Introduction to Rational Numbers

Rational numbers are numbers that can be written as a fraction, where the numerator (top number) and denominator (bottom number) are both integers. This means that numbers like ( \frac{1}{2} ), ( -\frac{3}{4} ), and ( 5 ) are all rational numbers.

When we compare and order rational numbers, we look at their value to decide which is bigger or smaller.

Comparing Rational Numbers

Key Rules for Comparing Rational Numbers

  1. Same Denominator: If the fractions have the same denominator, compare the numerators. The bigger numerator means a bigger fraction.
    • Example: ( \frac{3}{5} ) and ( \frac{4}{5} ). Here, ( 4 > 3 ), so ( \frac{4}{5} > \frac{3}{5} ).
  2. Different Denominators: If the fractions have different denominators, you can find a common denominator or convert them to decimal form to compare.
    • Example: ( \frac{1}{2} ) and ( \frac{1}{3} ). Convert to decimals: ( 0.5 ) and ( 0.33 ). So, ( \frac{1}{2} > \frac{1}{3} ).
  3. Negative Numbers: Remember that negative numbers are always less than positive numbers.
    • Example: ( -1 ) is less than ( 0 ) because all positive numbers are greater than negative numbers.

Tips and Tricks

  • Convert to Decimals: Sometimes it’s easier to compare numbers as decimals.
  • Draw Number Lines: Visualising numbers on a number line can help understand their order.
  • Use Cross Multiplication: For comparing two fractions ( \frac{a}{b} ) and ( \frac{c}{d} ), you can check if ( a \times d ) is greater than ( b \times c ).

Ordering Rational Numbers

Ordering means putting numbers in a sequence, either from smallest to largest or largest to smallest.

Steps to Order Rational Numbers

  1. Convert all numbers to the same format (either all fractions or all decimals).
  2. Compare each number using the rules above.
  3. List them in order based on their value.

Example Word Problem

Imagine you have the following numbers: ( \frac{2}{3} ), ( -\frac{1}{2} ), ( 0.75 ), and ( \frac{1}{4} ).

To order them:

  1. Convert ( \frac{2}{3} ) to a decimal: ( 0.67 ).
  2. Compare: ( -0.5 < 0.25 < 0.67 < 0.75 ).
  3. So, the order is: ( -\frac{1}{2}, \frac{1}{4}, \frac{2}{3}, 0.75 ).

Practice Questions

Easy Level (20 Questions)

  1. Compare: ( \frac{1}{2} ) and ( \frac{1}{4} )
  2. Order: ( 0.5, 0.25, 0.75 )
  3. Compare: ( \frac{3}{5} ) and ( \frac{2}{5} )
  4. Order: ( -0.1, 0.1, 0 )
  5. Compare: ( \frac{1}{3} ) and ( \frac{1}{6} )
  6. Order: ( -\frac{1}{2}, -\frac{1}{4}, 0 )
  7. Compare: ( \frac{2}{8} ) and ( \frac{1}{4} )
  8. Order: ( -2, -1, 0 )
  9. Compare: ( 0.2 ) and ( 0.5 )
  10. Order: ( \frac{1}{2}, \frac{3}{2}, \frac{5}{2} )
  11. Compare: ( -\frac{3}{4} ) and ( -\frac{1}{2} )
  12. Order: ( \frac{1}{8}, \frac{3}{8}, \frac{5}{8} )
  13. Compare: ( 0.9 ) and ( 0.99 )
  14. Order: ( -1, -\frac{1}{2}, \frac{1}{2}, 1 )
  15. Compare: ( \frac{4}{5} ) and ( \frac{3}{5} )
  16. Order: ( 0, 0.1, 0.01 )
  17. Compare: ( \frac{5}{6} ) and ( \frac{4}{6} )
  18. Order: ( -2, -1.5, -1.8 )
  19. Compare: ( 0.25 ) and ( 0.75 )
  20. Order: ( \frac{1}{10}, \frac{1}{2}, \frac{3}{10} )

Medium Level (20 Questions)

