In demography and actuarial science, there is a relationship between the probability of dying within an interval ($q_x$) and the central mortality rate ($m_x$). The central mortality rate is the rate of mortality during an interval $(x, x+1)$ based on the population at the midpoint of the interval.A common approximation relating these two is: $m_x \approx \frac{q_x}{1 - \frac{1}{2}q_x} = \frac{2q_x}{2-q_x}$This formula assumes that the force of mortality is constant throughout the age interval. If $q_x$ is small, $m_x \approx q_x$. However, for a more accurate relationship, the formula $\frac{q_x}{2-q_x}$ is used. The options provided suggest a slightly different derivation or approximation might be expected, or a specific convention. Let's re-evaluate the options based on common actuarial approximations.A common approximation is indeed $m_x \approx \frac{q_x}{1 - (1-a)q_x}$ where $a$ is the fraction of the interval passed at the beginning of the interval. For $m_x$, we consider the midpoint. Another very common relation is derived by considering the number of person-years lived. If $L_x$ is the number of person-years lived between ages $x$ and $x+1$, and $D_x$ is the number of deaths between ages $x$ and $x+1$, then $m_x = D_x / L_x$. We know $q_x = D_x / P_x$, where $P_x$ is the population at age $x$. And $L_x$ is approximately the average population in the interval, $L_x \approx P_x - D_x/ 2$. So, $m_x = \frac{D_x}{P_x - D_x/2}$. Dividing numerator and denominator by $P_x$: $m_x = \frac{D_x/P_x}{1 - (D_x/P_x)/2} = \frac{q_x}{1 - q_x/2} = \frac{2q_x}{2-q_x}$.Let's check the provided options. A) $\frac{q_x}{1-q_x}$ (This is related to odds) B) $\frac{2q_x}{1+q_x}$ (This doesn't directly relate $m_x$ and $q_x$) C) $ \frac{q_x}{2-q_x}$ (This is missing a factor of 2 in the numerator) D) $\frac{2q_x}{2-q_x}$ (This matches our derivation)Upon re-checking the options and common actuarial formulas, the closest and most standard approximation for $m_x$ in terms of $q_x$ is $\frac{2q_x}{2-q_x}$. However, the question has an option that is $\frac{q_x}{2- q_x}$. It's possible there's a slight variation in notation or the question expects a specific, perhaps less common, approximation.Let's assume option C is intended to be $\frac{q_x}{1 - q_x/2}$. No, option C is $\frac{q_x}{2-q_x}$.Let's look at the relationship from another angle. If $m_x$ is constant over the interval $[x, x+1]$, then $P(t) = P_x e^{-\int_x^t m_s ds}$. $P_{x+1} = P_x e^{- m_x}$. $q_x = 1 - P_{x+1}/P_x = 1 - e^{-m_x}$. So, $e^{-m_x} = 1 - q_x$. $-m_x = \ln(1-q_x)$. $m_x = -\ln(1-q_x)$. Using Taylor series expansion for $\ln(1-q_x) \approx -q_x - q_x^2/2 - ...$ $m_x \approx -(-q_x - q_x^2/2) = q_x + q_x^2/2$.Now let's look at the given options and see which one fits this. If $m_x = \frac{q_x}{2-q_x}$, then $m_x(2- q_x) = q_x \implies 2m_x - m_x q_x = q_x \implies 2m_x = q_x(1+m_x) \implies m_x = \frac{q_x} {1+m_x}$. This is not helpful.Let's reconsider the person-years lived approach. $m_x = \frac{\text{Number of deaths in (x, x+1)}} {\text{Person-years lived in (x, x+1)}}$ $q_x = \frac{\text{Number of deaths in (x, x+1)}}{P_x}$ (Population at exact age x)Let $N_x$ be the number of deaths in the interval $(x, x+1)$. Let $L_x$ be the person-years lived in the interval $(x, x+1)$. $m_x = N_x / L_x$. $q_x = N_x / P_x$.A common approximation is that $L_x \approx P_x - N_x/2$. So, $m_x = \frac{N_x}{P_x - N_x/2}$. Divide numerator and denominator by $P_x$: $m_x = \frac{N_x/P_x}{1 - (N_x/P_x)/2} = \frac{q_x}{1 - q_x/2} = \frac{2q_x}{2-q_x}$.This derivation strongly suggests that option D, $\frac{2q_x}{2- q_x}$, is the correct answer. However, if option C is $\frac{q_x}{2-q_x}$, it's possible it's a simplified approximation or a typo.Let's assume there might be a confusion with notation. If $q_x$ refers to the probability of dying in the interval for a person alive at the beginning of the interval, and $m_x$ refers to the rate for a person alive at the midpoint, then the relationship derived is standard.Given the options, and the standard derivation, $\frac{2q_x}{2-q_x}$ is the most theoretically sound relation. If option C is indeed $\frac{q_x}{2- q_x}$ and is the provided correct answer, it implies a different derivation or a simplification.Let's try to work backwards from option C: If $m_x = \frac{q_x}{2-q_x}$, then $m_x(2-q_x) = q_x \implies 2m_x - m_x q_x = q_x$. This does not seem to lead to a common actuarial relation easily.Let's re-examine the problem statement and options very carefully. $q_x$ is the probability of a person dying in the age interval $(x, x+1)$. This usually implies starting at age $x$. $m_x$ is the central mortality rate.If we assume the problem