Exercises: Word problems with equations

If we add $12$ to three times a number, we get $48$. What is the number?

We want to put a wooden frame around a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If framing it required a strip of wood with a total length (perimetre) of $70 \text{ cm}$, what are the dimensions of the photograph?

In a video game shop, they have sold twice as many handheld consoles as desktop consoles. If they have sold a total of $18$ consoles, how many have they sold of each type?

Seán is $4$ years older than his sister Aoife. If the sum of their ages is $26$ years, how old is each sibling?

A shepherd has a mix of sheep and chickens in his pen. If we count heads there is a total of $20$, and if we count legs there are $64$. How many sheep and how many chickens are in the pen? (Remember that sheep have $4$ legs and chickens $2$).

Aoife has bought $3$ identical T-shirts for the athletics team and a pair of sports trousers costing $€15$. If she spent a total of $€60$, how much did each T-shirt cost?

Find three consecutive integers whose sum is equal to $54$.

The perimetre of an isosceles triangle is $40 \text{ cm}$. If the unequal side measures $10 \text{ cm}$, how long is each of the equal sides?

In a classroom problem, you are asked to calculate in how many years a father's age will be three times his daughter's age. The daughter is currently $12$ years old and the father is $40$ years old. Cian has proposed the following equation to solve it: $40 + x = 3 \times 12 + x$.

1. a) Point out the conceptual error that Cian made and explain in your own words why his approach contradicts the fact that time passes equally for both people.
2. b) Write the correct equation that represents the problem and solve it to find out in how many years the condition will be met.

At your high school, they are going to fence a rectangular plot intended for the school garden. Due to the shape of the available land, the length must be exactly three times the width. In the storage room, they have a $64\text{ m}$ roll of wire fencing that will be used entirely to enclose the whole perimeter.

1. a) Before setting up any equation, make a reasoned estimate: is it possible for the width of the plot to be $10\text{ m}$ or more? Justify your deduction without calculating the exact value.
2. b) Define the unknown, set up the equation corresponding to the perimeter, and find the exact dimensions of the plot (width and length).
3. c) Check the obtained solution and confirm whether it fits your previous estimate.

On a social network, Aoife has three times as many followers as her brother Seán. If Aoife were to lose $150$ followers and Seán were to gain $50$, both would have exactly the same amount. How many followers does each currently have?

A rectangular padel court has a perimetre of $60$ metres. Knowing that the length of the court is $6$ metres more than the width, calculate the dimensions (length and width) of this court.

A mother's age is currently three times that of her son. $6$ years ago, the sum of their ages was $40$ years. How old are the son and the mother currently?

Cian has bought a video game and headphones for his games console. The headphones cost him $12$ € less than the video game. If the video game had been discounted to half its original price, Cian would have paid $48$ € for both items together. What is the original price of each item?

A farmer wants to fence off a triangular plot for his sheep. Knowing that it is an isosceles triangle, that the unequal side measures $5$ metres less than each of the equal sides and that a total of $40$ metres of fencing was needed, how long is each side of the plot?

In a café, they have mixed Colombian coffee, which costs $12$ €/kg, with Brazilian coffee, which costs $8$ €/kg. If they obtained $20$ kg of mixture and its final price is $9$ €/kg, how many kilograms of coffee of each type did they use?

Find three consecutive integers such that the sum of the first two is $15$ units greater than the third number.

Calculate the dimensions of a rectangle knowing that its base is $3$ cm greater than its height and that, if we increase the height by $2$ cm, the area of the new rectangle increases by $16$ $cm^2$.

In the urban transport of Dublin, a standard daytime bus ticket costs $1.40$ € and a night bus ticket costs $2.10$ €. Last month, Aoife used the bus a total of $35$ times and her travel card statement showed a total expenditure of exactly $57.40$ €.

1. a) Define the main unknown, clearly stating what it represents, and express the number of night trips in terms of that unknown.
2. b) Set up a linear equation relating the tickets used to the total expenditure, and solve it to find out how many trips of each type Aoife took.
3. c) Check whether the obtained numerical values make sense in real life, considering the type of solution required by this problem.

You are going to design an algebra problem for your group's exam. The statement must begin as follows: «Currently, Seán is twice as old as his sister Aoife...». Your task is to add a second sentence describing a relationship between the ages both will have in $6$ years (based on the sum of their future ages), such that solving the equation necessarily results in Aoife being $8$ years old today.

1. a) Determine how old Seán and Aoife will be in $6$ years knowing that Aoife is $8$ years old today, and calculate what their ages will add up to at that future moment.
2. b) Write the complete and coherent statement of the problem, incorporating that numerical data.
3. c) Set up the equation from the statement you wrote, solve it step by step, and show that the solution is indeed $x = 8$.

A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $120$ €/litre and essence B costs $75$ €/litre. They have decided that to get a $250$ ml bottle, the cost of the mixture must be exactly $24.75$ €. What amount of each essence, measured in millilitres, should the bottle contain?

