Problems with equations

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

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?

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?

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

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

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?

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?

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.

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?

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 $2$).

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?

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?

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 $16$ $cm^2$.

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 $10x + y$).

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.

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.

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?

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).