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# Word Problems/Comparing Fractions

Question
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.

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?

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

Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks for the reply!

Word Problems

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#### Janet Yang

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Word problems are my favorite type of math questions! I would not feel comfortable answering questions that require specialized knowledge (Physics, Statistics, etc.) because I have not studied these in depth.

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