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summed numerator and summed denominator
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(Theorem)
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If $\displaystyle\frac{a_1}{b_1},\,\ldots,\,\frac{a_n}{b_n}$ are any real fractions with positive denominators and $$m:= \min\left\{\frac{a_1}{b_1},\,\ldots,\,\frac{a_n}{b_n}\right\}, \quad M:= \max\left\{\frac{a_1}{b_1},\,\ldots,\,\frac{a_n}{b_n}\right\}$$ are the least and the greatest of the fractions, then
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(1) |
The equality signs are valid if and only if all fractions are equal; in this case one has $$\frac{a_1}{b_1} = \ldots = \frac{a_n}{b_n} = \frac{a_1\!+\!\ldots+\!a_n}{b_1\!+\!\ldots\!+\!b_n}.$$ Proof. Set $\displaystyle q_1 := \frac{a_1}{b_1}$ , ..., $\displaystyle q_n := \frac{a_n}{b_n}$ . Then we have $a_1\!+\!\ldots\!+\!a_n = b_1q_1\!+\!\ldots\!+\!b_nq_n$ , which apparently has the lower bound $(b_1\!+\cdots+\!b_n)m$ and the upper bound $(b_1\!+\ldots+\!b_n)M$ . Dividing the three last expressions by the sum $b_1\!+\ldots+\!b_n$ yields the asserted double inequality (1).
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"summed numerator and summed denominator" is owned by pahio.
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Cross-references: inequality, sum, expressions, upper bound, lower bound, proof, valid, equality, denominators, positive, fractions, real
There is 1 reference to this entry.
This is version 6 of summed numerator and summed denominator, born on 2005-07-01, modified 2007-03-14.
Object id is 7203, canonical name is SummedNumeratorAndSummedDenominator.
Accessed 1570 times total.
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Pending Errata and Addenda
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