Problems with equations
Wilkin goes to the corner store and buys 3 sodas and a pack of cookies that costs $\text{RD}\$90$. In total he pays $\text{RD}\$330$. How much does each soda cost? You already know how to solve an equation. The hard part is setting it up, turning what the problem says into an equality. With a method and a few templates, almost every problem becomes routine.
The four-step method
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Choose the unknown and write down what it means. For example: "$x$ = price of one soda."
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Translate each piece of information in the problem using that letter.
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Set up the equation using the condition the problem states, and solve it.
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Check the result in the problem (not just in the equation) and answer with a sentence and units.
If we add $12$ to three times a number, we get $48$. What is the number?
If we add $12$ to three times a number, we get $48$. What is the number?
Marta has bought $3$ identical T-shirts for the athletics team and a pair of sports trousers costing $\text{RD\$}\ 15$. If she spent a total of $\text{RD\$}\ 60$, how…
Marta has bought $3$ identical T-shirts for the athletics team and a pair of sports trousers costing $\text{RD\$}\ 15$. If she spent a total of $\text{RD\$}\ 60$, how much did each T-shirt cost?
In a video game shop, they have sold twice as many handheld consoles as desktop consoles. If they have sold a total of $18$ consoles, how many have they sold of each…
In a video game shop, they have sold twice as many handheld consoles as desktop consoles. If they have sold a total of $18$ consoles, how many have they sold of each type?
Consecutive numbers and digits
Consecutive numbers: if the first is $x$, the next ones are $x+1$, $x+2$, etc.
Two-digit number: if the tens digit is $x$ and the ones digit is $y$, the number is worth $10x + y$. For example, $47 = 10 \cdot 4 + 7$. If the digits are reversed, the number is $10y + x$.
Find three consecutive integers whose sum is equal to $54$.
Find three consecutive integers whose sum is equal to $54$.
Find three consecutive integers such that the sum of the first two is $15$ units greater than the third number.
Find three consecutive integers such that the sum of the first two is $15$ units greater than the third number.
A number has two digits that add up to $12$. If we reverse the order of its digits, we get a new number that is $18$ units larger than the original number. Find the…
A number has two digits that add up to $12$. If we reverse the order of its digits, we get a new number that is $18$ units larger 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 $10x + y$).
Age problems
In age problems, use a table with "today" and "in / ago" a certain number of years. Time passes the same for everyone: you add (or subtract) the same amount to each age.
| Today | In 8 years | 5 years ago | |
|---|---|---|---|
| Son | $x$ | $x + 8$ | $x - 5$ |
| Father (three times as old) | $3x$ | $3x + 8$ | $3x - 5$ |
Álvaro is $4$ years older than his sister Lucía. If the sum of their ages is $26$ years, how old is each sibling?
Álvaro is $4$ years older than his sister Lucía. If the sum of their ages is $26$ years, how old is each sibling?
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?
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?
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 had at that time. How many years must pass for the…
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 had at that time. How many years must pass for the sum of the father's and the son's ages to be exactly $100$ years?
Comparing quantities
Many problems say things like "three times…", "12 less than…", or "if one gains and the other loses, they end up equal." They are translated like this:
| In words | In algebra |
|---|---|
| twice $x$ | $2x$ |
| 12 less than $x$ | $x - 12$ |
| half of $x$ | $\dfrac{x}{2}$ |
| $A$ and $B$ end up equal | $A = B$ |
On a social network, Alba has three times as many followers as her brother Javier. If Alba were to lose $150$ followers and Javier were to gain $50$, both would have…
On a social network, Alba has three times as many followers as her brother Javier. If Alba were to lose $150$ followers and Javier were to gain $50$, both would have exactly the same amount. How many followers does each currently have?
Marcos has bought a video game and headphones for his games console. The headphones cost him $720$ pesos less than the video game. If the video game had been discounted…
Marcos has bought a video game and headphones for his games console. The headphones cost him $720$ pesos less than the video game. If the video game had been discounted to half its original price, Marcos would have paid $2\,880$ pesos for both items together. What is the original price of each item?
A courier company uses vans and motorbikes. Vans have $4$ wheels and motorbikes have $2$. In total there are $24$ drivers (one per vehicle) and $64$ wheels. Due to…
A courier company uses vans and motorbikes. Vans have $4$ wheels and motorbikes have $2$. In total there are $24$ drivers (one per vehicle) and $64$ wheels. Due to logistical problems, the daily maintenance cost is $15$ pesos per van and $6$ pesos per motorbike. If the total daily maintenance budget is $200$ pesos, how many van drivers should switch, at a minimum, to driving motorbikes so as not to exceed the budget without changing the total number of vehicles?
Two quantities with a single unknown
Sometimes there are two quantities that together add up to a known total (heads, kilograms, liters). We call one of them $x$ and the other is total $- x$. Then we use the second condition (legs, money, price) to set up the equation.
