Find a rational between 2/3 and 3/4

General
1

find a rational between 2/3 and 3/4

2

Find a rational number between \frac{2}{3} and \frac{3}{4}

To find a rational number between two given fractions, such as \frac{2}{3} and \frac{3}{4} , we can use several methods, but one of the straightforward approaches is to convert them to a common denominator, compare them, and then find a number between them.

1. Convert to a Common Denominator

First, let’s find the least common denominator of the fractions \frac{2}{3} and \frac{3}{4} . The denominators here are 3 and 4. The least common multiple (LCM) of 3 and 4 is 12. So, we’ll convert both fractions to this common denominator:

  • Convert \frac{2}{3} :
    To find an equivalent fraction with a denominator of 12, multiply both the numerator and denominator by 4:
    $$ \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} $$

  • Convert \frac{3}{4} :
    To find an equivalent fraction with a denominator of 12, multiply both the numerator and denominator by 3:
    $$ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} $$

Now our two fractions are \frac{8}{12} and \frac{9}{12} .

2. Find a Rational Number Between Them

When the fractions are expressed with a common denominator, finding a rational number between \frac{8}{12} and \frac{9}{12} becomes simple. A straightforward rational number between these two is \frac{17}{24} . To verify, let’s ensure this number lies between the given two fractions:

  • Convert \frac{17}{24} to have a denominator of 48:
    $$ \frac{17}{24} = \frac{17 \times 2}{24 \times 2} = \frac{34}{48} $$

  • Compare \frac{8}{12} and \frac{34}{48} :
    Convert \frac{8}{12} to have a denominator of 48:
    $$ \frac{8}{12} = \frac{8 \times 4}{12 \times 4} = \frac{32}{48} $$
    Here, \frac{34}{48} is greater than \frac{32}{48} .

  • Compare \frac{34}{48} and \frac{9}{12} :
    Convert \frac{9}{12} to have a denominator of 48:
    $$ \frac{9}{12} = \frac{9 \times 4}{12 \times 4} = \frac{36}{48} $$
    Here, \frac{34}{48} is less than \frac{36}{48} .

Thus, \frac{17}{24} is definitely a rational number between \frac{2}{3} and \frac{3}{4} .

3. Another Method: Averaging

An alternative method to find a rational number between two given fractions is to average them.

Calculate the average:
$$ \frac{\frac{2}{3} + \frac{3}{4}}{2} = \frac{\frac{8}{12} + \frac{9}{12}}{2} = \frac{17/12}{2} = \frac{17}{24} $$

This shows that \frac{17}{24} is the average and, thus, rational number lying between the two fractions. This method reaffirms our earlier calculation.

Thus, we’ve demonstrated two methods to find the rational number \frac{17}{24} , and we verified it does indeed lie between \frac{2}{3} and \frac{3}{4} . Keep experimenting with these strategies for more practice! Happy learning! @anonymous6