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?

Álvaro is $4$ years older than his sister Lucía. 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 $2$).

Marta has bought $3$ identical T-shirts for the athletics team and a pair of sports trousers costing $\text{RD\$}\ 15$. If she spent a total of $\text{RD\$}\ 60$, 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?

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

A rectangular padel court has a perimeter of $60$ meters. Knowing that the length of the court is $6$ meters 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?

Marcos has bought a video game and headphones for his games console. The headphones cost him $720$ pesos less than the video game. If the video game had been discounted to half its original price, Marcos would have paid $2\,880$ pesos 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 $5$ meters less than each of the equal sides and that a total of $40$ meters of fencing was needed, how long is each side of the plot?

In a café, they have mixed Colombian coffee, which costs $720$ pesos/kg, with Brazilian coffee, which costs $480$ pesos/kg. If they obtained $20$ kg of mixture and its final price is $540$ pesos/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 $16$ $cm^2$.

A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $120$ pesos/liter and essence B costs $75$ pesos/liter. They have decided that to get a $250$ ml bottle, the cost of the mixture must be exactly $24.75$ pesos. What amount of each essence, measured in milliliters, 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 $150$ meters 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 $2$ meters. 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 $15$ pesos per van and $6$ pesos per motorbike. If the total daily maintenance budget is $200$ pesos, 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?