Why this matters: A calculator can solve an equation in a second.
But it cannot read a real situation and decide which equation to write.
That is the human part — and the most important part of mathematics.
Worked Example
A candle is 20 cm tall. It burns down by 3 cm every hour.
After how many hours is it 8 cm tall?
Step 1 — decide what x stands for
Let $x$ = the number of hours.
Step 2 — describe both sides in terms of x
Height after $x$ hours: $20 - 3x$
Target height: $8$
Step 3 — set them equal
$$20 - 3x = 8$$
That's it. We stop here. Solving is a separate step.
Your Turn — write only the equation, on paper
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Problem 1
A cinema ticket costs 9 €. Sofia buys some tickets and pays a total of 36 €.
How many tickets did she buy?
Let $x$ = number of tickets
your equation →
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Problem 2
A water tank holds 80 litres. A pipe fills it at 5 litres per minute.
After how many minutes is it full?
Let $x$ = number of minutes
your equation →
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Problem 3
Three friends share a restaurant bill equally. Each person pays 14 €.
What was the total bill?
Let $x$ = total bill in €
your equation →
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Problem 4
Anna is 5 years older than twice her brother's age. Anna is 23.
How old is her brother?
Let $x$ = brother's age
your equation →
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Problem 5
Two taxis charge differently. Taxi A charges 3 € per km plus a 2 € booking fee.
Taxi B charges 1 € per km plus a 10 € booking fee.
At what distance do both taxis cost the same?
Let $x$ = distance in km
your equation →
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Problem 6
A rectangle is 3 cm longer than it is wide.
Its perimeter is 38 cm. What is the width?
Let $x$ = width in cm
your equation →
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Problem 7
Mia and Leo are both saving money.
Mia has 12 € and saves 4 € per week.
Leo has 30 € and spends 2 € per week.
After how many weeks will they have the same amount?
define x yourself, then write the equation →
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Problem 8 Challenge
A bag contains red and blue marbles. There are 4 more red marbles than blue.
There are 22 marbles in total. How many blue marbles are there?
define x yourself, then write the equation →
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Problem 9 Challenge
A factory produces 3 boxes every 2 minutes. Another factory produces 2 boxes
every minute but started 6 minutes later. After how many minutes from the
start of the first factory do both factories have the same number of boxes?
define x carefully, then write the equation →
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Problem 10 Write your own
Make up a real situation of your own — something from daily life —
where you need to find an unknown number. Write the situation in one or two sentences,
then write the equation that goes with it.
your situation →
your equation →
Show equations
Problem 1
$9x = 36$
Problem 2
$5x = 80$
Problem 3
$\dfrac{x}{3} = 14$
or $x = 3 \times 14$
Problem 4
$2x + 5 = 23$
Problem 5
$3x + 2 = x + 10$
Problem 6
$2(x + x + 3) = 38$
or $2x + 2(x+3) = 38$
Problem 7
$12 + 4x = 30 - 2x$
x = number of weeks
Problem 8
$x + (x + 4) = 22$
x = blue marbles
Problem 9
$\dfrac{3}{2}x = 2(x - 6)$
x = minutes from start of factory 1
Problem 10
your own — no single answer
Coming up next →
You've practised writing equations. Now back to solving — but this time with
brackets to expand and coefficients that are fractions. The | method handles
both cleanly.