# Situation involving the application of addition or subtraction of algebraic fraction

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

• 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. Jan 8 '18 at 3:12
• 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. Jan 8 '18 at 8:10
• 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. Jan 8 '18 at 18:13
• 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. Jan 8 '18 at 22:16

• 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.