  1. Compare: ( \frac{7}{8} ) and ( \frac{3}{4} )
  2. Order: ( -\frac{2}{3}, \frac{1}{3}, \frac{2}{3} )
  3. Compare: ( 0.45 ) and ( 0.5 )
  4. Order: ( \frac{4}{5}, \frac{3}{10}, \frac{1}{2} )
  5. Compare: ( \frac{2}{9} ) and ( \frac{1}{3} )
  6. Order: ( \frac{5}{8}, \frac{1}{2}, \frac{3}{8} )
  7. Compare: ( 1.1 ) and ( 1.01 )
  8. Order: ( -1, -1.5, 0, 1 )
  9. Compare: ( \frac{5}{12} ) and ( \frac{4}{12} )
  10. Order: ( \frac{2}{5}, \frac{1}{5}, \frac{3}{5} )
  11. Compare: ( \frac{3}{7} ) and ( \frac{2}{7} )
  12. Order: ( -0.3, -0.1, 0.3 )
  13. Compare: ( 0.6 ) and ( \frac{2}{3} )
  14. Order: ( \frac{1}{4}, \frac{3}{4}, \frac{1}{2} )
  15. Compare: ( \frac{7}{10} ) and ( \frac{3}{4} )
  16. Order: ( \frac{1}{6}, \frac{5}{6}, \frac{4}{6} )
  17. Compare: ( -0.5 ) and ( -0.75 )
  18. Order: ( 1.2, 0.9, 0.8 )
  19. Compare: ( \frac{2}{7} ) and ( \frac{3}{8} )
  20. Order: ( -2, -3, -1 )

Hard Level (20 Questions)

  1. Compare: ( \frac{19}{20} ) and ( \frac{3}{4} )
  2. Order: ( -\frac{5}{6}, \frac{0}{1}, \frac{5}{6} )
  3. Compare: ( 1.25 ) and ( \frac{5}{4} )
  4. Order: ( \frac{7}{10}, \frac{3}{5}, \frac{4}{10} )
  5. Compare: ( \frac{11}{12} ) and ( \frac{10}{12} )
  6. Order: ( \frac{5}{3}, \frac{4}{3}, \frac{1}{3} )
  7. Compare: ( 0.333 ) and ( \frac{1}{3} )
  8. Order: ( -\frac{1}{3}, -\frac{2}{3}, -\frac{1}{2} )
  9. Compare: ( \frac{4}{5} ) and ( 0.8 )
  10. Order: ( 1.5, 1.1, 2.1 )
  11. Compare: ( \frac{8}{9} ) and ( 0.9 )
  12. Order: ( \frac{1}{10}, \frac{2}{10}, \frac{3}{10} )
  13. Compare: ( -\frac{4}{5} ) and ( -\frac{3}{5} )
  14. Order: ( 0.25, 0.5, 0.75 )
  15. Compare: ( \frac{5}{8} ) and ( \frac{5}{9} )
  16. Order: ( -1, -2, 0 )
  17. Compare: ( \frac{3}{2} ) and ( 1.5 )
  18. Order: ( \frac{9}{10}, \frac{1}{2}, \frac{7}{10} )
  19. Compare: ( -0.2 ) and ( -0.3 )
  20. Order: ( 0, \frac{1}{4}, \frac{3}{4} )

Answers and Explanations

Easy Level Answers

  1. ( \frac{1}{2} > \frac{1}{4} ) because ( 1 > 0.5 ).
  2. The order is ( 0.25, 0.5, 0.75 ) because ( 0.25 < 0.5 < 0.75 ).
  3. ( \frac{3}{5} > \frac{2}{5} ) because ( 3 > 2 ).
  4. The order is ( -0.1, 0, 0.1 ) since ( -0.1 < 0 < 0.1 ).
  5. ( \frac{1}{3} > \frac{1}{6} ) because ( 1 > 0.5 ).
  6. The order is ( -\frac{1}{2}, 0, \frac{1}{3} ) since ( -0.5 < 0 < 0.33 ).
  7. ( \frac{2}{8} = \frac{1}{4} ) so they are equal.
  8. The order is ( -2, -1, 0 ) since ( -2 < -1 < 0 ).
  9. ( 0.5 < 0.5 ) so they are equal.
  10. The order is ( \frac{1}{2}, \frac{3}{2}, \frac{5}{2} ) since ( 0.5 < 1.5 < 2.5 ).
  11. ( -\frac{3}{4} < -\frac{1}{2} ) because all negative numbers are in reverse order.
  12. The order is ( \frac{1}{8}, \frac{3}{8}, \frac{5}{8} ) since ( 0.125 < 0.375 < 0.625 ).
  13. ( 0.9 > 0.99 ) because ( 0.99 ) is closer to ( 1 ).
  14. The order is ( -2, -1.5, -1, 0, 1 ) since they are all negative and zero is larger.
  15. ( \frac{4}{5} > \frac{3}{5} ) because ( 4 > 3 ).
  16. The order is ( 0, 0.1, 0.01 ) as ( 0 < 0.1 < 0.01 ).
  17. ( \frac{5}{6} > \frac{4}{6} ) since ( 5 > 4 ).
  18. The order is ( -2 < -1.5 < -1 ).
  19. ( 0.75 > 0.25 ) so ( 0.75 > 0.25 ).
  20. The order is ( \frac{1}{10}, \frac{3}{10}, \frac{1}{2} ) since ( 0.1 < 0.3 < 0.5 ).

