Imagine you're buying hay for a horse. One kilogram costs €2.40. Two kilograms costs €4.80. Five kilograms costs €12.00. The price always grows in exactly the same way: double the hay, double the price. This is a proportional relationship — and it shows up everywhere.
Proportional relationship: Two quantities $x$ and $y$ are proportional when $y = kx$ for some fixed number $k$, called the constant of proportionality. The ratio $\frac{y}{x}$ is always the same.
Worked Example
Worked Example — Finding k and writing the equation
Hay costs €2.40 per kilogram. Let $x$ = kilograms, $y$ = cost in €.
| $x$ (kg) | $y$ (€) | $y \div x$ |
|---|
| 1 | 2.40 | 2.40 |
| 3 | 7.20 | 2.40 |
| 7 | 16.80 | 2.40 |
The ratio $\frac{y}{x}$ is always 2.40, so $k = 2.40$ and the equation is:
$$y = 2.40x$$
To find the cost of 11 kg: $y = 2.40 \times 11 = €26.40$.
To find how many kg cost €19.20: $19.20 = 2.40x \quad \mid \div 2.40 \quad \Rightarrow x = 8$ kg.
Warm-Up — Complete the ratio table (equation given)
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Problem 1
Stable rent costs €15 per day. Complete the table and write the equation.
Equation: $y = 15x$ where $x$ = days, $y$ = cost in €
| Days ($x$) | Cost € ($y$) |
|---|
| 1 | 15 |
| 3 | ? |
| 7 | ? |
| ? | 90 |
| 14 | ? |
fill in the table →
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Problem 2
Mia earns €12.50 per hour giving riding lessons. Complete the table and confirm the ratio is constant.
Equation: $y = 12.50x$ where $x$ = hours, $y$ = earnings in €
| Hours ($x$) | Earnings € ($y$) | $y \div x$ |
|---|
| 1 | ? | ? |
| 4 | 50.00 | ? |
| 6 | ? | ? |
| ? | 100.00 | ? |
fill in the table →
Core Problems — find $k$ and write the equation yourself, then solve
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Problem 3
Veterinary visits cost €85 per visit at a stable near Seville. A table shows: 2 visits cost €170, 5 visits cost €425. Find $k$, write the equation $y = kx$, then find the cost of 9 visits.
find k → write equation → solve →
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Problem 4
The table below shows the cost of horse blankets at a market in Munich. Find the constant of proportionality and write the equation. How many blankets can Mia buy with €280?
| Blankets ($x$) | Cost € ($y$) |
|---|
| 2 | 70 |
| 5 | 175 |
| 8 | 280 |
find k → write equation → answer →
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Problem 5
In a horse trailer, the weight of feed bags is proportional to the number of bags. 4 bags weigh 48 kg. Write the equation. How many bags weigh 84 kg? Is this proportional? How do you know?
find k → write equation → solve →
Inverse Proportion — more of one means less of the other
Concept — Inverse Proportion
In direct proportion: more $x$ → more $y$ (hay: more kg → more cost).
In inverse proportion: more $x$ → less $y$. The product $x \times y$ is constant.
Example: 2 horses eat a hay bale in 6 days. If there are 3 horses, the bale lasts: $2 \times 6 = 12$, so $3 \times ? = 12 \Rightarrow ? = 4$ days.
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Problem 6
At a stable near Kyoto, 3 grooms can clean all the stalls in 8 hours. How long would it take 4 grooms working at the same pace? How long for 6 grooms?
product = constant → solve →
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Problem 7
A 120 km trail ride is planned. At 15 km/h the ride takes 8 hours. Complete the table for different speeds. What type of proportion is this?
| Speed (km/h) | Time (hours) |
|---|
| 15 | 8 |
| 20 | ? |
| 24 | ? |
| 30 | ? |
direct or inverse? → fill in table →
Open Challenge Open
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Problem 8 Open
Mia wants to set up a horse-grooming service. Design a proportional pricing table: choose your own rate per horse (€ per horse), make a table with at least 5 entries, write the equation $y = kx$, and use it to answer: "How much does Mia earn if she grooms 11 horses?" Then decide — is €/horse a fair rate, or would you charge differently? Explain.
choose k → build table → write equation → answer → explain →
Show answers
Problem 1
$y = 15x$ · Days 3→€45, 7→€105, 6 days→€90, 14→€210
Problem 2
$y = 12.50x$ · 1hr→€12.50, 4hr→€50 (✓), 6hr→€75, 8hr→€100 · $k = 12.50$ throughout
Problem 3
$k = 85$, equation $y = 85x$ · 9 visits: $y = 85 \times 9 = €765$
Problem 4
$k = 35$ (€35 per blanket) · Equation $y = 35x$ · €280 ÷ 35 = 8 blankets
Problem 5
$k = 12$ kg/bag · Equation $y = 12x$ · $84 = 12x \Rightarrow x = 7$ bags · Yes, proportional because $y/x = 12$ always
Problem 6
Product = $3 \times 8 = 24$ · 4 grooms: $24 \div 4 = 6$ hours · 6 grooms: $24 \div 6 = 4$ hours
Problem 7
Inverse proportion (speed × time = 120) · 20 km/h → 6 h · 24 km/h → 5 h · 30 km/h → 4 h
Problem 8
Open — check that table has constant ratio $y/x = k$, equation is $y = kx$, and answer uses the equation correctly.
Coming up next → Lesson 13: Angles
You've seen that proportionality keeps a constant ratio. In geometry, the angles inside every triangle also keep a constant sum — always 180°. Understanding angles is the first step toward triangles, and triangles are the key to understanding why slope never changes along a line.