Exercises: Problems with equations

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

We want to put a wooden frame on a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If we needed a strip of $70 \text{ cm}$ total length (perimeter) to frame it, what are the dimensions of the photograph?

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

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

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

Emma bought $3$ identical t-shirts for the track team and a pair of sweatpants that cost $15 \text{ \$}$. If she spent $60 \text{ \$}$ in total, how much did each t-shirt cost?

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

The perimeter 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, Ashley's number of followers is three times that of her brother Ethan. If Ashley lost $150$ followers and Ethan gained $50$, both would have exactly the same amount. How many followers does each one currently have?

A rectangular tennis court has a perimeter of $60$ feet. Knowing that the length of the court is $6$ feet more than the width, calculate the dimensions (length and width) of the 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?

Marcus bought a video game and some headphones for his console. The headphones cost him $\$12$ less than the video game. If the video game were reduced to half its original price, Marcus would have paid $\$48$ for both items together. What is the original price of each item?

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

In a coffee shop, they mixed Colombian coffee, which costs $\$12$/lb, with Brazilian coffee, which costs $\$8$/lb. If they obtained $20$ pounds of mixture and its final price is $\$9$/lb, how many pounds of each type of coffee 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 $3$ in greater than its height and that, if we increase the height by $2$ in, 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 $\$120$/liter and essence B costs $\$75$/liter. They have decided that to obtain a $250$ mL bottle, the cost of the mixture must be exactly $\$24,75$. What quantity of each essence, measured in milliliters, must the bottle contain?

In a robotics competition, a robot leaves the starting line at a constant speed of $2$ ft/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3,5$ ft/s. If the track has a total length of $150$ feet before reaching an obstacle zone, will the second robot manage to catch the first one before it enters the obstacle zone? Set up the equation and justify the answer by calculating the catch-up distance.

A rectangular garden is to be designed surrounded by a tiled path of a constant width of $2$ feet. The length of the garden is twice its width. If we know that the total area occupied by the tiled path is $112$ ft$^2$, 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 obtain a new number that is $18$ units greater 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 was at that time. How many years must pass for the sum of the father's and son's ages to be exactly $100$ years?

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

Calculate the perimeter of a right triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use the Pythagorean theorem $a^2 + b^2 = c^2$, where $c$ is the longest side).

A courier company uses vans and motorcycles. The vans have $4$ wheels and the motorcycles $2$. 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 motorcycle. If the total daily maintenance budget is $\$250$, how many van drivers should switch to driving motorcycles so as not to exceed the budget without changing the total number of vehicles?