The Sustainability Index (SI) of a country at a point in time is an integer between 1 and 100.

This question is related to SI of six countries - A, B, C, D, E, and F - at three different points in time - 2016, 2020, and 2024.

The plot represents the exact changes in their SI, with X-coordinate representing % increase in 2020 from 2016, i.e., (SI in 2020 minus SI in 2016) / (SI in 2016), and Y-coordinate representing % increase in 2024 from 2020. At any point in time, the country with highest SI is ranked 1, while the country with the lowest SI is ranked 6.

The following additional facts are known.

1. In 2016, B, C, E, and A had ranks 1, 2, 3, and 4 respectively.

2. F had lower SI than any other country in 2016, 2020, and 2024.

3. In 2024, E was the only country with SI of 90.

4. The range of SI of the six countries was 60 in 2016 as well as in 2024.

What was the SI of C in 2024?

The correct answer is 84.

To solve this step-by-step, we need to extract information from the scatter plot and combine it with the given constraints. Let's define the Sustainability Index for any given country in a specific year as ${\mathrm{SI}}_{\mathrm{Year}}$.

The scatter plot shows two axes:

1. The X-axis represents the percentage increase in 2020 from 2016. Mathematically, this means:

$X=\frac{{\mathrm{SI}}_{2020}-{\mathrm{SI}}_{2016}}{{\mathrm{SI}}_{2016}}$

Which simplifies to:

${\mathrm{SI}}_{2020}={\mathrm{SI}}_{2016}\times (1+X)$

2. The Y-axis represents the percentage increase in 2024 from 2020. Similarly, this translates to:

${\mathrm{SI}}_{2024}={\mathrm{SI}}_{2020}\times (1+Y)$

Our first goal is to find absolute values we can anchor to. Let's look at Country E. From the scatter plot image, Point E is located at an X-coordinate of 25% (or 0.25) and a Y-coordinate of 20% (or 0.20). Let's calculate its multiplier from 2016 to 2024:

${\mathrm{SI}}_{2024}\left(E\right)={\mathrm{SI}}_{2016}\left(E\right)\times (1+0.25)\times (1+0.20)$

${\mathrm{SI}}_{2024}\left(E\right)={\mathrm{SI}}_{2016}\left(E\right)\times 1.25\times 1.2$

${\mathrm{SI}}_{2024}\left(E\right)={\mathrm{SI}}_{2016}\left(E\right)\times 1.5$

Fact 3 states that in 2024, E was the only country with an SI of 90. Substituting this into our equation gives us E's initial SI in 2016:

$90={\mathrm{SI}}_{2016}\left(E\right)\times 1.5$

${\mathrm{SI}}_{2016}\left(E\right)=60$

Now, according to Fact 1, the ranks in 2016 were: B (Rank 1), C (Rank 2), E (Rank 3). Because a lower rank corresponds to a higher SI, their indices must follow this inequality:

${\mathrm{SI}}_{2016}\left(B\right)>{\mathrm{SI}}_{2016}\left(C\right)>{\mathrm{SI}}_{2016}\left(E\right)$

Since we found that ${\mathrm{SI}}_{2016}\left(E\right)=60$, it means ${\mathrm{SI}}_{2016}\left(C\right)$ must be strictly greater than 60.

Next, we turn our attention to Country C to find its SI in 2024. Looking at the scatter plot, Point C is located at an X-coordinate of -20% (or -0.20) and a Y-coordinate of 40% (or +0.40). Let's write the equation mapping its 2016 SI to its 2024 SI:

${\mathrm{SI}}_{2024}\left(C\right)={\mathrm{SI}}_{2016}\left(C\right)\times (1-0.20)\times (1+0.40)$

${\mathrm{SI}}_{2024}\left(C\right)={\mathrm{SI}}_{2016}\left(C\right)\times 0.8\times 1.4$

${\mathrm{SI}}_{2024}\left(C\right)={\mathrm{SI}}_{2016}\left(C\right)\times 1.12$

To analyze this mathematically, let's convert the decimal multiplier 1.12 into a simple fraction:

${\mathrm{SI}}_{2024}\left(C\right)={\mathrm{SI}}_{2016}\left(C\right)\times \frac{112}{100}$

${\mathrm{SI}}_{2024}\left(C\right)={\mathrm{SI}}_{2016}\left(C\right)\times \frac{28}{25}$

The problem states that the Sustainability Index is always an integer between 1 and 100. For ${\mathrm{SI}}_{2024}\left(C\right)$ to be an integer, ${\mathrm{SI}}_{2016}\left(C\right)$ must be perfectly divisible by 25 (it has to be a multiple of 25).

Earlier, we established that ${\mathrm{SI}}_{2016}\left(C\right)>60$. The only multiples of 25 that are greater than 60 and within the maximum limit of 100 are 75 and 100. However, remember the ranking constraint: Country B scored higher than Country C. If ${\mathrm{SI}}_{2016}\left(C\right)$ was 100, then Country B would need an SI greater than 100, which violates the rule that SI cannot exceed 100. Therefore, the only valid number for C in 2016 is 75.

Finally, substitute 75 back into our equation to determine Country C's SI in 2024:

${\mathrm{SI}}_{2024}\left(C\right)=75\times 1.12$

${\mathrm{SI}}_{2024}\left(C\right)=84$

This confirms that the SI of C in 2024 was 84.