Mixtures. The total money value of the mixture is the sum of what each ingredient is worth: (amount × price) of one + (amount × price) of the other $=$ total amount × price of the mixture.
A shepherd has a mix of sheep and chickens in his pen. If we count heads there is a total of $20$, and if we count legs there are $64$. How many sheep and how many…
A shepherd has a mix of sheep and chickens in his pen. If we count heads there is a total of $20$, and if we count legs there are $64$. How many sheep and how many chickens are in the pen? (Remember that sheep have $4$ legs and chickens $2$).
In a café, they have mixed Colombian coffee, which costs $720$ pesos/kg, with Brazilian coffee, which costs $480$ pesos/kg. If they obtained $20$ kg of mixture and its…
In a café, they have mixed Colombian coffee, which costs $720$ pesos/kg, with Brazilian coffee, which costs $480$ pesos/kg. If they obtained $20$ kg of mixture and its final price is $540$ pesos/kg, how many kilograms of coffee of each type did they use?
A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $120$ pesos/liter and essence B costs $75$ pesos/liter. They have decided…
A perfume laboratory wants to create a new fragrance by mixing two essences. Essence A costs $120$ pesos/liter and essence B costs $75$ pesos/liter. They have decided that to get a $250$ ml bottle, the cost of the mixture must be exactly $24.75$ pesos. What amount of each essence, measured in milliliters, should the bottle contain?
Figures: perimeters
In geometry, draw the figure and write its measurements using $x$. The perimeter is the sum of all the sides.
The perimetre of an isosceles triangle is $40 \text{ cm}$. If the unequal side measures $10 \text{ cm}$, how long is each of the equal sides?
The perimetre of an isosceles triangle is $40 \text{ cm}$. If the unequal side measures $10 \text{ cm}$, how long is each of the equal sides?
We want to put a wooden frame around a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If framing it required a…
We want to put a wooden frame around a rectangular photograph. We know that the length of the photograph is $5 \text{ cm}$ more than the width. If framing it required a strip of wood with a total length (perimetre) of $70 \text{ cm}$, what are the dimensions of the photograph?
A farmer wants to fence off a triangular plot for his sheep. Knowing that it is an isosceles triangle, that the unequal side measures $5$ meters less than each of the…
A farmer wants to fence off a triangular plot for his sheep. Knowing that it is an isosceles triangle, that the unequal side measures $5$ meters less than each of the equal sides and that a total of $40$ meters of fencing was needed, how long is each side of the plot?
Figures: areas and Pythagoras
The area of a rectangle is base $\times$ height. For a path surrounding a courtyard, the area of the path is the total area minus the area of the courtyard. In a right triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
If after expanding you are left with an $x^2$ term that does not cancel, you have a quadratic equation. Move everything to one side, set it equal to 0, and solve with the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ or by factoring. Then discard the solution that doesn't make sense (a negative length, for example).
A rectangular padel court has a perimeter of $60$ meters. Knowing that the length of the court is $6$ meters more than the width, calculate the dimensions (length and…
A rectangular padel court has a perimeter of $60$ meters. Knowing that the length of the court is $6$ meters more than the width, calculate the dimensions (length and width) of this court.
Calculate the dimensions of a rectangle knowing that its base is $3$ cm greater than its height and that, if we increase the height by $2$ cm, the area of the new…
Calculate the dimensions of a rectangle knowing that its base is $3$ cm greater than its height and that, if we increase the height by $2$ cm, the area of the new rectangle increases by $16$ $cm^2$.
Calculate the perimetre of a right-angled triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use Pythagoras' Theorem…
Calculate the perimetre of a right-angled triangle knowing that its three sides are expressed by three consecutive natural numbers. (Hint: use Pythagoras' Theorem $a^2 + b^2 = c^2$, where $c$ is the longest side).
Motion and work
Uniform motion: distance = speed $\times$ time ($d = vt$). When one catches up to the other, both have traveled the same distance.
Work: if a faucet fills the tank in $x$ hours, in one hour it fills $\dfrac{1}{x}$ of the tank. If two faucets run together, what they fill in one hour adds up.
In a robotics competition, a robot leaves the starting line at a constant speed of $2$ m/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3.5$…
In a robotics competition, a robot leaves the starting line at a constant speed of $2$ m/s. Thirty seconds later, a second robot sets off in pursuit at a speed of $3.5$ m/s. If the track has a total length of $150$ meters before reaching an obstacle zone, will the second robot manage to catch up with the first before it enters the obstacle zone? Set up the equation and justify your answer by calculating the catching-up distance.
A water tank is filled by two taps. Tap A takes $6$ hours longer than tap B to fill the tank on its own. If both are opened at the same time, the tank fills in exactly…
A water tank is filled by two taps. Tap A takes $6$ hours longer than tap B to fill the tank on its own. If both are opened at the same time, the tank fills in exactly $4$ hours. Calculate how long tap B would take to fill the tank on its own. (This problem requires solving a quadratic equation).