Question 116890
Given:
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{{{r[t] = 1/(1/r[1] + 1/r[2])}}}
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This is not the way this formula is usually written, but it is correct.
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You can simplify this by multiplying the right side by {{{(r[1]*r[2])/(r[1]*r[2])}}}
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Since the numerator of this multiplier is equal to the denominator, this is the same as
multiplying the right side by 1. This means that this multiplication doesn't change the
value of the right side.
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The multiplication leads to:
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{{{r[t] = (1/(1/r[1] + 1/r[2]))*((r[1]*r[2])/(r[1]*r[2]))}}}
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A little difficult to see, maybe, but the very top "1" gets multiplied by {{{r[1]*r[2]}}} and in
the denominator the numerators of the two fractions (both numerators are "1") also get multiplied
by {{{r[1]*r[2]}}}. This changes the equation to:
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{{{r[t] = ((1*r[1]*r[2])/((1*r[1]*r[2])/r[1] + (1*r[1]*r[2])/r[2]))}}}
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In this, in the three places where a "1" is a multiplier, you can simplify the work by just 
erasing the "1". This simplifies the equation to:
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{{{r[t] = ((r[1]*r[2])/((r[1]*r[2])/r[1] + (r[1]*r[2])/r[2]))}}}
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Then in the fractions in the denominator you can cancel the denominator of the fraction with
the corresponding term in the numerator of the fraction as follows:
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{{{r[t] = ((r[1]*r[2])/((cross(r[1])*r[2])/cross(r[1]) + (r[1]*cross(r[2]))/cross(r[2])))}}}
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Note that the cancellations shown reduce the equation to:
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{{{r[t] = r[1]*r[2]/(r[2] + r[1])}}}
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or by simply reversing the order of the terms in the denominator, which changes nothing, you 
get the more familiar form:
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{{{r[t] = r[1]*r[2]/(r[1] + r[2])}}}
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This is the answer to the problem. It shows that two resistors connected in parallel can
be replaced by a single resistor {{{r[t]}}} whose value is equal to the product of the two
resistors divided by the sum of the two resistors. 
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Regarding my comment above, this problem does not list the formula for {{{r[t]}}} the way that it
is usually written. The way it is usually given is:
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{{{1/r[t] = 1/r[1] + 1/r[2]}}}
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If you solve this equation for {{{r[t]}}} you will get the same answer that we got for this problem ...
namely you will again get:
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{{{r[t] = r[1]*r[2]/(r[1] + r[2])}}}
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Hope this helps you to understand the problem and how to solve it.
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