each fraction = 400, we have
$\frac{15}{16}$ = $\frac{15\times 25}{16\times 25}=$$\frac{375}{400}$ and the other fraction = $\frac{24\times 16}{25\times 16}=$$\frac{384}{400}$
Clearly, 384 > 375. So, $\frac{384}{400}$ > $\frac{375}{400}$.
Therefore, $\frac{24}{25}$>$\frac{15}{16}$.
(iii) Writing the ratios as fractions, we have
4 : 7 = $\frac{4}{7}$ and 5 : 8 = $\frac{5}{8}$
Now, LCM of 7 and 8 = 56.
Making the denominator of each fraction = 56, we have
$\frac{4\times 8}{7\times 8}=$$\frac{32}{56}$ and the other fraction = $\frac{5\times 7}{8\times 7}=$$\frac{35}{56}$
Clearly, 36 > 32. So, $\frac{35}{56}$ > $\frac{32}{56}$.
Therefore, $\frac{5}{8}$>$\frac{4}{7}$.
(iv) Writing the ratios as fractions, we have
9 : 20 = $\frac{9}{20}$ and 8 : 13 = $\frac{8}{13}$
Now, LCM of 20 and 13 = 260.
Making the denominator of each fraction = 260, we have
$\frac{9\times 13}{20\times 13}=$$\frac{117}{260}$ and the other fraction = $\frac{8\times 20}{13\times 20}=$$\frac{160}{260}$
Clearly, 160 > 117. So, $\frac{160}{260}$ > $\frac{117}{260}$.
Therefore, $\frac{8}{13}$>$\frac{9}{20}$.
(v) Writing the ratios as fractions, we have
1 : 2 = $\frac{1}{2}$ and 13 : 27 = $\frac{13}{27}$
Now, LCM of 2 and 27 = 54.
Making the denominator of each fraction = 54, we have
$\frac{1\times 27}{2\times 27}=$$\frac{27}{54}$ and the other fraction = $\frac{13\times 2}{27\times 2}=$$\frac{26}{54}$
Clearly, 27 > 26. So, $\frac{27}{54}$ > $\frac{26}{54}$.
Therefore, $\frac{1}{2}$>$\frac{13}{27}$.