We follow six countries, $A$ to $F$, across the years $2016$, $2020$ and $2024$. The plot supplies two percentages for each country: $X$ = percentage change in SI from $2016$ to $2020$ $Y$ = percentage change in SI from $2020$ to $2024$ Reading these off the plot: | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | | | | | B | $-25$ | $-40$ | | | | | C | $+5$ | $0$ | | | | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | | | | | F | $0$ | $-25$ | | | | Since $\text{SI}_{2020} = \text{SI}_{2016}\,(1 + X/100)$ and $\text{SI}_{2024} = \text{SI}_{2020}\,(1 + Y/100)$, fixing any one year for a country fixes all three.

The anchors we will use: 2016 ranks give $B>C>E>A>D>F$ (since $B,C,E,A$ are ranks $1,2,3,4$ and $F$ is lowest, so $D$ is rank $5$). $F$ is the lowest SI in every year. $E$ is the only country at $90$ in $2024$. The range of SI is $60$ in $2016$ and again in $2024$. —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | | | | | B | $-25$ | $-40$ | | | | | C | $+5$ | $0$ | | | | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | | | | $E$ in $2024$ is $90$. E's overall multiplier is $(1+0.20)(1+0.25) = 1.20 \times 1.25 = 1.5$. So $\text{SI}_{2016} = 90 / 1.5 = 60$ $\text{SI}_{2020} = 90 / 1.25 = 72$. The value $60$ also matches $E$ being rank $3$ in $2016$. —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | | | | | B | $-25$ | $-40$ | | | | | C | $+5$ | $0$ | | | | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | $40$ | $40$ | $30$ | In $2024$ the six SIs span a range of $60$. $E$ sits at $90$, and no country exceeds it (the completed table confirms this), so $90$ is the maximum. $F$ is always the lowest, so $F_{2024} = 90 - 60 = 30$.

Working back through F's percentages: $F_{2020} = 30 / (1 - 0.25) = 30/0.75 = 40$ $F_{2016} = 40 / (1+0) = 40$. —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | | | | | B | $-25$ | $-40$ | $100$ | $75$ | $45$ | | C | $+5$ | $0$ | | | | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | $40$ | $40$ | $30$ | In $2016$ the range is $60$. $B$ is rank $1$ (the maximum) and $F$ is the minimum at $40$, so $B_{2016} = 40 + 60 = 100$. Applying B's percentages: $B_{2020} = 100 \times (1 - 0.25) = 75$ $B_{2024} = 75 \times (1 - 0.40) = 45$. —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | | | | | B | $-25$ | $-40$ | $100$ | $75$ | $45$ | | C | $+5$ | $0$ | $80$ | $84$ | $84$ | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | $40$ | $40$ | $30$ | $C$ is rank $2$ in $2016$, between $B=100$ and $E=60$, so $60 < C_{2016} < 100$. C's first step is $+5\%$: $C_{2020} = C_{2016}\times 1.05 = C_{2016}\times \tfrac{21}{20}$. For this to be a whole number, $C_{2016}$ must be a multiple of $20$. The only multiple of $20$ strictly between $60$ and $100$ is $80$. So $C_{2016} = 80,\quad C_{2020} = 84,\quad C_{2024} = 84$ (since $Y = 0$). —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | $50$ | $60$ | $72$ | | B | $-25$ | $-40$ | $100$ | $75$ | $45$ | | C | $+5$ | $0$ | $80$ | $84$ | $84$ | | D | $+20$ | $0$ | | | | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | $40$ | $40$ | $30$ | $A$ is rank $4$ in $2016$, below $E=60$ and above $F=40$, so $40 < A_{2016} < 60$. A's overall multiplier is $1.20\times1.20 = 1.44 = \tfrac{36}{25}$. For $A_{2024}$ to be a whole number, $A_{2016}$ must be a multiple of $25$. The only one in $(40, 60)$ is $50$. So $A_{2016} = 50,\quad A_{2020} = 60,\quad A_{2024} = 72$. —-- | Country | $X\%$ | $Y\%$ | SI 2016 | SI 2020 | SI 2024 | |---|---|---|---|---|---| | A | $+20$ | $+20$ | $50$ | $60$ | $72$ | | B | $-25$ | $-40$ | $100$ | $75$ | $45$ | | C | $+5$ | $0$ | $80$ | $84$ | $84$ | | D | $+20$ | $0$ | $45$ | $54$ | $54$ | | E | $+20$ | $+25$ | $60$ | $72$ | $90$ | | F | $0$ | $-25$ | $40$ | $40$ | $30$ | $D$ is rank $5$ in $2016$, between $A=50$ and $F=40$, so $40 < D_{2016} < 50$. D's first step is $+20\% = \tfrac{6}{5}$, so $D_{2016}$ must be a multiple of $5$. The only one in $(40,50)$ is $45$. So $D_{2016} = 45,\quad D_{2020} = 54,\quad D_{2024} = 54$.

Checking the facts: $2016$: $100>80>60>50>45>40$, ranks $B,C,E,A = 1,2,3,4$, range $= 100-40 = 60$. $2024$: maximum $90$ ($E$, the unique $90$), minimum $30$ ($F$), range $= 90-30 = 60$. $F$ is the lowest in all three years. Every entry is now fixed, so the table is fully determined.

CAT (DILR) PYQs

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