Copper needs exactly the right amount of feed — not a whole bag, sometimes just a fraction. Fractions in equations work the same way as whole numbers: the balance rule still holds. The key insight for today: dividing both sides by $n$ is exactly the same as multiplying both sides by $\frac{1}{n}$. These are two names for the same move.
Worked Examples
Worked Example 1
Mia paid 35 € for riding lessons this month. The fee included a 5 € booking charge, plus two-thirds of her competition prize money. How much prize money did she win?
Let $x$ = prize money in €. Two-thirds of it, plus 5 €, equals 35 €:
Why ×3/2? Because $\frac{2}{3}x = 30$ means we need to undo ×2/3. The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$, so multiplying by $\frac{3}{2}$ isolates $x$.
Mia won 45 € in prize money.
Two ways to say the same thing:
$\frac{x}{4} = 9 \quad \mid \div \frac{1}{4}$ is the same as $\frac{x}{4} = 9 \quad \mid \times 4$
Dividing by $\frac{1}{4}$ = multiplying by $4$. Dividing by $n$ = multiplying by $\frac{1}{n}$.
Worked Example 2
A bag of supplement weighs $\frac{2}{5}$ of a bale of hay, plus 3 kg. The supplement bag weighs 13 kg. How heavy is one full bale?
Problems 1–4: the equation is shown. Solve on paper using the | method.
Copper's daily ration. Copper's daily hay ration is one-third of a bale. Mia adds 4 kg of grain supplement. The total daily feed weighs 10 kg. How many kg does a full bale weigh?
Equation: $\dfrac{x}{3} + 4 = 10$
Riding gloves. Mia spent three-quarters of her prize money on new riding gloves. She spent $18. What was the full prize?
Equation: $\dfrac{3}{4}x = 18$
Stable rent discount. The stable gives Mia a discount because she helps clean. She pays three-fifths of the listed weekly rent, which comes to 120 €. What is the listed rent?
Equation: $\dfrac{3}{5}x = 120$
Lesson fee. A riding lesson costs three-quarters of the hourly rate, plus a 6 € arena booking fee. Mia pays 21 € total. What is the hourly rate?
Equation: $\dfrac{3}{4}x + 6 = 21$
From here — write the equation yourself, then solve.
Competition entry. Mia pays for competition entry online. She uses two-thirds of her saved pocket money, and pays an additional $5 late registration fee. She pays $37 in total. How much pocket money did she have saved?
Define $x$, write the equation, then solve.
Jumping practice. Copper can jump two-thirds of his maximum height when warming up. Mia sets the warm-up pole 10 cm higher than his warm-up height. He clears it at 90 cm. What is Copper's full maximum jump height?
Group entry. Three riders split the group entry fee equally. Each person also pays $8 for parking. Each person ends up paying $23 in total. What was the full group entry fee?
Copper's summer weight. In September, Copper weighed five-quarters of his May weight, minus 10 kg (he lost some water weight on the journey home). In September he weighed 490 kg. What did he weigh in May?
Tack shop sale. A bridle is on sale at two-fifths of its original price, minus a further $12 loyalty discount. Mia pays $28. What was the original price?
Make your own. Write a real-world scenario — anything involving a fraction of something — that leads to an equation of the form $\dfrac{ax}{b} + c = d$. Write and solve your equation.
What's next → You can now handle equations where $x$ is multiplied by a fraction. The next challenge is equations where $x$ has a negative coefficient — like $-3x + 9 = 21$. These show up whenever something is decreasing: a horse losing weight, points being deducted, a balance going down. The rule is the same, but we need to keep careful track of signs.
Show answers
$\dfrac{x}{3} + 4 = 10$ → $x = 18$ kg per bale
$\dfrac{3}{4}x = 18$ → $x = \$24$ prize money
$\dfrac{3}{5}x = 120$ → $x = 200 €$ listed rent
$\dfrac{3}{4}x + 6 = 21$ → $x = 20 €$ hourly rate
$\dfrac{2}{3}x + 5 = 37$ → $x = \$48$ saved
$\dfrac{2}{3}x + 10 = 90$ → $x = 120$ cm maximum height