Exercises: Algebraic fractions

Simplify the algebraic fraction $\frac{12x^4y}{4x^2}$.

Simplify the algebraic fraction $\frac{3x + 6}{3}$.

Simplify the algebraic fraction $\frac{x^2 - 4}{x + 2}$.

Simplify the algebraic fraction $\frac{x^2 + 2x}{x^2}$.

Check whether the algebraic fractions $\frac{x - 1}{x^2 - 1}$ and $\frac{1}{x + 1}$ are equivalent. Justify your answer.

Calculate and simplify $\frac{2x}{3} \cdot \frac{6}{x^2}$.

Calculate and simplify $\frac{x}{x + 1} : \frac{x^2}{x + 1}$.

A student claims that $\frac{x + 5}{5} = x + 1$, because "the $5$ on top and bottom cancel out". State whether her claim is true or false by checking it with a specific value of $x$, explain the error, and write the correct simplification.

Marcos, a 4th-year ESO student, has to simplify the algebraic fraction $\dfrac{2x + 6}{x^2 - 9}$ for a class exercise. In his notebook, he does the following:

"I cross out the $+6$ with the $-9$ by dividing both by $3$, and I simplify the $x$ in the numerator with the $x^2$ in the denominator. I get $\dfrac{2 + 2}{x - 3} = \dfrac{4}{x - 3}$. Therefore, the only restricted value of the expression is $x = 3$".

1. a) Identify and explain in detail the two conceptual mistakes Marcos made: one in the algebraic simplification procedure and the other concerning the domain of the fraction.
2. b) Carry out the simplification in a mathematically correct way and determine all the values that must be excluded from the original domain.

A group of students from a secondary school in Córdoba is organising a cultural bus trip to visit Medina Azahara. The transport company charges a flat fee of $360$ € for hiring the vehicle, and the admission ticket to the archaeological site costs $4$ € per student.

1. a) Write the algebraic fraction $C(x)$ that represents the average cost per student as a function of the number $x$ of attendees participating in the trip.
2. b) Does it make sense to calculate $C(0)$? Justify your answer both from an algebraic point of view and from a practical perspective. If the bus has $55$ seats and the school requires a minimum of $15$ participants to authorise the trip, state the realistic domain of this expression.
3. c) Decompose the fraction into the sum of a fixed quantity and another that depends on $x$. Based on that decomposition, explain what happens to the average cost per student if the number of attendees $x$ becomes very large.

Simplify the algebraic fraction $\frac{x^2 - 6x + 9}{x^2 - 9}$.

Simplify the algebraic fraction $\frac{x^2 + x - 6}{x^2 - x - 2}$ by first factorising the numerator and the denominator.

Work out and simplify $\frac{2}{x} + \frac{3}{x + 1}$.

Work out and simplify $\frac{x}{x - 2} - \frac{1}{x + 2}$.

Work out and simplify $\frac{3}{x^2 - 1} + \frac{1}{x + 1}$.

Work out and simplify $\frac{x^2 - 1}{x} \cdot \frac{x^2}{x + 1}$.

Work out and simplify $\frac{x^2 - 4}{x + 3} : \frac{x - 2}{x^2 + 3x}$.

A tap takes $x$ minutes to fill a tank and another tap takes $5$ minutes less to fill the same tank. a) Express, in terms of $x$, the fraction of the tank filled by both taps together in one minute, simplifying the expression. b) What fraction of the tank do they fill in one minute if the first tap takes $20$ minutes?

Construct an algebraic fraction $\dfrac{P(x)}{Q(x)}$ that simultaneously satisfies these three conditions:

1. Its only restricted values (those for which the original fraction is undefined) are exactly $x = 3$ and $x = -2$.

2. When simplified by cancelling common factors, it yields the irreducible fraction $\dfrac{5}{x + 2}$.

3. The original numerator $P(x)$ is a degree $1$ polynomial whose constant term is $-15$.

1. a) Write the polynomials $P(x)$ and $Q(x)$ in expanded form (as sums and differences of monomials, not factorised).
2. b) Provide a reasoned justification as to whether the fraction you constructed, $\dfrac{P(x)}{Q(x)}$, and the reduced fraction, $\dfrac{5}{x + 2}$, are strictly equivalent for every real value of $x$, indicating what happens in particular if both are evaluated at $x = 3$.

A Renfe Cercanías commuter train makes a round trip of $60\text{ km}$ each way ($120\text{ km}$ in total). On the outbound journey, it travels at a constant average speed of $x\text{ km/h}$. On the return journey, due to infrastructure maintenance works according to Adif data, it must reduce its average speed by $15\text{ km/h}$, travelling at $(x - 15)\text{ km/h}$.

1. a) Knowing that time is calculated by dividing distance by speed, write the sum of algebraic fractions that models the total time spent (in hours) and add them to obtain a single reduced algebraic fraction $T(x)$.
2. b) Estimate and check: imagine that the train usually travels at $x = 75\text{ km/h}$.

• Before substituting into your reduced formula, make a reasoned estimate: if on the outbound trip it takes $60/75 = 0,8\text{ hours}$ ($48\text{ minutes}$) and on the return trip it travels at $60\text{ km/h}$, will the total round-trip time be greater than, less than, or equal to $1,7\text{ hours}$? Explain why the time cannot simply be calculated by assuming an overall average speed of $(75 + 60)/2 = 67,5\text{ km/h}$.

