Jamar invested $\$ 110$ in an account paying an interest rate of $2 \frac{1}{8} \%$ compounded quarterly. Malika invested $\$ 110$ in an account paying an interest rate of $1 \frac{1}{2} \%$ compounded monthly. After 7 years, how much more money would Jamar have in his account than Malika, to the nearest dollar?

Step 1 ：Given that Jamar and Malika both invested $110 in their respective accounts, we need to calculate the future value of their investments after 7 years using the formula for compound interest: \[A = P(1 + \frac{r}{n})^{nt}\] where A is the amount of money accumulated after n years, including interest, P is the principal amount (the initial amount of money), r is the annual interest rate (in decimal), n is the number of times that interest is compounded per year, and t is the time the money is invested for in years.

Step 2 ：For Jamar, the principal amount P is $110, the annual interest rate r is 0.02125 (or 2.125%), the number of times interest is compounded per year n is 4 (quarterly), and the time t is 7 years. Substituting these values into the formula, we get \[A_{jamar} = 110(1 + \frac{0.02125}{4})^{4*7} = 127.5918685044927\]

Step 3 ：For Malika, the principal amount P is $110, the annual interest rate r is 0.015 (or 1.5%), the number of times interest is compounded per year n is 12 (monthly), and the time t is 7 years. Substituting these values into the formula, we get \[A_{malika} = 110(1 + \frac{0.015}{12})^{12*7} = 122.17015613491466\]

Step 4 ：To find out how much more money Jamar would have in his account than Malika, we subtract Malika's future value from Jamar's: \[difference = A_{jamar} - A_{malika} = 127.5918685044927 - 122.17015613491466 = 5.421712369578032\]

Step 5 ：Rounding this difference to the nearest dollar, we get \[\boxed{5}\]

Step 6 ：Therefore, Jamar would have \(\boxed{5}\) dollars more in his account than Malika, to the nearest dollar.