Question: Madeline wants to buy a violin that costs £430. She puts £345 into a savings account that pays compound interest of 1% per month. Using trial and improvement, work out the smallest number of whole months that Madeline will have to wait to have enough money in this account to buy the violin.
1. **State the problem:**
Madeline wants to buy a violin costing $430$. She deposits $345$ into a savings account with compound interest of $1\%$ per month. We need to find the smallest whole number of months $n$ such that the amount in the account is at least $430$.
2. **Formula for compound interest:**
The amount $A$ after $n$ months with monthly compound interest rate $r$ and initial principal $P$ is:
$$A = P(1 + r)^n$$
where $P = 345$, $r = 0.01$ (1\%), and $A \geq 430$.
3. **Set up inequality:**
$$345(1.01)^n \geq 430$$
4. **Trial and improvement:**
Calculate $345(1.01)^n$ for increasing whole values of $n$:
- For $n=7$:
$$345 \times (1.01)^7 = 345 \times 1.071858 = 369.79 < 430$$
- For $n=20$:
$$345 \times (1.01)^{20} = 345 \times 1.22019 = 420.46 < 430$$
- For $n=25$:
$$345 \times (1.01)^{25} = 345 \times 1.28243 = 442.94 \geq 430$$
5. **Check $n=24$ to confirm smallest $n$:**
$$345 \times (1.01)^{24} = 345 \times 1.26973 = 438.06 \geq 430$$
6. **Check $n=23$:**
$$345 \times (1.01)^{23} = 345 \times 1.25716 = 433.97 \geq 430$$
7. **Check $n=22$:**
$$345 \times (1.01)^{22} = 345 \times 1.24471 = 429.68 < 430$$
8. **Conclusion:**
The smallest whole number of months $n$ for which the amount is at least $430$ is $23$ months.
**Final answer:**
$$\boxed{23}$$ months