Word Problems with Equations
Solving an equation is the easy part: it's mechanical. The hard part, and the one that really counts, is setting it up: turning a statement into an equation with letters. With a clear method and a few templates, most problems become routine.
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 with the same letter if you can.
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Translate each piece of data into algebraic language (review the table of algebraic expressions).
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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 the equation, which might have been set up wrong) and answer with a sentence and units.
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 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?
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 the same for everyone:
| Today | In 6 years | |
|---|---|---|
| Hugo | $x$ | $x + 6$ |
| Marta | $x + 4$ | $x + 10$ |
Aisha is $4$ years older than her younger brother, Iker. In $6$ years' time, the sum of their ages will be $32$ years. Calculate the current age of each person by…
Aisha is $4$ years older than her younger brother, Iker. 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.
The sum of three race numbers from a race, which turn out to be three consecutive integers, is equal to $63$. What numbers are those three runners wearing on their…
The sum of three race numbers from a race, which turn out to be three consecutive integers, is equal to $63$. What numbers are those three runners wearing on their shirts?
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?
Geometry problems
Always draw the figure and write its measurements using $x$. Then use the right formula: perimeter, area, Pythagorean theorem…
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 $5$ meters 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 $5$ meters more than the width and that $38$ meters 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 $2$ meters, its area would increase by $16$ square meters.
Calculate the side length of a square billboard, knowing that if its side length were increased by $2$ meters, its area would increase by $16$ square meters.
Problems with a quadratic equation
Many problems about areas, numbers, or sharing lead to an equation with $x^2$. The method is the same four steps; only two things change: you have to put the equation in standard form $ax^2 + bx + c = 0$ to solve it (by factoring or with the quadratic formula), and there are almost always two solutions, of which sometimes only one makes sense.
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 new rectangular play area is being designed in a municipal park. It is known that the length of the area must be $5$ meters 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 $5$ meters greater than the width. If the total area allocated for play is $150$ $m^2$, calculate the dimensions (length and width) of the area.
A graphic designer is working on a new company logo. The design consists of a central rectangle for the name, surrounded by a decorative border of uniform thickness $x$.…
A graphic designer is working on a new company logo. 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 should the thickness $x$ of the border be?
Motion, mixtures, and sharing
These are the classic arithmetic problems, but set up with $x$. The templates:
| Type | Typical equation |
|---|---|
| Meeting (starting toward each other) | $v_1 t + v_2 t = \text{distance}$ |
| Catching up (same direction) | $v_1 t = \text{head start} + v_2 t$ |
| Mixture | $p_1 x + p_2 (T - x) = p_m \times T$ |
| Sharing among people | (price per person before) − (after) = difference |
Two cyclists, Lucía and Marcos, are at a distance of $60$ $km$ apart and decide to meet at an intermediate point. Lucía sets off from $A$ towards Marcos at a constant…
Two cyclists, Lucía and Marcos, are at a distance of $60$ $km$ apart and decide to meet at an intermediate point. Lucía sets off from $A$ towards Marcos at a constant speed of $20$ $km/h$. At the same time, Marcos sets off from $B$ towards Lucía 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 $240$ pesos/kg and "Robusta" coffee costs $160$ pesos/kg. If we want to obtain a mixture…
In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $240$ pesos/kg and "Robusta" coffee costs $160$ pesos/kg. If we want to obtain a mixture of $20$ $kg$ priced at $210$ pesos/kg, how many kilograms of each type of coffee should we use?
A group of friends wants to buy a birthday present that costs $2\,400$ pesos. At first there was a certain number of people, but at the last minute $2$ more friends…
A group of friends wants to buy a birthday present that costs $2\,400$ pesos. 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 $100$ pesos less. How many friends were originally in the group?