Lesson 05 · Competition Day

Story Assignment · Mia

It's the morning of the Regional Young Riders Championship. Mia is up at 5:30 am to groom Copper, load the trailer, and drive 32 km to the showground. She has been saving up for this for three months.

The entry fee was 85 €, but Mia had earned discount vouchers worth 15 € by volunteering at last month's qualifier. The trailer hire costs 2.50 € per kilometre each way. At the showground, Mia also needs to buy a bag of show shavings for Copper's temporary stable — these cost 8 € per bag and she needs 3 bags.

The jumping round has 12 fences. Each fence cleared earns 5 points. Each knockdown deducts 4 points. There is also a time bonus: finishing under the target time earns 3 extra points per second saved. Mia cleared 10 fences, knocked 2, and finished 4 seconds under the target time.

Mia's friend Lea also competed. Lea cleared all 12 fences, but went overtime by 2 seconds — which costs 2 points per second over. Their final scores turned out to be equal.

On the drive home, Mia was already planning her training for next month. She wants to save 220 € for a new saddle pad and entry fees for two upcoming shows. She earns 15 € per week doing stable yard work and has 40 € already saved.

Questions

Work through each question in order. Show all working — write your equation, then solve.

  1. Concrete — direct calculation
    What did Mia actually pay for entry after applying the discount?
  2. Concrete — direct calculation
    What was the total cost of the trip? Include: net entry fee, trailer hire (both ways), and show shavings.
  3. Equation — set it up and solve
    What was Mia's final score? Write an expression using the scoring rules (points for fences cleared, deductions for knockdowns, time bonus).
  4. Equation — x on both sides
    Verify that Lea's score matched Mia's. Let $s$ = Lea's score. Write an expression for Lea's score and check it equals Mia's score from Question 3.
  5. Equation — write and solve
    How many weeks until Mia has saved enough for the saddle and entries? Let $w$ = number of weeks. Write an equation and solve for $w$.
  6. Equation — fractional coefficient
    The saddle pad alone costs two-thirds of the total 220 € target. How much does it cost? Write and solve an equation.
  7. Equation — negative coefficient
    Copper started the competition day weighing 492 kg. After the event Mia noticed he had sweated. The vet said Copper loses approximately the same amount of weight per hour of competition. After 3 hours he weighed 489 kg. How much does he lose per hour?
  8. Open — write your own equation
    Mia is thinking about a second competition next month. She wonders: if she does extra chores to earn more money, how many extra chores at $6 each would she need to afford the 35 € entry fee, given she has 11 € already? Write and solve the equation. Then write a second question of your own based on the story, and solve it.
What's next → Every equation so far has had exactly one answer. But many real situations don't ask "how much exactly?" — they ask "how much is enough?" or "what's the maximum?" Lesson 06 introduces inequalities: the same tools, but instead of $=$ we use $\leq$, $\geq$, $<$, and $>$. Mia will need this when planning her budget — she doesn't need to know exactly what she'll spend, just that it won't exceed what she has.
Show answers
  1. 85 € − 15 € = 70 € entry fee after discount
  2. Trailer: $2 \times 32 \times 2.50 € = 160 €$. Shavings: $3 \times 8 € = 24 €$. Total: $70 € + 160 € + 24 € = $ 254 €
  3. Fences: $10 \times 5 = 50$. Knockdowns: $2 \times (-4) = -8$. Time bonus: $4 \times 3 = 12$. Score: $50 - 8 + 12 = $ 54 points
  4. Lea: $12 \times 5 - 2 \times 2 = 60 - 4 = $ 56 points. Note: this does not equal 54 — there is a discrepancy in the story. A great discussion point: did Lea and Mia score the same? What would Lea's overtime need to be to match 54? ($12 \times 5 - 2t = 54 \Rightarrow t = 3$ seconds over.)
  5. $40 + 15w = 220$  →  $15w = 180$  →  $w = $ 12 weeks
  6. $\frac{2}{3}x = 220$  →  $x = $ 330 € — wait, the saddle pad costs two-thirds of the 220 € target, so $\frac{2}{3} \times 220 \approx $ 146.67 € (no equation needed, direct multiplication)
  7. $492 - 3x = 489$  →  $3x = 3$  →  $x = $ 1 kg per hour
  8. $11 + 6x = 35$  →  $6x = 24$  →  $x = $ 4 extra chores
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