Which of the following fractions is smaller than $\dfrac{1}{4}$ ?
(1) $\dfrac{6}{25}$
(2) $\dfrac{7}{12}$
(3) $\dfrac{6}{9}$
(4) $\dfrac{7}{11}$
(1) $\dfrac{6}{25}$
Convert to decimals$$\dfrac{1}{4} = 0.25$$
$$\dfrac{6}{25} = 0.24$$
$$\dfrac{7}{12} \approx 0.583$$
$$\dfrac{6}{9} = \dfrac{2}{3} \approx 0.667$$
$$\dfrac{7}{11} \approx 0.636$$
(1) $\dfrac{6}{25}$
Common Denominator Method
Compare $\dfrac{1}{4}$ with each option:
Option (1): $\dfrac{6}{25}$$$\dfrac{1}{4} = \dfrac{1 \times 25}{4 \times 25} = \dfrac{25}{100}$$ $$\dfrac{6}{25} = \dfrac{6 \times 4}{25 \times 4} = \dfrac{24}{100}$$ $$\bm{\dfrac{24}{100} < \dfrac{25}{100}}$$
Option (2): $\dfrac{7}{12}$$$\dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12}$$ $$\dfrac{7}{12}$$ $$\dfrac{7}{12} > \dfrac{3}{12} \quad \Rightarrow \quad \dfrac{7}{12} \not< \dfrac{1}{4}$$
Option (3): $\dfrac{6}{9}$$$\dfrac{6}{9} = \dfrac{2}{3}$$ $$\dfrac{1}{4} = \dfrac{1 \times 3}{4 \times 3} = \dfrac{3}{12}$$ $$\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$$ $$\dfrac{8}{12} > \dfrac{3}{12} \quad \Rightarrow \quad \dfrac{6}{9} \not< \dfrac{1}{4}$$
Option (4): $\dfrac{7}{11}$$$\dfrac{1}{4} = \dfrac{1 \times 11}{4 \times 11} = \dfrac{11}{44}$$ $$\dfrac{7}{11} = \dfrac{7 \times 4}{11 \times 4} = \dfrac{28}{44}$$ $$\dfrac{28}{44} > \dfrac{11}{44} \quad \Rightarrow \quad \dfrac{7}{11} \not< \dfrac{1}{4}$$
