The initial rate of hydrolysis of methyl acetate (1 M) by a weak acid (HA, 1 M) is 1/100th of that of a strong acid (HX, 1 M), at 25°C. The Ka of HA is:

The rate of the hydrolysis of methyl acetate is directly proportional to the hydrogen ion concentration.
In case of weak acid, the equilibrium constant is given by the expression
Ka = [H⁺][A⁻]/[HA] = [H⁺]²/[HA]
Hence, [H⁺] = √(Ka[HA])
Hence, the expression for the ratio of the rate of the hydrolysis of methyl acetate in presence of weak acid and strong acid becomes
rweak acid/rstrong acid = √(Ka[HA])/[HA] = 1/100
Substitute [HA] = 1 M
Hence, √Ka × 1/1 = 1/100
Ka = 1 × 10⁻⁸