Sure! Let's go through each question step by step:

To calculate the value of Tom's account at the end of $3$ years with a $3\mathrm{.}7$ percent interest rate compounded semi$-$annually, we can use the formula for compound interest: A $=$ P$(1+$ r$/$n$)^($nt$),$ where A is the final amount, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

Using this formula, the calculation would be: A $=\pounds 8,000(1+0\mathrm{.}037/2)^(2*3)=\pounds 8,000(1\mathrm{.}0185)^6\approx \pounds 8,000(1\mathrm{.}1270)\approx \pounds 9,016.$

So$,$ the value of Tom's account at the end of $3$ years, compounded semi$-$annually, would be approximately $\pounds 9,016.$

To calculate the value of Tom's account at the end of $3$ years with a $3\mathrm{.}7$ percent interest rate compounded continuously, we can use the formula: A $=$ P$*$e$^($rt$),$ where A is the final amount, P is the principal amount, r is the annual interest rate, and t is the number of years.

Using this formula, the calculation would be: A $=\pounds 8,000$ e$^(0\mathrm{.}0373)\approx \pounds 8,000$ e$^(0\mathrm{.}111)\approx \pounds 8,0001\mathrm{.}117\approx \pounds 8,936.$

So$,$ the value of Tom's account at the end of $3$ years, compounded continuously, would be approximately $\pounds 8,936.$

To calculate how much Tom should deposit today if he requires $\pounds 12,000$ at the end of $3$ years with a semi$-$annual compounding interest rate of $3\mathrm{.}7$ percent, we can rearrange the formula from question $1$ to solve for P$.$

The calculation would be: P $=$ A $/(1+$ r$/$n$)^($nt$)=\pounds 12,000/(1+0\mathrm{.}037/2)^(2*3)\approx \pounds 12,000/(1\mathrm{.}0185)^6\approx \pounds 12,000/1\mathrm{.}1270\approx \pounds 10,642.$

So$,$ Tom should deposit approximately $\pounds 10,642$ today to have $\pounds 12,000$ at the end of $3$ years, compounded semi$-$annually.

To find the interest rate at which Tom's account will double in $3$ years with quarterly compounding, we can use the formula: A $=$ P$(1+$ r$/$n$)^($nt$),$ where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

We want the final amount to be double the principal amount, so we can set up the equation: $\pounds 8,000*2=\pounds 8,000(1+$ r$/4)^(4*3).$

Simplifying the equation, we get: $2=(1+$ r$/4)^12.$

To find the interest rate, we can take the $12$th root of both sides: $(1+$ r$/4)=2^(1/12).$

Solving for r$,$ we get: r $=4(2^(1/12)-1).$

Using a calculator, we find that r $\approx 0\mathrm{.}0304.$

So$,$ the interest rate at which Tom's account will double in $3$ years with quarterly compounding is approximately $3\mathrm{.}04$ percent.

To calculate how long it will take for Jonathan's account to grow to $\pounds 22,000$ with a $12$ percent interest rate, we can use the formula: A $=$ P$(1+$ r$)^$t$,$ where A is the final amount, P is the principal amount, r is the interest rate, and t is the number of years.

Using this formula, the calculation would be: $\pounds 22,000=$ P$(1+0\mathrm{.}12)^$t$.$

To solve for t$,$ we can divide both sides by P and take the logarithm: log$(\pounds 22,000/$P$)=$ t $*$ log$(1\mathrm{.}12).$

Dividing both sides by log$(1\mathrm{.}12),$ we get: t $=$ log$(\pounds 22,000/$P$)/$ log$(1\mathrm{.}12).$

Since we don't know the initial principal amount P$,$ we can't determine the exact time it will take for the account to grow to $\pounds 22,000.$ However, once we have the value of P$,$ we can substitute it into the equation to find t$.$

I hope that helps! Let me know if you have any further questions or if there's anything else I can assist you with.