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I am preparing a unit on operations with algebraic fractions; But, particularly addition/subtraction of algebraic fraction such as

$$\frac{3}{b} + \frac{2}{a} ; \frac{x+1}{3}- \frac{2x+3}{4}; \frac{2}{x+1}+ \frac{2+x}{x+4} + \frac{x}{2x}$$

I am researching on realistic word problem for a performance assessment related to those for a student level equivalent of US middle school or 1st year high school.

I was thinking: considering that 5 girls in the class like basketball and 3 boys like basketball, evaluate the fraction of students who like basketball in the class if we have

Situation 1: 10 girls and 15 boys

Situation 2: x girls and 10 boys

Situation 3: x girls and 3x boys

Situation 4: there are twice more boys than girls

Situation 5: we have 10 more girls than boys

What could be other interesting word problem, situation that could involve the addition or subtraction of algebraic fraction? Any other suggestion greatly appreciated

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    $\begingroup$ This doesn't answer your question, but usually now I motivate this topic ("rational expressions") along the lines of, "you need this in calculus because of the variable dy/dx calculation for slope on any nonlinear curve". This gets some traction if it's after the unit on slope on a line. $\endgroup$ – Daniel R. Collins Jan 8 '18 at 3:12
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    $\begingroup$ Maybe try looking over common problem-types that you can find in textbooks and appropriately adapt them to your needs. For example: "During a $50$-mile trip, John drove the first $x$ miles at $40$ miles per hour and the remaining miles at $60$ miles per hour. In terms of $x,$ what is the total time that John drove on the trip?" Of course, a slight drawback to this is that depending on what one might later do with the result, there may be no compelling reason to rewrite the resulting sum as a single algebraic expression, a drawback that also applies to your examples. $\endgroup$ – Dave L Renfro Jan 8 '18 at 8:10
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    $\begingroup$ Please avoid phrasing that combines a multiplier with "more than", as in "there are twice more boys than girls", because it's confusing. A literal reading of this would be $number\ of\ girls = g;\ number\ of\ boys = g + 2g = 3g$, while a very common interpretation is $number\ of\ girls = g;\ number\ of\ boys = 2g$. Due to the lack of agreement, it's best to avoid this construction altogether. $\endgroup$ – shoover Jan 8 '18 at 18:13
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    $\begingroup$ This might not be helpful, but your question made me think of the lens formula: en.wikipedia.org/wiki/Lens_(optics)#Imaging_properties. You might be able to design a question around it. $\endgroup$ – A. Goodier Jan 8 '18 at 22:16
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Addition in reciprocals (I would call this harmonic addition or harmonic sum, by generalisation from harmonic mean, but both of those terms seem already to be used) comes up in some situations in physics.

  • A. Goodier has mentioned the simple lens formula $\frac{1}{f} = \frac{1}{s} + \frac{1}{o}$. You can place two lenses in a row to complicate things if you think this won't lead to too much focus on the physics to the detriment of the mathematics. Basic practical zoom lenses use three in a row, but the algebra starts getting hairy.
  • Resistor networks: resistors $R_1$ and $R_2$ in series behave like a single resistor with resistance $R_1 + R_2$. Resistors $R_1$ and $R_2$ in parallel behave like a single resistor with resistance $\left(\frac{1}{R_1} + \frac{1}{R_2}\right)^{-1}$. You can combine the two to create expressions of arbitrary complexity.
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