Question

Put the $ > , < $ or $ = $ signs without changing into equivalent fractions. Apply the rule.
 $ \dfrac{8}{19} \boxed{} \dfrac{8}{11} $ \[\]

Answer 1

Hint: We recall the definition of a fraction, the numerator, and denominator. We use the rule that if two fractions have equal numerators and different denominators then the fraction with the smaller denominator is greater of the fractions to put the appropriate sign from the given $ >, < $ or $ = $ signs.\[\]

Complete step by step answer:
We know that a fraction represents part of a whole. A fraction is always represented in the form of $ \dfrac{p}{q} $ where the positive number $ p $ above the line is called the numerator and the positive number $ q $ below the line is called the denominator. Here $ \dfrac{p}{q} $ tells us that $ p $ is being divided $ q $ equal parts, it means in $ \dfrac{3}{4} $ apples means 3 apples are being divided into 4 equal parts. \[\]

We are asked to find the sign of inequalities $ > , < $ or equality $ = $ between the given fractions $ \dfrac{8}{19} \boxed{} \dfrac{8}{11} $ . We know the rule that if two fractions have equal numerators and different denominators then the fraction with the smaller denominators is greater than the fractions. We see that in both given fractions the numerator is 8 and the denominators are 19 and 11. Since 11 is the smaller denominator the fraction $ \dfrac{8}{11} $ will be greater than the fraction $ \dfrac{8}{19} $ . So we have;
\[\dfrac{8}{19} < \dfrac{8}{11}\]
We can understand the rule in the following way. If we divide 8 into 19 equal parts or 11 equal part then the parts of 8 which will be larger when we divide 8 into 11 equal parts.\[\]

Note:
We note that $ \dfrac{p}{q} $ also means $ p $ parts out of $ q $ equal parts. When we multiply the same number in numerator and denominator, we get equivalent fractions. We call two fractions like fractions when they have the same denominator but may or may not have different numerators. We can convert two fractions into like fractions by converting into equivalent fractions. When we compare two numbers we convert them into like fractions by changing both of their denominators to the least common multiple of denominators. Here in the given fractions the lcm of 19 and 11 is $ 19\times 11=209 $ and the equivalent fractions are $ \dfrac{8\times 11}{19\times 11}=\dfrac{88}{209},\dfrac{8\times 19}{11\times 19}=\dfrac{144}{209} $ . We are told not to convert into equivalent fractions, otherwise, we can compare the numerators to get the greater fraction.