Directions: Based on the statements answer the question that follows: Is the…

Concept: When two quantities grow at their own compounded annual rates starting from a known ratio, only that starting ratio and the two growth rates decide whether the ratio changes over time — the actual absolute starting values never matter. So if GDP(X)/GDP(Y) starts at ratio k and the two grow at rates rx and ry for n years, the final ratio is k × (1 + rx/100)n / (1 + ry/100)n.

Application:

1. From statement ii, 5 years ago GDP(X) was 1.2 times GDP(Y). Any base values in this ratio work, so take GDP(Y) = 10000; then GDP(X) = 1.2 × 10000 = 12000.

2. From statement i, apply the compound-growth formula P(1 + r/100)T: GDP(Y) after 5 years = 10000 × (1 + 6/100)5 = 10000 × 1.3382 = 13382.25.

3. GDP(X) after 5 years = 12000 × (1 + 5/100)5 = 12000 × 1.2763 = 15315.37.

4. Compare the two compounded values: 15315.37 > 13382.25, so GDP(X) stays higher than GDP(Y) after 5 years.

Cross-check: Confirm this using the ratio form directly, without picking any base values: GDP(X)/GDP(Y) after 5 years = 1.2 × (1.05)5 / (1.06)5 ≈ 1.2 × 0.954 ≈ 1.144, which is greater than 1 — the same conclusion regardless of which base values were used, since only the ratio and the two rates decide it.

Answer: Because statement i (the two growth rates) and statement ii (the starting ratio) together fix the final ratio to a single definite value above 1, the two statements combined give one certain answer — GDP(X) stays higher than GDP(Y) even after 5 years.