What are Rational Numbers?

Rational numbers are numbers that can be written as a fraction, where the numerator (top number) and the denominator (bottom number) are both integers. This means any number that can be expressed as $$\frac{a}{b}$$, where $$a$$ and $$b$$ are whole numbers and $$b \neq 0$$, is a rational number. For example, $$\frac{1}{2}$$, $$\frac{3}{4}$$, and even whole numbers like $$5$$ (which can be written as $$\frac{5}{1}$$) are all rational.

Comparing Rational Numbers

When we compare rational numbers, we want to find out which one is bigger or smaller. Here are some key steps to help you compare:

Step 1: Convert to the Same Denominator

Sometimes, fractions have different denominators. To compare them easily, we can convert them to the same denominator. For example, to compare $$\frac{1}{4}$$ and $$\frac{1}{2}$$, we can convert $$\frac{1}{2}$$ to a fraction with a denominator of 4.

$$\frac{1}{2} = \frac{2}{4}$$

Now we can compare $$\frac{1}{4}$$ and $$\frac{2}{4}$$.

Step 2: Compare Numerators

Once the fractions have the same denominator, we can compare their numerators. The larger numerator means the larger fraction. In our example, $$\frac{2}{4}$$ is bigger than $$\frac{1}{4}$$ since 2 is greater than 1.

Step 3: Use Decimal Form

If fractions are tricky, you can also convert them to decimal form. For example, $$\frac{1}{4} = 0.25$$ and $$\frac{1}{2} = 0.5$$. Now it’s easy to see that 0.5 is greater than 0.25.

Tips and Tricks

  1. Cross-Multiply: When comparing two fractions, you can cross-multiply. For example, to compare $$\frac{1}{3}$$ and $$\frac{2}{5}$$, calculate $$1 \times 5$$ and $$2 \times 3$$. Compare the results: if $$5 > 6$$, then $$\frac{1}{3} < \frac{2}{5}$$.
  2. Use a Number Line: Sometimes drawing a number line can help visualize where the rational numbers fall, making it easier to compare them.
  3. Practice with Common Fractions: Get familiar with common fractions and their decimal equivalents. Knowing that $$\frac{1}{2} = 0.5$$ helps when comparing fractions quickly.

Questions

Easy Level Questions

  1. Compare $$\frac{1}{2}$$ and $$\frac{1}{4}$$.
  2. Which is greater: $$\frac{3}{5}$$ or $$\frac{2}{5}$$?
  3. Compare $$\frac{1}{3}$$ and $$\frac{2}{3}$$.
  4. Which is smaller: $$\frac{5}{6}$$ or $$\frac{3}{6}$$?
  5. Compare $$\frac{2}{8}$$ and $$\frac{1}{4}$$.
  6. Which is greater: $$\frac{7}{10}$$ or $$\frac{5}{10}$$?
  7. Compare $$\frac{4}{5}$$ and $$\frac{3}{5}$$.
  8. Which is smaller: $$\frac{2}{7}$$ or $$\frac{3}{7}$$?
  9. Compare $$\frac{1}{5}$$ and $$\frac{2}{5}$$.
  10. Which is greater: $$\frac{1}{8}$$ or $$\frac{1}{9}$$?
  11. Compare $$\frac{3}{4}$$ and $$\frac{1}{2}$$.
  12. Which is smaller: $$\frac{0}{1}$$ or $$\frac{1}{1}$$?
  13. Compare $$\frac{1}{6}$$ and $$\frac{1}{3}$$.
  14. Which is greater: $$\frac{2}{2}$$ or $$\frac{1}{2}$$?
  15. Compare $$\frac{4}{10}$$ and $$\frac{5}{10}$$.
  16. Which is smaller: $$\frac{3}{8}$$ or $$\frac{4}{8}$$?
  17. Compare $$\frac{2}{3}$$ and $$\frac{1}{3}$$.
  18. Which is greater: $$\frac{5}{5}$$ or $$\frac{4}{5}$$?
  19. Compare $$\frac{3}{12}$$ and $$\frac{1}{4}$$.
  20. Which is smaller: $$\frac{2}{9}$$ or $$\frac{3}{9}$$?

