How do you compare each pair of fractions with <, > or = given #5/4, 7/6#?

The easiest way to compare two fractions is to rewrite one (or both) of them so that their denominators match.

You can think about the denominator of a fraction like a unit, similar to inches, centimeters, feet, etc. If you were asked to compare 5 inches to 7 centimeters, you'd first need to convert one value (or both) so that their units matched.

In fractions, the "unit" is the denominator. The first value is 5 "fourths", and the second value is 7 "sixths". These aren't in the same unit, so we can't just compare 5 to 7. But, we can convert the numbers to the same unit.

For every fourth, there are 3 twelfths. We have 5 fourths, so we have:

This was just for explanation. There's a much more concise way to write this change of "unit":

Similarly, if we have 7 sixths, and there are 2 twelfths in a sixth, then we have 14 twelfths:

There's a trick to comparing fractions where the numerator is 1 more than the denominator:

This works because the numerators are getting proportionally closer to their denominators.

This is a little taste of calculus for you. It's not so bad. ;)

To compare the fractions ( \frac{5}{4} ) and ( \frac{7}{6} ), we can find a common denominator and then compare their numerators.

First, we find a common denominator, which is the least common multiple (LCM) of 4 and 6, which is 12.

[ \frac{5}{4} = \frac{15}{12} ] [ \frac{7}{6} = \frac{14}{12} ]

Now, comparing the numerators: [ 15 > 14 ]

So, ( \frac{5}{4} > \frac{7}{6} ).