1. **Stating the problem:** Mary starts saving with 136 coins worth 51 euros. Each week, she receives 8 more coins worth 3 euros (1 coin of 1 euro, 2 coins of 0.5 euro, and 5 coins of 0.2 euro). We analyze the total coins and money over weeks, check if coins equal money, find the number of coins in each box, and calculate USC rates. 2. **Part (a): Completing the table** - Total coins start at 136 and increase by 8 each week: $$\text{Total coins} = 136 + 8n$$ - Total money starts at 51 euros and increases by 3 euros each week: $$\text{Total money} = 51 + 3n$$ - This matches the given table: - After 1 week: 136 + 8(1) = 144 coins, 51 + 3(1) = 54 euros - After 2 weeks: 136 + 8(2) = 152 coins, 51 + 3(2) = 57 euros - After 3 weeks: 160 coins, 60 euros - After 4 weeks: 168 coins, 63 euros 3. **Part (b): When will total coins equal total money?** Set total coins equal to total money: $$136 + 8n = 51 + 3n$$ Subtract 3n from both sides: $$136 + \cancel{8n} - \cancel{3n} = 51 + \cancel{3n} - \cancel{3n}$$ $$136 + 5n = 51$$ Subtract 136 from both sides: $$\cancel{136} + 5n - \cancel{136} = 51 - 136$$ $$5n = -85$$ Divide both sides by 5: $$\frac{5n}{\cancel{5}} = \frac{-85}{\cancel{5}}$$ $$n = -17$$ Since $n$ represents weeks and must be a natural number, $n = -17$ is not possible. Therefore, **there will never be a time when the total number of coins equals the total amount of money in euros.** 4. **Part (c): Number of coins in each money box at start** Let: - $x$ = number of €1 coins (Box 1) - $y$ = number of 50-cent coins (Box 2) - $z$ = number of 20-cent coins (Box 3) We have two equations: - Total coins: $$x + y + z = 136$$ - Total money: $$1 \cdot x + 0.5 \cdot y + 0.2 \cdot z = 51$$ Since each box contains the same amount of money: $$1 \cdot x = 0.5 \cdot y = 0.2 \cdot z$$ Let this common amount be $A$: $$x = A$$ $$0.5y = A \Rightarrow y = \frac{A}{0.5} = 2A$$ $$0.2z = A \Rightarrow z = \frac{A}{0.2} = 5A$$ Substitute into total coins: $$x + y + z = A + 2A + 5A = 8A = 136$$ $$A = \frac{136}{8} = 17$$ Therefore: - Box 1: $x = 17$ coins - Box 2: $y = 2 \times 17 = 34$ coins - Box 3: $z = 5 \times 17 = 85$ coins 5. **Part (d)(i): USC calculation** Mary's mother pays USC at three rates: - 0.5% on first 12012 euros - 2% on next 8472 euros (20484 - 12012) - 4.5% on amount above 20484 euros Calculate USC at 0.5% rate: $$\text{USC}_{0.5\%} = 0.005 \times 12012 = 60.06$$ Calculate USC at 2% rate: $$\text{USC}_{2\%} = 0.02 \times (20484 - 12012) = 0.02 \times 8472 = 169.44$$