You already know how to solve systems. What makes the difference now is setting them up: turning a problem written in words into two correct equations. This is the part where most mistakes happen, and the one most worth practising.
The difference between two numbers is $14$. If we add twice the smaller number to the larger number, the result is $53$. Find both numbers by setting up a system of…
The difference between two numbers is $14$. If we add twice the smaller number to the larger number, the result is $53$. Find both numbers by setting up a system of equations.
The key is to write "$n$ years ago" (you subtract $n$ from the current age) and "in $n$ years' time" (you add $n$) carefully.
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The sum of Salma's and her grandfather's ages is $72$ years. $6$ years ago, the grandfather's age was five times Salma's age. Set up a system of equations and find their…
The sum of Salma's and her grandfather's ages is $72$ years. $6$ years ago, the grandfather's age was five times Salma's age. Set up a system of equations and find their current ages.
Here there are usually two equations: one for the total quantity and another for the total price or concentration.
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A merchant wants to prepare $50 \text{ kg}$ of a coffee blend. He uses one coffee costing $6 \text{ €/kg}$ and another costing $8 \text{ €/kg}$. If he wants the price of…
A merchant wants to prepare $50 \text{ kg}$ of a coffee blend. He uses one coffee costing $6 \text{ €/kg}$ and another costing $8 \text{ €/kg}$. If he wants the price of the blend to be exactly $7,20 \text{ €/kg}$, how many kilograms of each type of coffee should he use?
An investor divides a capital of $20\,000 \text{ €}$ between two investment funds. The first offers an annual yield of $4\%$ and the second of $6\%$. At the end of one…
An investor divides a capital of $20\,000 \text{ €}$ between two investment funds. The first offers an annual yield of $4\%$ and the second of $6\%$. At the end of one year, the total profit from both funds amounts to $1\,050 \text{ €}$. How much money did she invest in each fund?
"Heads and legs", coins, tickets... the pattern is: one equation counts units and the other counts something that depends on each unit (legs, value, points).
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In a car park there are motorbikes and cars. In total there are $35$ vehicles and $114$ wheels (not counting spare wheels). Calculate how many motorbikes and how many…
In a car park there are motorbikes and cars. In total there are $35$ vehicles and $114$ wheels (not counting spare wheels). Calculate how many motorbikes and how many cars are in the car park.
A money box contains $45$ coins that add up to a total of $14,40$ euros. If all the coins are $20 \text{ cents}$ and $50 \text{ cents}$, find out how many coins of each…
A money box contains $45$ coins that add up to a total of $14,40$ euros. If all the coins are $20 \text{ cents}$ and $50 \text{ cents}$, find out how many coins of each type there are.
In a multiple-choice exam of $30$ questions, $5$ points are awarded for each correct answer and $2$ points are deducted for each incorrect or blank answer. A student…
In a multiple-choice exam of $30$ questions, $5$ points are awarded for each correct answer and $2$ points are deducted for each incorrect or blank answer. A student scored $94$ points. How many questions did they answer correctly?
With speed, time and distance, remember $d = v \cdot t$. When there is a current, wind or a delayed start, each situation gives a different equation with the same distance or the same meeting point.
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A tank can be filled using two taps, A and B. If both are turned on at the same time, the tank fills in $6$ hours. Tap A alone takes $5$ hours less than tap B to fill…
A tank can be filled using two taps, A and B. If both are turned on at the same time, the tank fills in $6$ hours. Tap A alone takes $5$ hours less than tap B to fill the full tank. Set up a system of equations and find how long each tap would take to fill it individually.
A cyclist leaves city A for city B at a constant speed. One hour later, a motorcyclist sets off from A in pursuit at a speed that is $15 \text{ km/h}$ higher than that…
A cyclist leaves city A for city B at a constant speed. One hour later, a motorcyclist sets off from A in pursuit at a speed that is $15 \text{ km/h}$ higher than that of the cyclist, catching up with him just as he arrives at B. If the distance between A and B is $60 \text{ km}$, set up a system of equations to find the speeds of both and the time the cyclist took to complete the journey.