What is the rational number equivalent to 3.24 with a bar over the 24?

by

To convert the repeating decimal 3.24̅ (where ’24’ is the repeating part) into a rational number, we can follow these steps:

  1. Identify the decimal: The decimal can be expressed as 3.24242424…
  2. Separate the whole number: The integer part is 3, and we will focus on the decimal part 0.242424…
  3. Set up an equation: Let x = 0.242424....
  4. Multiply to eliminate the repeating part: Since the repeating part has 2 digits (24), we’ll multiply by 100:
    5. 100x = 24.242424...
  5. Set up another equation: Now write the first equation:
    6. x = 0.242424...
  6. Subtract the two equations: Now, we can subtract the second equation from the first:
    7. 100x - x = 24.242424... - 0.242424...
  7. This simplifies to:
    8. 99x = 24
  8. Solve for x: Dividing both sides by 99 gives:
    9. x = 24/99
  9. Simplify the fraction: We can simplify 24/99 by dividing the numerator and denominator by 3:
    10. x = 8/33
  10. Combine the whole number and the decimal part: Since we started with 3.242424…, we add the whole number back in:
    11. 3 + 8/33 = 3 8/33

Therefore, the rational number equivalent to 3.24̅ is 3 8/33 or, as an improper fraction, 3.2424̅ = 107/33.

Leave a Comment