Word problems with equations
Your class is organising a Transition Year trip and the teacher says: "The bus costs €450 and we have 30 students, so what does each of us pay?" Solving an equation is the easy part: it is mechanical. The difficult part, and the one that really counts, is setting it up: turning a word problem into an equation with letters. With a clear method and a few templates, most problems become routine. We will come back to the trip at the end.
The four-step method
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Choose the unknown and write down what it means: "$x$ = Hugo's age today". If there are several quantities, express them all using the same letter if you can.
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Translate each piece of data into algebraic language ("three times as many" means $3x$, "5 more than" means $x + 5$).
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Set up the equation using the missing condition and solve it.
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Check against the problem statement (not against the equation, which might be set up wrongly) and answer with a sentence and units.
Working backwards
Sometimes it is quicker to start from the end result and undo each operation in reverse order. This strategy is called working backwards.
Back to the trip: with 30 students, $30x = 450$, so each pays €15.
In an eSports tournament, Alex's team scored three times as many points as Salma's team. If together the two teams scored a total of $152$ points, how many points did…
In an eSports tournament, Alex's team scored three times as many points as Salma's team. If together the two teams scored a total of $152$ points, how many points did each team get?
A group of friends buys tickets for a concert. "Standing" tickets cost $15$ € more than "seated" tickets. If they buy one of each type and the total bill is $85$ €, what…
A group of friends buys tickets for a concert. "Standing" tickets cost $15$ € more than "seated" tickets. If they buy one of each type and the total bill is $85$ €, what is the price of each ticket?
A food delivery driver carries twice as many pizzas as burgers in their backpack. If they are carrying $12$ orders in total between pizzas and burgers, how many of each…
A food delivery driver carries twice as many pizzas as burgers in their backpack. If they are carrying $12$ orders in total between pizzas and burgers, how many of each type are there?
Kenia has started saving to buy a tablet costing $240$ €. She already has $60$ € saved and, from now on, she is going to save $12$ € of her pocket money each week. How…
Kenia has started saving to buy a tablet costing $240$ €. She already has $60$ € saved and, from now on, she is going to save $12$ € of her pocket money each week. How many weeks will it take her to have enough money for the purchase?
Numbers and ages
Consecutive numbers: $x,\ x + 1,\ x + 2$. Consecutive even (or odd) numbers: $x,\ x + 2,\ x + 4$.
Ages: make a table with "today", "ago" and "in". Time passes at the same rate for everyone:
| Today | In 6 years' time | |
|---|---|---|
| Cian | $x$ | $x + 6$ |
| Grace | $x + 4$ | $x + 10$ |
Aoife is $4$ years older than her younger brother, Cian. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each person by…
Aoife is $4$ years older than her younger brother, Cian. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each person by setting up a linear equation.
To solve the problem: "Aoife is $38$ years old and her son Darragh is $10$. In how many years will Aoife's age be exactly twice her son's age?", Seán wrote on his exam:…
To solve the problem: "Aoife is $38$ years old and her son Darragh is $10$. In how many years will Aoife's age be exactly twice her son's age?", Seán wrote on his exam:
$38 + x = 2 \times 10 + x$
- a) Explain in your own words the two conceptual errors that Seán made when translating the problem statement into algebraic language.
- b) Define precisely what the unknown $x$ represents and set up the correct equation.
- c) Solve the equation step by step and verify that the solution makes sense as an answer to the question in the problem statement.
Currently, a father's age is three times that of his daughter. In $12$ years' time, the father's age will be only double that of his daughter. How old is each of them…
Currently, a father's age is three times that of his daughter. In $12$ years' time, the father's age will be only double that of his daughter. How old is each of them now?
Find a natural number such that the difference between its square and three times its value is equal to $40$. Is the solution to this problem unique in the set of…
Find a natural number such that the difference between its square and three times its value 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 mother's current age is the square of her daughter's age. In $9$ years' time, the mother's age will be only three times the age her daughter will be then. Find their…
A mother's current age is the square of her daughter's age. In $9$ years' time, the mother's age will be only three times the age her daughter will be then. Find their current ages.
Geometry problems
Always draw the figure and write its measurements using $x$. Then use the appropriate formula: perimeter, area, Pythagoras…
When setting up a quadratic equation, you move everything to one side and factorise. Problems that lead to a quadratic equation in geometry are Higher Level only, so the examples below that lead to a quadratic equation are marked "Higher Level only".
A school vegetable garden of rectangular shape needs to be fenced off to protect the crops. We know that the length of the garden is $5$ metres more than the width and…
A school vegetable garden of rectangular shape needs to be fenced off to protect the crops. We know that the length of the garden is $5$ metres more than the width and that $38$ metres of fencing were needed to completely enclose it. What are the dimensions of the garden?
Calculate the side length of a square billboard, knowing that if its side length were increased by $2$ metres, its area would increase by $16$ square metres.
Calculate the side length of a square billboard, knowing that if its side length were increased by $2$ metres, its area would increase by $16$ square metres.
