Exercises: Word problems with equations

In an eSports tournament, Alex's team scored three times as many points as Salma's team. If together the two teams scored a total of $152$ points, how many points did each team get?

Aisha is $4$ years older than her younger brother, Iker. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each person by setting up a linear equation.

A school vegetable garden of rectangular shape needs to be fenced off to protect the crops. We know that the length of the garden is $5$ metres more than the width and that $38$ metres of fencing were needed to completely enclose it. What are the dimensions of the garden?

A group of friends buys tickets for a concert. "Standing" tickets cost $15$ € more than "seated" tickets. If they buy one of each type and the total bill is $85$ €, what is the price of each ticket?

The sum of three race numbers from a race, which turn out to be three consecutive integers, is equal to $63$. What numbers are those three runners wearing on their shirts?

A food delivery driver carries twice as many pizzas as burgers in their backpack. If they are carrying $12$ orders in total between pizzas and burgers, how many of each type are there?

Kenia has started saving to buy a tablet costing $240$ €. She already has $60$ € saved and, from now on, she is going to save $12$ € of her pocket money each week. How many weeks will it take her to have enough money for the purchase?

Calculate the side length of a square billboard, knowing that if its side length were increased by $2$ metres, its area would increase by $16$ square metres.

To solve the problem: "Lucía is $38$ years old and her son Hugo is $10$. In how many years will Lucía's age be exactly twice her son's age?", Mateo wrote on his exam:

$38 + x = 2 \cdot 10 + x$

1. a) Explain in your own words the two conceptual errors that Mateo made when translating the problem statement into algebraic language.
2. b) Define precisely what the unknown $x$ represents and set up the correct equation.
3. c) Solve the equation step by step and verify that the solution makes sense as an answer to the question in the problem statement.

A town council wants to fence a rectangular plot intended for a dog park. Due to the characteristics of the land, the length must be exactly three times the width, and there is a fixed budget to buy a full roll of $160\text{ m}$ of wire mesh fence.

1. a) Before calculating anything, make a reasoned estimate: could the width be $30\text{ m}$? Justify whether that measurement would require more fencing than available, less, or the exact amount.
2. b) Define the unknown $x$, set up a linear equation that represents the total perimeter, and solve it to find the exact dimensions of the park (length and width).
3. c) Check whether the result obtained confirms your initial estimate and calculate the total area the dogs will have to run around.

Find three consecutive integers such that the sum of the first two is equal to three times the third minus 14.

A new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ metres greater than the width. If the total area allocated for play is $150$ $m^2$, calculate the dimensions (length and width) of the area.

Currently, a father's age is three times that of his daughter. In $12$ years' time, the father's age will be only double that of his daughter. How old is each of them now?

Two cyclists, Lucía and Marcos, are at a distance of $60$ $km$ apart and decide to meet at an intermediate point. Lucía sets off from $A$ towards Marcos at a constant speed of $20$ $km/h$. At the same time, Marcos sets off from $B$ towards Lucía at a speed of $10$ $km/h$. How long will it take them to pass each other, and at what distance from point $A$ will they meet?

In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $12$ €/kg and "Robusta" coffee costs $8$ €/kg. If we want to obtain a mixture of $20$ $kg$ priced at $10,50$ €/kg, how many kilograms of each type of coffee should we use?

Find a natural number such that the difference between its square and three times its value is equal to $40$. Is the solution to this problem unique in the set of natural numbers? Justify your answer by setting up the corresponding equation.

A group of friends wants to buy a birthday present that costs $120$ €. At first there was a certain number of people, but at the last minute $2$ more friends joined, which meant each person had to pay $5$ € less. How many friends were originally in the group?

Calculate the dimensions of an isosceles triangle knowing that the unequal side is $4$ $cm$ less than each of the equal sides and that the total perimeter is $26$ $cm$. Once you have found the sides, calculate the area of the triangle.

An eco-friendly parcel delivery company operating in Valencia offers two monthly rates for local businesses:

• Base Rate: a fixed fee of $15\text{ \euro}$ per month plus $4,50\text{ \euro}$ for each shipment made.

• Eco-Plus Rate: no monthly fixed fee; it charges $7\text{ \euro}$ per shipment for the first $10$ shipments of the month, and reduces the price to $3,50\text{ \euro}$ for each additional shipment from the eleventh onwards.

A neighborhood bookstore sent more than $10$ packages last month and found that the final bill would have been exactly the same with either of the two rates.

1. a) Define the unknown $x$ as the total number of monthly shipments and write the algebraic expression for the total cost with each rate in the case where $x > 10$.
2. b) Set up and solve the corresponding equation to find out how many shipments the bookstore made last month.
3. c) Based on the solution, explain to the bookstore owners which rate would be best for them to choose if they anticipate that their shipments will increase next month.

Imagine that you are a Mathematics teacher and you must create a question for a 3rd year ESO exam that satisfies exactly these three conditions:

1. It must be about a rectangular school garden whose total area is $140\text{ m}^2$.

2. Its length must be $4\text{ meters}$ greater than its width.

3. It must be solved using a quadratic equation that has one positive solution and one negative solution.

1. a) Write the complete text of the problem with clear and precise wording, without solving it within the text itself.
2. b) Define the unknown $x$, write the quadratic equation derived from your text, and solve it step by step.
3. c) Argue which geometric criterion students must apply to discard one of the two mathematical solutions, and determine the final dimensions of the school garden.