In a robotics competition, a robot leaves the starting line at a constant speed of $2$ m/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3.5$ m/s. If the track has a total length of $150$ metres before reaching an obstacle zone, will the second robot manage to catch up with the first before it enters the obstacle zone? Set up the equation and justify your answer by calculating the catching-up distance.

We want to design a rectangular garden surrounded by a paved path of a constant width of $2$ metres. The length of the garden is double its width. If we know that the total area occupied by the paved path is $112$ m$^2$, what are the dimensions of the garden (width and length)?

A number has two digits that add up to $12$. If we reverse the order of its digits, we get a new number that is $18$ units larger than the original number. Find the original number using an equation with a single unknown (Hint: remember that a two-digit number can be written as $10x + y$).

A father's age today is three times that of his son. $6$ years ago, the father's age was five times the age his son had at that time. How many years must pass for the sum of the father's and the son's ages to be exactly $100$ years?

A water tank is filled by two taps. Tap A takes $6$ hours longer than tap B to fill the tank on its own. If both are opened at the same time, the tank fills in exactly $4$ hours. Calculate how long tap B would take to fill the tank on its own. (This problem requires solving a quadratic equation).

Calculate the perimetre of a right-angled triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use Pythagoras' Theorem $a^2 + b^2 = c^2$, where $c$ is the longest side).

A courier company uses vans and motorbikes. Vans have $4$ wheels and motorbikes have $2$. In total there are $24$ drivers (one per vehicle) and $64$ wheels. Due to logistical problems, the daily maintenance cost is $15$ € per van and $6$ € per motorbike. If the total daily maintenance budget is $200$ €, how many van drivers should switch, at a minimum, to driving motorbikes so as not to exceed the budget without changing the total number of vehicles?

During the high school's end-of-year party, a total of $60$ sandwiches were sold at the canteen to raise funds for a trip. The omelette sandwiches were sold for $2.20$ € each and the Iberian ham ones for $2.80$ €. The total revenue was $147.60$ €.

To find out how many sandwiches of each type were sold, two students propose different approaches:

• Sara says: «I prefer to use only the unknown $x$ for the omelette ones, express the ham ones as $60 - x$, and solve a single linear equation».

• David says: «It is neater to use two unknowns, $x$ for the omelette ones and $y$ for the ham ones, and set up a system of two linear equations».

1. a) State and solve the problem following Sara's proposed method from start to finish.
2. b) State and solve the problem following David's proposed method from start to finish.
3. c) Compare both procedures: what advantages and disadvantages does each approach have when translating the problem statement data and when carrying out the algebraic operations? Justify which one you would recommend to a classmate who usually hesitates when isolating variables.

The quarterly water bill for a household in an Irish town includes a fixed service charge of $18.50$ € and a cost based on the volume consumed. To encourage water conservation due to drought, the first $30\text{ m}^3$ of water consumed in the quarter are charged at $1.35$ €/$\text{m}^3$, while any additional cubic metre above that limit is billed at a penalised rate of $2.10$ €/$\text{m}^3$.

A family has received a quarterly bill with a total amount payable of $71.60$ €.

1. a) Demonstrate numerically, before formulating the overall equation, why this family's consumption must necessarily have exceeded $30\text{ m}^3$.
2. b) Define the unknown $x$, formulate the linear equation describing the bill, and calculate how many cubic metres of water the family used during the quarter.
3. c) If during the following quarter the family manages to reduce their consumption volume in $\text{m}^3$ by half, will the total bill amount in euros also be reduced by exactly half? Justify your answer by algebraically analysing the influence of the fixed charge and the change in tariff tier.

Unit Challenge. How would you optimize the space and energy consumption of your study area? Become a home efficiency consultant by solving this challenge at home.

1. Data collection: Measure your desk space with a tape measure and check the SEAI (Sustainable Energy Authority of Ireland) website for the average price of a kWh (approx. $0.14$ €/kWh).

2. Energy and time: Set up a problem using a 1st-degree equation to calculate how many hours of use it takes for an LED bulb to pay for itself compared to a traditional one. Then, solve a system of two linear equations (for example, with $x+y=24$) to split the hours of the day between study and leisure according to the estimated energy cost.

3. Geometric design: Design a rectangular floor protector knowing that its length must be $0.5$ m longer than its width ($x$). Set up and solve a 2nd-degree equation of the type $x(x+0.5)=A$, setting a realistic area $A$ suitable for your room.

4. Final report: Prepare a brief report (supported by GeoGebra for the floor plan or a spreadsheet for the costs) justifying your decisions with the mathematical solutions obtained.

Assessment criteria:

• Rigor in the algebraic formulation and solving of 1st- and 2nd-degree equations and systems.

• Consistent use of real data and justification of contextualized solutions.

• Quality of the report and use of digital tools (GeoGebra or Calc/Excel).