Medium Level Questions

  1. Compare $$\frac{3}{4}$$ and $$\frac{5}{8}$$.
  2. Which is greater: $$\frac{7}{12}$$ or $$\frac{3}{8}$$?
  3. Compare $$\frac{2}{5}$$ and $$\frac{1}{2}$$.
  4. Which is smaller: $$\frac{4}{9}$$ or $$\frac{5}{12}$$?
  5. Compare $$\frac{1}{10}$$ and $$\frac{3}{20}$$.
  6. Which is greater: $$\frac{9}{16}$$ or $$\frac{5}{8}$$?
  7. Compare $$\frac{7}{10}$$ and $$\frac{2}{3}$$.
  8. Which is smaller: $$\frac{3}{5}$$ or $$\frac{4}{7}$$?
  9. Compare $$\frac{5}{6}$$ and $$\frac{2}{3}$$.
  10. Which is greater: $$\frac{1}{2}$$ or $$\frac{3}{8}$$?
  11. Compare $$\frac{2}{7}$$ and $$\frac{3}{10}$$.
  12. Which is smaller: $$\frac{5}{12}$$ or $$\frac{4}{9}$$?
  13. Compare $$\frac{8}{15}$$ and $$\frac{1}{2}$$.
  14. Which is greater: $$\frac{3}{8}$$ or $$\frac{4}{10}$$?
  15. Compare $$\frac{1}{6}$$ and $$\frac{2}{5}$$.
  16. Which is smaller: $$\frac{2}{3}$$ or $$\frac{3}{4}$$?
  17. Compare $$\frac{5}{8}$$ and $$\frac{7}{10}$$.
  18. Which is greater: $$\frac{2}{5}$$ or $$\frac{3}{7}$$?
  19. Compare $$\frac{1}{2}$$ and $$\frac{5}{12}$$.
  20. Which is smaller: $$\frac{4}{11}$$ or $$\frac{3}{8}$$?

Hard Level Questions

  1. Compare $$\frac{11}{18}$$ and $$\frac{5}{9}$$.
  2. Which is greater: $$\frac{7}{15}$$ or $$\frac{3}{7}$$?
  3. Compare $$\frac{4}{5}$$ and $$\frac{7}{12}$$.
  4. Which is smaller: $$\frac{8}{21}$$ or $$\frac{4}{9}$$?
  5. Compare $$\frac{2}{3}$$ and $$\frac{5}{8}$$.
  6. Which is greater: $$\frac{11}{20}$$ or $$\frac{5}{9}$$?
  7. Compare $$\frac{3}{5}$$ and $$\frac{9}{16}$$.
  8. Which is smaller: $$\frac{10}{21}$$ or $$\frac{5}{11}$$?
  9. Compare $$\frac{13}{24}$$ and $$\frac{7}{12}$$.
  10. Which is greater: $$\frac{4}{7}$$ or $$\frac{5}{8}$$?
  11. Compare $$\frac{7}{10}$$ and $$\frac{1}{2}$$.
  12. Which is smaller: $$\frac{3}{5}$$ or $$\frac{7}{14}$$?
  13. Compare $$\frac{8}{15}$$ and $$\frac{5}{9}$$.
  14. Which is greater: $$\frac{9}{20}$$ or $$\frac{11}{25}$$?
  15. Compare $$\frac{1}{2}$$ and $$\frac{3}{5}$$.
  16. Which is smaller: $$\frac{4}{9}$$ or $$\frac{5}{11}$$?
  17. Compare $$\frac{2}{3}$$ and $$\frac{4}{7}$$.
  18. Which is greater: $$\frac{5}{8}$$ or $$\frac{3}{5}$$?
  19. Compare $$\frac{7}{15}$$ and $$\frac{2}{5}$$.
  20. Which is smaller: $$\frac{10}{18}$$ or $$\frac{7}{12}$$?

Answers

Easy Level Answers

  1. $$\frac{1}{2} > \frac{1}{4}$$. The numerator of $$\frac{1}{2}$$ is greater.
  2. $$\frac{3}{5} > \frac{2}{5}$$. Both fractions have the same denominator, so we compare the numerators.
  3. $$\frac{2}{3} > \frac{1}{3}$$. Again, the numerators tell us which is larger.
  4. $$\frac{5}{6} > \frac{3}{6}$$. The larger numerator indicates that $$\frac{5}{6}$$ is greater.
  5. $$\frac{1}{4} = \frac{2}{8}$$. They are equal.
  6. $$\frac{7}{10} > \frac{5}{10}$$. The numerator of $$\frac{7}{10}$$ is greater than that of $$\frac{5}{10}$$.
  7. $$\frac{4}{5} > \frac{3}{5}$$. Compare the numerators to see which is larger.
  8. $$\frac{3}{7} > \frac{2}{7}$$. The larger numerator indicates that $$\frac{3}{7}$$ is greater.
  9. $$\frac{2}{5} > \frac{1}{5}$$. Both fractions have the same denominator.
  10. $$\frac{1}{8} > \frac{1}{9}$$. The larger denominator means the fraction is smaller.
  11. $$\frac{3}{4} > \frac{1}{2}$$. Compare the numerators once they have the same denominator.
  12. $$0 < 1$$. The numerator of $$\frac{1}{1}$$ is greater.
  13. $$\frac{1}{3} > \frac{1}{6}$$. The numerator comparison shows which is larger.
  14. $$\frac{2}{2} > \frac{1}{2}$$. The numerator is greater.
  15. $$\frac{5}{10} > \frac{4}{10}$$. Compare the numerators directly.
  16. $$\frac{3}{8} < \frac{4}{8}$$. The numerator comparison shows that $$\frac{3}{8}$$ is smaller.
  17. $$\frac{2}{3} > \frac{1}{3}$$. Compare the numerators.
  18. $$\frac{5}{5} > \frac{4}{5}$$. The larger numerator indicates that $$\frac{5}{5}$$ is greater.
  19. $$\frac{1}{4} = \frac{3}{12}$$. They are equal.
  20. $$\frac{2}{9} < \frac{3}{9}$$. The numerator comparison shows which is smaller.