A town council wants to fence a rectangular plot intended for a dog park. Due to the characteristics of the land, the length must be exactly three times the width, and…
A town council wants to fence a rectangular plot intended for a dog park. Due to the characteristics of the land, the length must be exactly three times the width, and there is a fixed budget to buy a full roll of $160\text{ m}$ of wire mesh fence.
- a) Before calculating anything, make a reasoned estimate: could the width be $30\text{ m}$? Justify whether that measurement would require more fencing than available, less, or the exact amount.
- b) Define the unknown $x$, set up a linear equation that represents the total perimeter, and solve it to find the exact dimensions of the park (length and width).
- c) Check whether the result obtained confirms your initial estimate and calculate the total area the dogs will have to run around.
A new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ metres greater than the width. If the total area…
A new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ metres greater than the width. If the total area allocated for play is $150$ $m^2$, calculate the dimensions (length and width) of the area.
Calculate the dimensions of an isosceles triangle knowing that the unequal side is $4$ $cm$ less than each of the equal sides and that the total perimeter is $26$ $cm$.…
Calculate the dimensions of an isosceles triangle knowing that the unequal side is $4$ $cm$ less than each of the equal sides and that the total perimeter is $26$ $cm$. Once you have found the sides, calculate the area of the triangle.
The area of a right-angled triangle is $30\text{ cm}^2$ and its hypotenuse is $13\text{ cm}$. Calculate the lengths of the two legs of the triangle.
The area of a right-angled triangle is $30\text{ cm}^2$ and its hypotenuse is $13\text{ cm}$. Calculate the lengths of the two legs of the triangle.
Moving objects, mixtures and sharing
These are the classic arithmetic problems, but set up using $x$. The templates:
| Type | Typical equation |
|---|---|
| Meeting (they set off towards each other) | $v_1 t + v_2 t = \text{distance}$ |
| Catching up (same direction) | $v_1 t = \text{head start} + v_2 t$ |
| Mixture | $p_1 x + p_2 (T - x) = p_m \cdot T$ |
| Money at two interest rates | $r_1 x + r_2 (T - x) = \text{total interest}$ |
| Working together (tank, pipes) | $\frac{1}{t_1} + \frac{1}{t_2} = \frac{1}{T}$ |
| Sharing among people | (price per person before) − (after) = difference |
Two cyclists, Aoife and Cian, are at a distance of $60$ $km$ apart and decide to meet at an intermediate point. Aoife sets off from $A$ towards Cian at a constant speed…
Two cyclists, Aoife and Cian, are at a distance of $60$ $km$ apart and decide to meet at an intermediate point. Aoife sets off from $A$ towards Cian at a constant speed of $20$ $km/h$. At the same time, Cian sets off from $B$ towards Aoife at a speed of $10$ $km/h$. How long will it take them to pass each other, and at what distance from point $A$ will they meet?
In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $12$ €/kg and "Robusta" coffee costs $8$ €/kg. If we want to obtain a mixture of $20$ $kg$…
In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $12$ €/kg and "Robusta" coffee costs $8$ €/kg. If we want to obtain a mixture of $20$ $kg$ priced at $10.50$ €/kg, how many kilograms of each type of coffee should we use?
A group of friends wants to buy a birthday present that costs $120$ €. At first there was a certain number of people, but at the last minute $2$ more friends joined,…
A group of friends wants to buy a birthday present that costs $120$ €. At first there was a certain number of people, but at the last minute $2$ more friends joined, which meant each person had to pay $5$ € less. How many friends were originally in the group?
An eco-friendly parcel delivery company operating in Cork offers two monthly rates for local businesses: Base Rate: a fixed fee of $15\text{ €}$ per month plus…
An eco-friendly parcel delivery company operating in Cork offers two monthly rates for local businesses:
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Base Rate: a fixed fee of $15\text{ €}$ per month plus $4.50\text{ €}$ for each shipment made.
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Eco-Plus Rate: no monthly fixed fee; it charges $7\text{ €}$ per shipment for the first $10$ shipments of the month, and reduces the price to $3.50\text{ €}$ for each additional shipment from the eleventh onwards.
A neighborhood bookstore sent more than $10$ packages last month and found that the final bill would have been exactly the same with either of the two rates.
- a) Define the unknown $x$ as the total number of monthly shipments and write the algebraic expression for the total cost with each rate in the case where $x > 10$.
- b) Set up and solve the corresponding equation to find out how many shipments the bookstore made last month.
- c) Based on the solution, explain to the bookstore owners which rate would be best for them to choose if they anticipate that their shipments will increase next month.
Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00 h, a cyclist departs from $A$ towards $B$ at a constant speed of $20\text{ km/h}$. An…
Two cities, $A$ and $B$, are separated by a distance of $155\text{ km}$. At 10:00 h, a cyclist departs from $A$ towards $B$ at a constant speed of $20\text{ km/h}$. An hour and a half later, a motorbike departs from $B$ towards $A$ at a speed of $80\text{ km/h}$. At what time will they pass each other, and at what distance from city $A$ will the meeting take place?