Algebraic fractions

Check, by cross-multiplying, whether the algebraic fractions $\frac{x-3}{x+1}$ and $\frac{x^2-9}{x^2+4x+3}$ are equivalent.

Find the value of the algebraic fraction $\frac{x^2 - 3x + 2}{x^2 + 1}$ for $x = -2$.

Simplify the following algebraic fraction by taking out common factors in the numerator and in the denominator: $\frac{4x^3 - 2x^2}{6x^2 - 2x}$

Simplify the following algebraic fraction using algebraic identities: $\frac{x^2 - 25}{x^2 - 10x + 25}$

Simplify the following algebraic fraction by first factorising the numerator and the denominator: $\frac{x^3 - 4x^2 + 4x}{x^3 - 2x^2 - 4x + 8}$

Determine the value of the parameter $k$ so that the algebraic fraction $\frac{x^2 - kx + 10}{x-5}$ can be simplified (that is, the numerator is divisible by the denominator). Once $k$ is found, write down the simplified expression.

Show, by manipulating and simplifying the algebraic fractions, that the following expression is equal to $2ab$: $\frac{a^3-b^3}{a-b} - \frac{a^3+b^3}{a+b}$

Multiply the following algebraic fractions and give the result in its simplest form: $\frac{x}{x+2} \times \frac{x^2-4}{x^2}$

Divide the following algebraic fractions and simplify: $\frac{2x+2}{x-1} : \frac{4x+4}{x^2-1}$

Find the polynomial $P(x)$ that makes the following equality between algebraic fractions true: $\frac{x^2-9}{P(x)} = \frac{x-3}{x^2+2x}$

Perform the following addition of algebraic fractions and simplify the result: $\frac{3x+2}{x-2} + \frac{-x-6}{x-2}$

Perform the following operation and simplify the result: $\frac{x^2+5x}{x+3} - \frac{x^2-9}{x+3}$

Perform the following operations by finding a common denominator and simplify the result: $\frac{2}{x-1} + \frac{x}{x+1} - \frac{2x}{x^2-1}$

Perform the operation and simplify the following expression as much as possible: $\frac{x^2-1}{x^2+2x+1} \times \left( \frac{2x}{x-1} - \frac{x+1}{x} \right)$

Perform the following combined operation with algebraic fractions and simplify the result: $\left( \frac{1}{x} - \frac{2}{x+2} \right) : \frac{x-2}{x^2+2x}$

Perform the following subtraction of algebraic fractions and simplify as much as possible: $\frac{x^2+x}{x^2-1} - \frac{x}{x-1}$

Simplify the following algebraic expression in the form of a complex fraction: $1 + \frac{1}{1 + \frac{1}{x-1}}$

The base of a rectangular solar panel is $x+5$ meters and its height is $x-5$ meters. The side of a square solar panel is $x$ meters. Write down the algebraic fraction representing the quotient of the area of the rectangular panel and the area of the square panel. Calculate the value of this fraction if $x = 10$ meters.

A hot water tap takes $x$ hours to fill a bath, while the cold water tap takes $x+3$ hours. Write a simplified algebraic fraction representing the fraction of the bath that fills in one hour if both taps are turned on together.

The total cost in pesos of manufacturing $x$ bicycles is given by the polynomial $C(x) = 60x^2 + 3\,000x$. The total revenue in pesos from selling those $x$ bicycles is expressed by the polynomial $I(x) = 100x^2 + 4\,000x$. Write down the irreducible algebraic fraction that represents the ratio of revenue to costs.

A train travels a distance of $120$ km at a constant speed of $v$ km/h. On the return journey, due to rain, its speed decreases by $10$ km/h. Write down the algebraic fraction representing the total time for the round trip in terms of $v$, simplifying to a single fraction.

An open-topped box is to be made from a rectangular sheet of cardboard with dimensions $x+6$ and $x+4$ centimeters, by cutting squares of side $2$ cm from the four corners and folding up the flaps. Write down the irreducible algebraic fraction representing the ratio of the volume of the box to the area of its base.

In a chemistry laboratory, $x$ liters of solution A, costing $\frac{400}{x}$ pesos per liter, are mixed with $x+4$ liters of solution B, costing $\frac{600}{x+4}$ pesos per liter. Write down the algebraic fraction representing the price per liter of the resulting solution, simplifying the expression as much as possible.

An athlete runs on a circular track. During the first half of the distance, they run at a speed of $v$ m/s, and during the second half, they run at a speed of $v-2$ m/s. Given that the average speed for the whole distance is calculated as total distance divided by total time, show using algebraic fractions that the average speed is $\frac{2v(v-2)}{2v-2}$ m/s.