Medium Level Answers

  1. ( \frac{19}{20} > \frac{3}{4} ) because ( 0.95 > 0.75 ).
  2. The order is ( -\frac{5}{6}, \frac{0}{1}, \frac{5}{6} ) since ( -0.833 < 0 < 0.833 ).
  3. ( 1.25 = \frac{5}{4} ) so they are equal.
  4. The order is ( \frac{3}{5}, \frac{4}{10}, \frac{7}{10} ) since ( 0.6 < 0.4 < 0.7 ).
  5. ( \frac{2}{9} < \frac{1}{3} ) because ( 0.222 < 0.333 ).
  6. The order is ( \frac{3}{8}, \frac{1}{2}, \frac{5}{8} ) since ( 0.375 < 0.5 < 0.625 ).
  7. ( 1.1 > 1.01 ) since ( 1.1 > 1.01 ).
  8. The order is ( -2, -1.5, 0 ) since ( -2 < -1.5 < 0 ).
  9. ( \frac{5}{12} > \frac{4}{12} ) since ( 5 > 4 ).
  10. The order is ( \frac{1}{5}, \frac{2}{5}, \frac{3}{5} ) since ( 0.2 < 0.4 < 0.6 ).
  11. ( \frac{3}{7} > \frac{2}{7} ) because ( 3 > 2 ).
  12. The order is ( -0.3 < -0.1 < 0.3 ).
  13. ( 0.6 > \frac{2}{3} ) since ( 0.666 < 0.666 ).
  14. The order is ( \frac{1}{4}, \frac{1}{2}, \frac{3}{4} ) since ( 0.25 < 0.5 < 0.75 ).
  15. ( \frac{7}{10} < 1.5 ) since ( 0.7 < 1.5 ).
  16. The order is ( \frac{1}{6}, \frac{4}{6}, \frac{5}{6} ) since ( 0.166 < 0.666 < 0.833 ).
  17. ( -0.5 < -0.75 ) since ( -0.5 > -0.75 ).
  18. The order is ( 0.8 < 0.9 < 1.2 ).
  19. ( \frac{2}{7} < \frac{3}{8} ) since ( 0.285 < 0.375 ).
  20. The order is ( -3 < -2 < -1 ).

Hard Level Answers

  1. ( \frac{19}{20} > \frac{3}{4} ) since ( 0.95 > 0.75 ).
  2. The order is ( -\frac{5}{6}, \frac{0}{1}, \frac{5}{6} ) since ( -0.833 < 0 < 0.833 ).
  3. ( 1.25 = \frac{5}{4} ) so they are equal.
  4. The order is ( \frac{4}{10}, \frac{3}{5}, \frac{7}{10} ) since ( 0.4 < 0.6 < 0.7 ).
  5. ( \frac{11}{12} > \frac{10}{12} ) since ( 11 > 10 ).
  6. The order is ( \frac{5}{3}, \frac{4}{3}, \frac{1}{3} ) since ( 1.666 > 1.333 > 0.333 ).
  7. ( 0.333 \approx \frac{1}{3} ) so they are very close.
  8. The order is ( -2 < -1.5 < -0.5 ).
  9. ( 0.9 = \frac{8}{9} ) so they are equal.
  10. The order is ( 1.1 < 1.5 < 2.1 ).
  11. ( \frac{8}{9} > 0.9 ) since ( 0.889 > 0.9 ).
  12. The order is ( \frac{1}{10}, \frac{2}{10}, \frac{3}{10} ) since ( 0.1 < 0.2 < 0.3 ).
  13. ( -\frac{4}{5} < -\frac{3}{5} ) since ( -0.8 < -0.6 ).
  14. The order is ( 0.25, 0.5, 0.75 ) since ( 0.25 < 0.5 < 0.75 ).
  15. ( \frac{5}{8} > \frac{5}{9} ) since ( 0.625 > 0.555 ).
  16. The order is ( -2 < -1 < 0 ).
  17. ( 1.5 = \frac{3}{2} ) so they are equal.
  18. The order is ( \frac{9}{10