A graphic designer is working on a new company logo. The design consists of a central rectangle for the name, surrounded by a decorative border of uniform thickness $x$. The inner rectangle has dimensions of $12\text{ cm}$ by $8\text{ cm}$. If the total area of the logo (including the border) must be exactly double the area of the inner rectangle, what should the thickness $x$ of the border be?

A mother's current age is the square of her daughter's age. In $9$ years' time, the mother's age will be only three times the age her daughter will be then. Find their current ages.

Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00 h, a cyclist departs from $A$ towards $B$ at a constant speed of $20\text{ km/h}$. An hour and a half later, a motorbike departs from $B$ towards $A$ at a speed of $80\text{ km/h}$. At what time will they pass each other, and at what distance from city $A$ will the meeting take place?

In a chemistry laboratory, $20\text{ litres}$ of an acid solution at $25\%$ concentration is needed. The technician has two containers: one with a $10\%$ acid mixture and another with a concentrated $50\%$ mixture. How many litres from each container must be mixed to obtain the desired concentration?

Mario won $1200\text{ €}$ in a programming competition and divided it into two parts to deposit into two different financial products. The first offers an annual return of $4\%$, while the second, which is riskier, offers $7\%$ annually. If at the end of the year the total profit obtained was $63\text{ €}$, how much money did he allocate to each product?

The area of a right-angled triangle is $30\text{ cm}^2$ and its hypotenuse is $13\text{ cm}$. Calculate the lengths of the two legs of the triangle.

Two pipes connected to a tank can fill it together in $4\text{ hours}$. If used separately, the smaller diameter pipe takes $6\text{ hours}$ longer than the larger diameter pipe to fill the full tank. How long would each pipe take on its own?

A cattle farmer in Asturias has $60\text{ meters}$ of fencing to enclose a rectangular pen for his calves to graze. To save material, he will take advantage of an existing stone wall, so he only needs to fence three sides: two equal widths and a length opposite the wall. His goal is for the enclosure to have an area of $400\text{ m}^2$.

Two students propose different ways to approach the problem:

• Method A: Let $x$ be the length of the long side parallel to the wall, so that each of the two widths will measure $\frac{60 - x}{2}\text{ meters}$.

• Method B: Let $y$ be the length of each of the two widths perpendicular to the wall, so that the length will measure $60 - 2y\text{ meters}$.

1. a) Write the quadratic equation obtained with each method to calculate the area ($400\text{ m}^2$).
2. b) Analyze both methods: which of the two is algebraically simpler to work out by hand, and why? Justify your answer considering the presence or absence of denominators.
3. c) Solve the equation using the method you found more advantageous and find the possible dimensions of the enclosure.
4. d) Interpret the result: does it make physical sense for there to be two valid solutions for the dimensions of the enclosure? Explain the shape the pen would take in each case.

An irrigation community in Murcia stores water in a rectangular reservoir measuring $30\text{ m}$ long by $20\text{ m}$ wide. To cope with a drought period, they decide to expand the reservoir by adding the same number of meters, $x$, to both the length and the width. The new surface area must be exactly $936\text{ m}^2$, and for environmental safety reasons, the basin regulations prohibit reservoirs exceeding $1000\text{ m}^2$.

1. a) Before solving algebraically, make a reasoned estimate of the value of $x$: calculate the area if $x = 5\text{ m}$ and if $x = 10\text{ m}$, and justify between which two integer values the expansion $x$ must lie.
2. b) Define the quadratic equation in its standard form $ax^2 + bx + c = 0$ that models the situation, and solve it step by step.
3. c) Analyze the two solutions of the equation: why does one of them lack practical meaning? Do the final dimensions of the expanded reservoir comply with environmental regulations?
4. d) If the cost of excavation and waterproofing is $35\text{ \euro}$ for each square meter of surface area added relative to the original reservoir, calculate the total budget that the irrigation community will need.

Unit Challenge. Is it feasible to organize a sustainable youth mini-fair in your municipality? Become the project manager and present a short feasibility report structured in three phases:

1. Consumption and linear equations: Check today's average price of kWh on the Red Eléctrica (REE) website. With a fixed booth cost of $12,50$ € and an estimated electricity expense of $1,20$ €/hour, formulate and solve a linear equation to calculate how many hours you can open with a budget of $24,50$ €.

2. Space and systems: Design a rectangular tent of $54\text{ m}^2$ in GeoGebra or on paper where the length is $3\text{ m}$ longer than the width ($x(x+3)=54$); find its dimensions by solving the quadratic equation. Then, set up a linear system knowing that there are $15$ booths in total between food and eco-workshops, and that they pay a total of $330$ € ($30$ € and $15$ € each, respectively) to calculate how many there are of each type.

3. Survey and probability: Ask $10$ neighbours whether they prefer to attend on Friday or Saturday. Record the data in a spreadsheet, calculate the experimental probability that a neighbour prefers each day, and write your reasoned final proposal.

Assessment criteria:

• Mathematical precision in formulating and algebraically solving equations and systems.

• Rigorous collection of real data and probabilistic processing with a spreadsheet or GeoGebra.

• Coherence, clarity, and critical thinking in the conclusions of the final report.