Question

Which ratio is greater?
\[11:21\] or \[19:28\].

Answer 1

Hint: In this problem we represent the given ratios in fractions. Using simplification or a calculator we can compare and we can tell which one is greater. Otherwise after representing the given ratios in fraction we can compare the fraction by making the denominator or numerator value the same in both the fractions.

Complete step-by-step answer:
Given,
\[11:21\] and \[19:28\].
We know that the ratios can be represented as fractions. That is
\[ \Rightarrow x:y = \dfrac{x}{y}\].
Then above becomes
 \[11:21 = \dfrac{{11}}{{21}}\] and \[19:28 = \dfrac{{19}}{{28}}\].
Thus we have
\[ \Rightarrow \dfrac{{11}}{{21}}\] and \[\dfrac{{19}}{{28}}\]. ----(1)
Now to make the denominators of the both fractions the same, we take LCM of 21 and 28 and we multiply to the numerator and denominator of the both fractions.
 Now prime factors of 21 are
\[ \Rightarrow 21 = 3 \times 7\]
The prime factors of 28 are
\[ \Rightarrow 28 = 2 \times 2 \times 7\]
Now we can see that the LCM of 21 and 28 is
 \[ \Rightarrow 3 \times {2^2} \times 7\]
\[ \Rightarrow 84\].
Now we multiply 84 in the numerator and the denominator of both fractions.
Then (1) becomes
\[ \Rightarrow \dfrac{{11 \times 84}}{{21 \times 84}}\] and \[\dfrac{{19 \times 84}}{{28 \times 84}}\]
\[ \Rightarrow \dfrac{{11 \times 4}}{{84}}\] and \[\dfrac{{19 \times 3}}{{84}}\]
\[ \Rightarrow \dfrac{{44}}{{84}}\] and \[\dfrac{{57}}{{84}}\].
Now we know that if we have two fractions with same denominator, that is \[\dfrac{a}{b}\] and \[\dfrac{c}{b}\] then we have
If \[a > c\] then \[\dfrac{a}{b} > \dfrac{c}{b}\]
If \[a < c\] then \[\dfrac{a}{b} < \dfrac{c}{b}\]
If \[a = c\] then \[\dfrac{a}{b} = \dfrac{c}{b}\]
Now from \[ \Rightarrow \dfrac{{44}}{{84}}\] and \[\dfrac{{57}}{{84}}\] we can tell that \[44 < 57\].
Then we can say that
\[ \Rightarrow \dfrac{{44}}{{84}} < \dfrac{{57}}{{84}}\]
Hence \[\dfrac{{57}}{{84}}\] is greater.
Thus \[19:28\] is greater.
So, the correct answer is “\[19:28\] ”.

Note: We can solve this if we have a calculator. Now we have \[ \Rightarrow \dfrac{{44}}{{84}} = 0.52380\] and \[ \Rightarrow \dfrac{{57}}{{84}} = 0.67857\]. We can easily see that \[\dfrac{{57}}{{84}}\] is greater. If we need to add two fractions we need to make sure that the denominator of both the fractions are equal. To make the denominator equal we use the LCM method as we did in above. Some numbers cannot be expressed as a fraction and they are called irrational. For example \[\sqrt 2 \].