Exercises: Problems with equations

In an eSports tournament, Alex's team has scored three times as many points as Lucy's team. If between the two teams they have scored a total of $152$ points, how many points has each team obtained?

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

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

A group of friends buys tickets for a concert. The "standing" tickets cost $\$15$ more than the "seated" ones. 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 bib numbers, which happen to be three consecutive integers, is equal to $63$. What numbers are these three runners wearing on their shirts?

A food delivery app rider is carrying twice as many pizzas as burgers in their backpack. If they are transporting a total of $12$ orders between pizzas and burgers, how many of each kind are there?

Lucy has started saving to buy a tablet that costs $\$240$. She has already saved $\$60$ and, from now on, she is going to put away $\$12$ of her allowance every week. How many weeks will it take her to have enough money for the purchase?

Calculate the side length of a square advertising panel knowing that if its side were to increase by $2$ feet, its area would increase by $16$ square feet.

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

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

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

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

In a gourmet shop, they want to mix two types of coffee. "Arabica" coffee is priced at $\$12$/lb and "Robusta" coffee costs $\$8$/lb. If we want to obtain a mixture of $20$ $kg$ that works out at a price of $\$10,50$/lb, how many pounds of each type of coffee must we use?

Find a natural number such that the difference between its square and three times the number 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 were a certain number of people, but at the last minute $2$ more friends joined in, which meant that everyone had to pay $\$5$ less. How many friends were there initially in the group?

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

A graphic designer is working on the logo for a new company. 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 must the thickness $x$ of this border be?

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

Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00, a cyclist leaves $A$ towards $E$ at a constant speed of $20\text{ km/h}$. An hour and a half later, a motorcycle leaves $E$ towards $A$ at a speed of $80\text{ km/h}$. At what time will they cross paths and at what distance from city $A$ will they meet?

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

An investor has $12\,000\text{ \$}$ which they divide 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 an annual return of $7\%$. If at the end of the year the total profits obtained were $630\text{ \$}$, what amount did they allocate to each product?

The area of a right triangle is $30\text{ cm}^2$ and its hypotenuse measures $13\text{ cm}$. Calculate the length of the two shorter sides of the triangle.

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

Find a two-digit number knowing that the sum of its digits is $12$ and that, if the order of its digits is reversed, the new number obtained exceeds the original by $18$ units. Represent the original number as $10x + y$.