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QUESTION: Hello:

If fractions with one (1) as the numerator are being compared to determine which one is larger than the other, just look at the denominator and the fraction with the smaller denominator is the larger fraction. For example, 1/7 is larger than 1/12.

However, if the numerator is larger than one (1) on both fractions and also different, it is not so easy to determine which fraction is larger. For example, 5/7 and 7/12.

Without changing the denominators to like denominators or without changing the two fractions to decimals, how can someone determine which fraction is larger? 5/7 is larger than 7/12.

I thank you for your reply.

ANSWER: a > b if and only if a·b⁻¹ > 1

Multiply one fraction by the reciprocal of the other fraction:

(5/7)·(7/12)⁻¹ = (5/7)(12/7) = 60/49 > 1

Therefore, (5/7) > (7/12).

---------- FOLLOW-UP ----------

QUESTION: Hello:

I have a follow-up question.

If 5/7 were less than 1 would 7/12 be larger?

I think you are asking about the converse case:

a < b if and only if ab⁻¹ < 1

For example, (7/12) < (5/7)

(7/12)·(5/7)⁻¹ = (7/12)·(7/5) = 49/60 < 1

However, I overlooked an important detail: These rules work only if a and b are both positive.

If a and b have different signs, you don't need to do any calculations because the positive term is obviously larger than the negative term.

-1/2 < 1/2

If a and b are both negative, then the direction of the inequality changes:

a < b iff a·b⁻¹ > 1

a > b iff a·b⁻¹ < 1

For example, (-3/4) < (-2/3)

(-3/4)(-2/3)⁻¹ = (-3/4)(-3/2) = 9/8 > 1

(-2/3)(-3/4)⁻¹ = (-2/3)(-4/3)⁻¹ = 8/9 < 1

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