• Substitute $x = 75$ into your algebraic fraction $T(x)$, obtain the exact result in hours and minutes, and check whether your previous estimate was correct.

Work out and simplify $\left( \frac{1}{x} - \frac{1}{x + 2} \right) : \frac{2}{x^2 + 2x}$.

Simplify the complex fraction $\frac{\frac{1}{x} + \frac{1}{y}}{\frac{1}{x} - \frac{1}{y}}$.

Find the highest common factor and lowest common multiple of the polynomials $x^2 - 9$ and $x^2 + 6x + 9$.

Solve the equation $\frac{x + 1}{x - 2} = \frac{x + 5}{x - 4}$. Before starting, state which values $x$ cannot take and explain why they must be discarded.

A cyclist covers $60$ km at an average speed of $v$ km/h and returns along the same route at $v + 10$ km/h. a) Express the total time of the trip in terms of $v$ as a single simplified algebraic fraction. b) If the complete trip lasted $5$ hours, set up the equation that would allow finding $v$ (you do not need to solve it) and state what type of equation it is.

1. a) Verify that $\frac{1}{x(x + 1)} = \frac{1}{x} - \frac{1}{x + 1}$. b) Use this identity to calculate, without adding all terms one by one, the value of $\frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + \frac{1}{3 \cdot 4} + \dots + \frac{1}{99 \cdot 100}$. c) What is the value of the analogous sum with $n$ terms?

A student simplifies like this: "$\frac{x^2 + 4}{x + 2} = x + 2$, because the numerator is the square of the denominator". Explain the conceptual error made (what is $(x + 2)^2$ actually equal to?), check it by substituting $x = 3$, and reason whether the fraction can be simplified in any way.

A greengrocer mixes $x$ kg of walnuts at $9\ \text{€}$ per kilo with $y$ kg of raisins at $5\ \text{€}$ per kilo. a) Express the price of one kilogram of the mixture in terms of $x$ and $y$. b) What is this price if twice as many walnuts as raisins are mixed? c) What relationship must there be between $x$ and $y$ for one kilogram of the mixture to cost exactly $7\ \text{€}$?

To calculate and simplify the following combined operation with algebraic fractions:

$E(x) = \left( \dfrac{3}{x - 2} - \dfrac{2}{x + 2} \right) \cdot \dfrac{x^2 - 4}{x + 10}$

two classmates propose different strategies:

• Alba's method: First work out the expression inside the brackets by finding a common denominator, simplify the resulting numerator, and finally perform the multiplication by factorising the second fraction.

• Bruno's method: Apply the distributive property by directly multiplying the fraction $\dfrac{x^2 - 4}{x + 10}$ by each of the two terms inside the brackets, simplifying each product before subtracting.

1. a) Solve the operation step by step following Alba's method.
2. b) Solve the operation step by step following Bruno's method.
3. c) Compare both approaches: analyse which of the two avoids dealing with higher-degree polynomials and minimises the risk of making sign errors. Justify which one you consider more advantageous.
4. d) Even though the final simplified result is a numerical constant, state the domain of definition of the initial expression and explain why these restrictions must be kept.

According to environmental management models used in Spain, the annual municipal cost $C_1$ (in thousands of euros) to collect and process $p\,\%$ of the plastic waste in a district is estimated using the algebraic fraction:

$C_1(p) = \dfrac{120p}{100 - p}$, with $0 \le p < 100$.

The city council is considering modernising the facilities with a semi-automated system funded by European funds, whose estimated annual cost would be:

$C_2(p) = 30 + \dfrac{60p}{100 - p}$ (thousands of euros).

1. a) Express $C_2(p)$ as a single reduced algebraic fraction with a common denominator.
2. b) Model the savings function $A(p) = C_1(p) - C_2(p)$ as a single irreducible algebraic fraction.
3. c) Analyse the domain of these expressions in the real-world context: why is it not possible to evaluate either formula at $p = 100$? What physical meaning does it have that the denominator approaches $0$ as the recycling percentage $p$ approaches $100\,\%$?
4. d) Study the sign of the numerator of $A(p)$ to determine from what recycling percentage $p$ onwards the automated plant becomes more economical than the traditional one. If the national Waste Law sets a district target of reaching at least $60\,\%$ recycling, explain which technological option the municipality should choose.

Unit Challenge. Optimize the packaging of a sustainable postal shipment.

1. Data collection: Measure an everyday object from home and check the rates and standard permitted dimensions on the Correos website. Model a box with a square base of side length $(x+3)\text{ cm}$ and height $x\text{ cm}$. Apply special products to express the base area.

2. Modeling: Obtain the volume $V(x)$ and the total cardboard surface area $S(x)$ through operations with monomials and polynomials. Formulate the algebraic fraction $\frac{S(x)}{V(x)}$ (material cost per unit of capacity) and analyze it.

3. Calculation and combinatorics: With GeoGebra or a spreadsheet, find the volume for $x=10\text{ cm}$ by applying the remainder theorem (evaluating the division by $x-10$). If you are going to combine $4$ different items inside the package, calculate how many linear arrangements they admit using $4!$. Find the roots of $V(x)-1500=0$ to achieve an exact capacity of $1,5\text{ liters}$.

4. Proposal: Write a brief report explaining the optimal dimensions to minimize costs.

Assessment criteria:

• Correctness when operating with polynomials, special products, and algebraic fractions.

• Proper application of the remainder theorem, roots, and factorial.

• Handling of real data and digital software (GeoGebra or spreadsheet) in the final report.