statement or options might have a typo and that option D is intended to be the correct one based on standard actuarial mathematics.However, let's consider a possibility where the question is based on a specific textbook or convention. Sometimes, simpler approximations are used.Let's assume option C is indeed the intended answer and try to find a justification. If $m_x = \frac{q_x}{2-q_x}$, then $1-m_x = 1 - \frac{q_x} {2-q_x} = \frac{2-q_x-q_x}{2-q_x} = \frac{2-2q_x}{2-q_x}$. This doesn't simplify well.Consider the case where the population at age $x$ is $P_x$, and at $x+1$ is $P_{x+1}$. $q_x = \frac{P_x - P_{x+1}}{P_x}$. Let the person-years lived in the interval be $L_x$. $m_x = \frac{P_x - P_{x+1}}{L_x}$. So, $q_x / m_x = L_x / P_x$. If we approximate $L_x \approx P_x - (P_x- P_{x+1})/2 = (P_x + P_{x+1})/2$. Then $q_x / m_x = \frac{(P_x+P_{x+1})/2}{P_x} = \frac{1} {2} (1 + P_{x+1}/P_x)$. From $q_x = 1 - P_{x+1}/P_x$, we have $P_{x+1}/P_x = 1 - q_x$. So, $q_x / m_x = \frac{1}{2} (1 + 1 - q_x) = \frac{1}{2} (2 - q_x)$. Therefore, $m_x = \frac{q_x}{\frac{1}{2}(2-q_x)} = \frac{2q_x}{2- q_x}$.This derivation consistently leads to option D.Let's check if there's any context where option C is correct. Sometimes, for very small $q_x$, approximations are used. If $m_x = \frac{q_x}{2- q_x}$, let's substitute $m_x = q_x + q_x^2/2$. $q_x + q_x^2/2 = \frac{q_x}{2-q_x}$. $1 + q_x/2 = \frac{1}{2-q_x}$. $(1+q_x/2)(2-q_x) = 1$. $2 - q_x + q_x - q_x^2/2 = 1$. $2 - q_x^2/2 = 1$. $1 = q_x^2/2 \implies q_x^2 = 2$. This is impossible for a probability.It seems highly likely that the intended correct answer is D, based on standard actuarial relationships. However, if the provided solution states C, then there might be a specific convention or a typo in the question or options that is not apparent.Given the prompt asks for the correct answer and explanation, and based on standard demographic/actuarial theory, option D is the most correct. However, if forced to choose from the options as presented, and assuming there might be a subtle point, let's reconsider.Let's assume there is a typo in the option, and it should be related to a standard formula.Could it be that $m_x$ is defined slightly differently? "central mortality rate" typically refers to the average rate over an interval based on the average population in that interval.Let's search for direct relationships between $q_x$ and $m_x$ in actuarial texts. Standard textbooks confirm that $m_x = \frac{q_x}{1 - q_x/2} = \frac{2q_x}{2-q_x}$.If the provided correct option is C ($\frac{q_x}{2-q_x}$), it's an unusual relationship. Let's check if there's any scenario where this might arise. It looks like option C is missing a factor of 2 in the numerator.Given the constraint to provide an answer, and the strong evidence for $ \frac{2q_x}{2-q_x}$, I will proceed with that derivation. If the question or options are flawed, the explanation will highlight it.Let's assume, for the sake of providing an answer that matches one of the options, that there is a misunderstanding of the formula or a specific convention is followed.Let's reconsider the relation $m_x \approx q_x$. This is true for very small $q_x$. Let's look at option C: $m_x = \frac{q_x}{2-q_x}$. If $q_x$ is small, $2- q_x \approx 2$. So $m_x \approx q_x/2$. This is the opposite of what we expect for small $q_x$.Let's assume there is a typo in the question, and the options are intended to test knowledge of common actuarial formulas. The most common formula relating $m_x$ and $q_x$ is $m_x = \frac{2q_x}{2-q_x}$.If forced to select from the options as written, and assuming there isn't a typo in the options themselves, there's a strong discrepancy with standard actuarial formulas. However, if this question appeared in an exam and one of these options was marked correct, it would be problematic.Let's stick to the most robust derivation. $m_x = \frac{\text{Number of deaths in } (x, x+1)} {\text{Person-years lived in } (x, x+1)}$ $q_x = \frac{\text{Number of deaths in } (x, x+1)}{P_x}$ (Population at exact age $x$) Let $N_x$ be deaths in $(x, x+1)$. Let $L_x$ be person-years lived. $m_x = N_x / L_x$. $q_x = N_x / P_x$. Assume $L_x = \int_x^{x+1} P(t) dt$. If $P(t)$ is linear, $L_x = (P_x + P_{x+1})/2$. $m_x = \frac{N_x} {(P_x + P_{x+1})/2}$. $q_x = N_x / P_x$. $m_x = \frac{2N_x}{P_x + P_{x+1}}$. $m_x = \frac{2(N_x/P_x)}{1 + P_{x+1}/P_x}$. We know $P_{x+1} = P_x - N_x$. So $P_{x+1}/P_x = 1 - N_x/P_x = 1 - q_x$. $m_x = \frac{2q_x}{1 + (1- q_x)} = \frac{2q_x}{2-q_x}$.This is consistently derived. Therefore, option D is the correct one. The provided options might be flawed if C is considered correct.Let's proceed assuming the standard formula is expected.