Medium Level Answers

  1. $$\frac{3}{4} > \frac{5}{8}$$. Convert $$\frac{5}{8}$$ to $$\frac{6}{8}$$ to compare.
  2. $$\frac{7}{12} > \frac{3}{8}$$. Convert to the same denominator to compare.
  3. $$\frac{2}{5} < \frac{1}{2}$$. Convert $$\frac{1}{2}$$ to $$\frac{2.5}{5}$$ to see it’s greater.
  4. $$\frac{4}{9} < \frac{5}{12}$$. Compare after converting to the same denominator.
  5. $$\frac{1}{10} < \frac{3}{20}$$. Convert to the same denominator to compare.
  6. $$\frac{9}{16} > \frac{5}{8}$$. Convert $$\frac{5}{8}$$ to $$\frac{10}{16}$$ for comparison.
  7. $$\frac{7}{10} > \frac{2}{3}$$. Convert to the same denominator.
  8. $$\frac{3}{5} < \frac{4}{7}$$. Compare after converting to the same denominator.
  9. $$\frac{5}{6} > \frac{2}{3}$$. Convert $$\frac{2}{3}$$ to $$\frac{4}{6}$$.
  10. $$\frac{1}{2} > \frac{3}{8}$$. Convert $$\frac{1}{2}$$ to $$\frac{4}{8}$$ for comparison.
  11. $$\frac{2}{7} < \frac{3}{10}$$. Compare after converting to the same denominator.
  12. $$\frac{5}{12} < \frac{4}{9}$$. Convert to the same denominator for comparison.
  13. $$\frac{8}{15} < \frac{1}{2}$$. Convert $$\frac{1}{2}$$ to $$\frac{7.5}{15}$$.
  14. $$\frac{4}{7} < \frac{5}{8}$$. Compare after converting to the same denominator.
  15. $$\frac{1}{6} < \frac{2}{5}$$. Convert to the same denominator for comparison.
  16. $$\frac{4}{9} < \frac{3}{4}$$. Convert to the same denominator.
  17. $$\frac{5}{8} < \frac{7}{10}$$. Compare after converting to the same denominator.
  18. $$\frac{2}{5} < \frac{3}{7}$$. Both fractions can be converted to the same denominator.
  19. $$\frac{1}{2} > \frac{5}{12}$$. Convert $$\frac{1}{2}$$ for easier comparison.
  20. $$\frac{4}{11} < \frac{3}{8}$$. Convert to the same denominator to compare.

Hard Level Answers

  1. $$\frac{11}{18} > \frac{5}{9}$$. Convert $$\frac{5}{9}$$ to $$\frac{10}{18}$$.
  2. $$\frac{7}{15} < \frac{3}{7}$$. Convert to the same denominator.
  3. $$\frac{4}{5} > \frac{7}{12}$$. Convert to the same denominator for comparison.
  4. $$\frac{8}{21} < \frac{4}{9}$$. Compare after converting.
  5. $$\frac{2}{3} > \frac{5}{8}$$. Convert to the same denominator.
  6. $$\frac{11}{20} < \frac{5}{9}$$. Compare after converting.
  7. $$\frac{3}{5} > \frac{9}{16}$$. Convert to the same denominator.
  8. $$\frac{10}{21} < \frac{5}{11}$$. Compare after converting.
  9. $$\frac{13}{24} < \frac{7}{12}$$. Convert to the same denominator.
  10. $$\frac{4}{7} < \frac{5}{8}$$. Compare after converting.
  11. $$\frac{7}{10} > \frac{1}{2}$$. Compare directly, since $$\frac{1}{2} = \frac{5}{10}$$.
  12. $$\frac{3}{5} = \frac{7}{14}$$. They are equal.
  13. $$\frac{5}{9} > \frac{8}{15}$$. Convert to the same denominator.
  14. $$\frac{11}{25} < \frac{9}{20}$$. Convert to the same denominator.
  15. $$\frac{1}{2} < \frac{3}{5}$$. Convert to the same denominator.
  16. $$\frac{4}{9} < \frac{5}{11}$$. Convert to the same denominator for comparison.
  17. $$\frac{2}{3} > \frac{4}{7}$$. Convert to the same denominator.
  18. $$\frac{5}{8} > \frac{3}{5}$$. Compare after converting to the same denominator.
  19. $$\frac{7}{15} > \frac{2}{5}$$. Convert $$\frac{2}{5}$$ to $$\frac{6}{15}$$.
  20. $$\frac{10}{18} < \frac{7}{12}$$. Convert to the same denominator for comparison.

By following these steps and using the tips provided, you should find it easier to compare rational numbers. Remember to practice with various examples to strengthen your understanding!