Problems with equations
Solving the equation is the mechanical part, the one you already know how to do. The hard part (and what really matters) is setting it up: turning a sentence into an equation with an $x$. With a clear method and a few models, most problems become routine.
The method in steps
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Choose the unknown and write down what it means: "$x$ = number of trading cards Marcos has". Don't just write the letter; note which quantity it refers to.
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Translate each piece of information in the sentence using that same $x$.
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Set up the equation using the missing piece of information (usually a total or an equality) and solve it.
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Check against the wording of the problem, not against the equation (it might have been set up wrongly), and answer with a sentence and units.
If you add a consecutive number (the number that comes right after) to a number, the result is $43$. Set up an equation to find out what the number is.
If you add a consecutive number (the number that comes right after) to a number, the result is $43$. Set up an equation to find out what the number is.
A group of friends buys a family-sized pizza that costs $12$ €. If they have $8$ € left over after paying for it from the money they had collected, how much money did…
A group of friends buys a family-sized pizza that costs $12$ €. If they have $8$ € left over after paying for it from the money they had collected, how much money did they have at the start? Represent the initial amount with the letter $x$.
Numbers, "twice as many as" and ages
Two types of information appear again and again:
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Consecutive numbers: if the first is $x$, the next is $x + 1$, and the one after that is $x + 2$.
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"Twice/three times as much as another quantity": if Marcos has $x$ trading cards and Lucía has twice as many, Lucía has $2x$. If Paula also has $10$ more than Marcos, Paula has $x + 10$.
For ages, it helps to make a small table with "today" and "in" a certain number of years: time passes equally for everyone, so you add the same amount to each age.
| Today | In 8 years' time | |
|---|---|---|
| Elena | $x$ | $x + 8$ |
| Héctor | $x + 5$ | $x + 13$ |
Amina is $3$ years older than her younger brother. If their combined age is $19$ years, how old is each of them? Let $x$ be the age of the younger brother.
Amina is $3$ years older than her younger brother. If their combined age is $19$ years, how old is each of them? Let $x$ be the age of the younger brother.
Marcos is playing a video game and has scored twice as many points in the current level as in the previous level. If he scores a total of $150$ points across both…
Marcos is playing a video game and has scored twice as many points in the current level as in the previous level. If he scores a total of $150$ points across both levels, how many points did he get in each level? Let $x$ be the points in the first level.
Three friends are comparing how many songs they have in their favourite playlist. Lucía has twice as many songs as Marcos, and Sara has 15 more songs than Marcos. If all…
Three friends are comparing how many songs they have in their favourite playlist. Lucía has twice as many songs as Marcos, and Sara has 15 more songs than Marcos. If all three have a total of 155 songs altogether, how many songs does each person have in their playlist? Set up an equation where $x$ is the number of Marcos's songs.
Problems with perimeters
In geometry, always draw the shape and write the measurements using $x$. Then use the perimeter formula (the sum of all the sides).
We are designing an esports team logo shaped like an isosceles triangle. We want the perimetre to be $30$ cm. If the base measures $10$ cm and the other two sides are…
We are designing an esports team logo shaped like an isosceles triangle. We want the perimetre to be $30$ cm. If the base measures $10$ cm and the other two sides are equal, how long must each of those sides ($x$) be?
We want to fence off a small rectangular vegetable patch to plant tomatoes. The length of the patch is twice its width. If we have used $18$ metres of fencing to…
We want to fence off a small rectangular vegetable patch to plant tomatoes. The length of the patch is twice its width. If we have used $18$ metres of fencing to completely surround it (perimetre), what are the width and length of the patch?
At Adrián's birthday party, two types of invitations were handed out: digital and paper. The number of digital invitations was three times the number of paper ones. If a…
At Adrián's birthday party, two types of invitations were handed out: digital and paper. The number of digital invitations was three times the number of paper ones. If a total of 48 invitations were sent out, how many people received a paper invitation and how many a digital one?
Sharing out money
When several people share a cost or a gift, call $x$ the share of one of them and write the others in terms of $x$. The equation comes from the fact that all the shares together make the total.
On a shelf in a comic book shop, there are three times as many action manga as mystery manga. If there are $48$ comics in total between both genres, how many are there…
On a shelf in a comic book shop, there are three times as many action manga as mystery manga. If there are $48$ comics in total between both genres, how many are there of each type? Let $x$ be the mystery comics.
Marta bought a video game on offer and a new controller. The controller cost her $25$ € less than the video game. If she paid a total of $89$ € for both items, what is…
Marta bought a video game on offer and a new controller. The controller cost her $25$ € less than the video game. If she paid a total of $89$ € for both items, what is the original price of each item without counting other discounts?
In a piggy bank, there are $0,50$ € and $1$ € coins. In total, there are $35$ coins with a total value of $26$ €. How many coins of each type are in the piggy bank?…
In a piggy bank, there are $0,50$ € and $1$ € coins. In total, there are $35$ coins with a total value of $26$ €. How many coins of each type are in the piggy bank? (Hint: if you let $x$ be the number of $1$ € coins, the number of $0,50$ € coins will be $35 - x$).