Problems with Equations
Yaritza wants to buy a cell phone that costs $\text{RD}\$15{,}000$. She already has $\text{RD}\$3{,}000$ saved and can put away $\text{RD}\$600$ each week. How many weeks does she need? Solving an equation is the mechanical part. The hard part 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. At the end of the page we come back to Yaritza's phone.
The four-step method
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Choose the unknown and write down what it means: "$x$ = Yeison's age today." If there are several quantities, express them with the same letter whenever you can.
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Translate each piece of information into algebraic language.
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Set up the equation using the condition that links the data, and solve it.
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Check against the word problem (not the equation, which might be 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?
A group of friends buys tickets for a concert. "Standing" tickets cost $900$ pesos more than "seated" tickets. If they buy one of each type and the total bill is…
A group of friends buys tickets for a concert. "Standing" tickets cost $900$ pesos more than "seated" tickets. If they buy one of each type and the total bill is $5\,100$ pesos, what is the price of each ticket?
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" and "in" (or "ago"). Time passes the same for everyone: the same amount is added (or subtracted) to all the ages.
| Today | In 6 years | |
|---|---|---|
| Yeison | $x$ | $x + 6$ |
| Pamela | $x + 4$ | $x + 10$ |
When one person's age is the square of the other's, the equation is second-degree. You solve it with the quadratic formula or by factoring, and you discard the solution that makes no sense (a negative age).
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?
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 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.
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.
Areas with borders and right triangles
Here are two situations that lead to second-degree equations.
A border of width $x$ around an $a \times b$ rectangle: the border is added on both sides, so the total dimensions are $(a + 2x)$ and $(b + 2x)$.
Right triangle: the area is $\dfrac{\text{leg}_1 \cdot \text{leg}_2}{2}$ and the Pythagorean theorem holds, $c_1^2 + c_2^2 = h^2$. In an isosceles triangle, the height to the unequal side splits it into two halves and forms a right triangle.
If the given information is the area and the hypotenuse, there is no direct relationship between the legs. Use two unknowns (a system) and the identity $(x + y)^2 = x^2 + y^2 + 2xy$.
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.
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?
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.
Motion: meeting and catching up
Remember that $\text{distance} = \text{speed} \times \text{time}$. Make a sketch of the road and write down what distance each vehicle or person travels.
| Type | Typical equation |
|---|---|
| Meeting (they head toward each other) | $v_1 t + v_2 t = \text{total distance}$ |
| Catching up (same direction) | $v_1 t = \text{head start} + v_2 t$ |
| Late departure | the one who leaves late travels $t - \text{delay}$ |
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?
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?
Mixtures, prices, and interest
In these problems a total amount $T$ is split into two parts: call one $x$ and the other $T - x$. Then you write a second condition about the price, the concentration, or the interest.
| Type | Typical equation |
|---|---|
| Mixture by price | $p_1 x + p_2 (T - x) = p_m \cdot T$ |
| Mixture by concentration | $c_1 x + c_2 (T - x) = c_m \cdot T$ |
| Two investments | $r_1 x + r_2 (T - x) = \text{total earnings}$ |
All three have the same form: each part is multiplied by its "value" (price, percent, or rate) and the total must match what is asked for. Percents are converted to decimals ($25\% = 0.25$).
You can also solve them with two unknowns ($x + y = T$ and the second condition) and a system by substitution: it is exactly the same thing.
In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $720$ pesos/kg and "Robusta" coffee costs $480$ pesos/kg. If we want to obtain a mixture…
In a gourmet shop, two types of coffee are to be mixed. "Arabica" coffee costs $720$ pesos/kg and "Robusta" coffee costs $480$ pesos/kg. If we want to obtain a mixture of $20$ $kg$ priced at $630$ pesos/kg, how many kilograms of each type of coffee should we use?
In a chemistry laboratory, $20\text{ liters}$ of an acid solution at $25\%$ concentration is needed. The technician has two containers: one with a $10\%$ acid mixture…
In a chemistry laboratory, $20\text{ liters}$ of an acid solution at $25\%$ concentration is needed. The technician has two containers: one with a $10\%$ acid mixture and another with a concentrated $50\%$ mixture. How many liters from each container must be mixed to obtain the desired concentration?
Mario won $\text{RD\$}\ 72\,000…
Mario won $\text{RD\$}\ 72\,000$ in a programming competition and divided it 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 $7\%$ annually. If at the end of the year the total profit obtained was $\text{RD\$}\ 3\,780$, how much money did he allocate to each product?
Sharing costs and teamwork
Some problems have the unknown in the denominator. You solve them by multiplying the whole equation by the common denominator.
Sharing costs: if a cost $G$ is divided among $x$ people, each person pays $\dfrac{G}{x}$. If more people join, each payment goes down.
Teamwork: if a faucet fills a tank in $x$ hours, in one hour it fills $\dfrac{1}{x}$ of the tank. The "work per hour" amounts add up: $\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{t}$, where $t$ is the time working together.
Another type of problem: a natural number satisfies a condition involving its square. You set up $x^2 - 3x = 40$, bring it to the form $x^2 - 3x - 40 = 0$, and solve it. If the problem asks for a natural number, discard negative or fractional solutions.
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 group of friends wants to buy a birthday present that costs $7\,200$ 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 $7\,200$ 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 $300$ pesos less. How many friends were originally in the group?
Two pipes connected to a tank can fill it together in $4\text{ hours}$. If used separately, the smaller diameter pipe takes $6\text{ hours}$ longer than the larger…
Two pipes connected to a tank can fill it together in $4\text{ hours}$. If used separately, the smaller diameter pipe takes $6\text{ hours}$ longer than the larger diameter pipe to fill the full tank. How long would each pipe take on its own?
Summary
Let's go back to Yaritza's phone. If $x$ is the number of weeks, $3{,}000 + 600x = 15{,}000$, so $600x = 12{,}000$ and $x = 20$. She needs 20 weeks. Check: $3{,}000 + 20 \cdot 600 = 15{,}000$ ✓