Adam invested $P in a savings account X with interest compounded quarterly at the rate of $1.5\%$ per annum. Ben invested $P in a savings account Y, paying simple interest at the rate of $x\%$ per year. At the end of 5 years, Ben made $10\%$ more than Adam. Mary would like to invest $P for 20 years. She believes that savings account Y is better as Ben made more money than Adam. Justify with clear mathematical working, whether Mary is correct.

1. Find Adam's amount and interest after 5 years in account X.

Quarterly rate $=\dfrac{1.5\%}{4}=0.375\%=0.00375$.

Number of quarters in 5 years $=5\times 4=20$.

Adam's amount after 5 years
$=P(1.00375)^{20}.$

Adam's interest after 5 years
$=P(1.00375)^{20}-P=P\left((1.00375)^{20}-1\right).$

2. Use Ben's 5-year simple interest to find $x$.

Ben's simple interest after 5 years is
$P\times \frac{x}{100}\times 5=\frac{5xP}{100}=0.05xP.$

Ben made $10\%$ more interest than Adam, so
$0.05xP=1.10P\left((1.00375)^{20}-1\right).$

Cancel $P$:
$0.05x=1.10\left((1.00375)^{20}-1\right).$

$x=22\left((1.00375)^{20}-1\right).$

Using a calculator,
$(1.00375)^{20}\approx 1.07774,$
so
$x\approx 22(0.07774)=1.71.$

3. Find the value of Mary's investment in account X after 20 years.

For 20 years, the number of quarters is
$20\times 4=80.$

Amount in account X after 20 years:
$P(1.00375)^{80}\approx 1.349P.$

4. Find the value of Mary's investment in account Y after 20 years and compare.

Using $x\approx 1.71$, amount in account Y after 20 years is
$P\left(1+\frac{20x}{100}\right).$

$=P\left(1+\frac{20(1.71)}{100}\right)$
$\approx 1.342P.$

Since
$1.349P>1.342P,$
account X gives more money after 20 years.