Fractions — Statement I-II Type Questions

Statement 1: $\dfrac{3}{5}$ is a fraction between $\dfrac{1}{2} \text{ and } \dfrac{2}{3}$.

Statement 2: If $\dfrac{a}{b} \text{ and } \dfrac{c}{d}$ are two fractions, then $\dfrac{a+c}{b+d}$ lies between $\dfrac{a}{b} \text{ and } \dfrac{c}{d}$.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

For Statement 1, the fraction between $\dfrac{1}{2} \text{ and } \dfrac{2}{3} \text{ is } \dfrac{1+2}{2+3} = \dfrac{3}{5}$.

Converting to a common denominator 30:

$\dfrac{1}{2} = \dfrac{15}{30}, \quad \dfrac{3}{5} = \dfrac{18}{30}, \quad \dfrac{2}{3} = \dfrac{20}{30}$

Since $\dfrac{15}{30} \lt \dfrac{18}{30} \lt \dfrac{20}{30}, \dfrac{3}{5}$ lies between $\dfrac{1}{2} \text{ and } \dfrac{2}{3}$. Statement 1 is true.

Statement 2 gives the general rule used above, which is true.

∴ Both the statements are true.

Hence, Option 1 is the correct option.

Statement 1: $\dfrac{7}{5}, \dfrac{2}{15}, \dfrac{5}{2}, \dfrac{1}{3}$ in descending order of magnitude is $\dfrac{7}{5} \gt \dfrac{5}{2} \gt \dfrac{2}{15} \gt \dfrac{1}{3}$.

Statement 2: After converting the given fractions into like fractions, the fraction with greater numerator is greater.

Which of the following options is correct?

1. Both the statements are true.

2. Both the statements are false.

3. Statement 1 is true, and statement 2 is false.

4. Statement 1 is false, and statement 2 is true.

Answer

LCM of denominators 5, 15, 2 and 3 = 30. Converting to like fractions:

$\dfrac{7}{5} = \dfrac{42}{30}, \quad \dfrac{2}{15} = \dfrac{4}{30}, \quad \dfrac{5}{2} = \dfrac{75}{30}, \quad \dfrac{1}{3} = \dfrac{10}{30}$

The correct descending order is $\dfrac{75}{30} \gt \dfrac{42}{30} \gt \dfrac{10}{30} \gt \dfrac{4}{30}$, i.e. $\dfrac{5}{2} \gt \dfrac{7}{5} \gt \dfrac{1}{3} \gt \dfrac{2}{15}$.

The order stated in Statement 1 is $\dfrac{7}{5} \gt \dfrac{5}{2} \gt \dfrac{2}{15} \gt \dfrac{1}{3}$, which is incorrect. Statement 1 is false.

Statement 2 correctly describes comparison using like fractions, so it is true.

∴ Statement 1 is false, and statement 2 is true.

Hence, Option 4 is the correct option.