In 2010, a swim club had a total of 35 swimmers, each classified as either advanced or intermediate. From 2010...
GMAT Algebra : (Alg) Questions
In 2010, a swim club had a total of \(\mathrm{35}\) swimmers, each classified as either advanced or intermediate. From 2010 to 2020, the number of advanced swimmers in the club increased by approximately \(\mathrm{53\%}\), and the number of intermediate swimmers in the club increased by approximately \(\mathrm{44\%}\). The total number of swimmers in the club increased by approximately \(\mathrm{49\%}\). Which equation best represents this situation, where \(\mathrm{a}\) represents the number of advanced swimmers in the club in 2010 and \(\mathrm{b}\) represents the number of intermediate swimmers in the club in 2010?
\(1.53\mathrm{a} + 1.49\mathrm{b} = 35(1.44)\)
\(1.49\mathrm{a} + 0.53\mathrm{b} = 35(1.44)\)
\(1.53\mathrm{a} + 1.44\mathrm{b} = 35(1.49)\)
\(1.44\mathrm{a} + 1.53\mathrm{b} = 35(1.49)\)
1. TRANSLATE the problem information
- Given information:
- 2010: 35 total swimmers (a = advanced, b = intermediate)
- Advanced swimmers increased by 53% from 2010 to 2020
- Intermediate swimmers increased by 44% from 2010 to 2020
- Total swimmers increased by 49% from 2010 to 2020
2. TRANSLATE percentage increases to mathematical expressions
- When something increases by a percentage, the new amount equals:
Original × (1 + percentage as decimal)
- So in 2020:
- Advanced swimmers: \(\mathrm{a × (1 + 0.53) = 1.53a}\)
- Intermediate swimmers: \(\mathrm{b × (1 + 0.44) = 1.44b}\)
- Total swimmers: \(\mathrm{35 × (1 + 0.49) = 35(1.49)}\)
3. INFER the relationship between parts and total
- In 2020, the total number of swimmers must equal the sum of advanced and intermediate swimmers
- Therefore: \(\mathrm{1.53a + 1.44b = 35(1.49)}\)
Answer: C. \(\mathrm{1.53a + 1.44b = 35(1.49)}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students mix up which percentage goes with which group of swimmers.
They might think the 53% increase applies to intermediate swimmers and 44% to advanced swimmers, leading them to write \(\mathrm{1.44a + 1.53b}\) on the left side. This may lead them to select Choice D (\(\mathrm{1.44a + 1.53b = 35(1.49)}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students incorrectly place the percentage increases in the equation structure.
They might confuse which side represents 2020 totals versus the individual group totals, or mix up the 44% and 49% values. This could lead them to select Choice A (\(\mathrm{1.53a + 1.49b = 35(1.44)}\)), where they've swapped the total percentage increase with one of the group increases.
The Bottom Line:
This problem tests your ability to carefully track multiple percentage increases and translate them into the correct mathematical structure. The key is methodically converting each English description into its mathematical equivalent while keeping track of which numbers belong to which groups.
\(1.53\mathrm{a} + 1.49\mathrm{b} = 35(1.44)\)
\(1.49\mathrm{a} + 0.53\mathrm{b} = 35(1.44)\)
\(1.53\mathrm{a} + 1.44\mathrm{b} = 35(1.49)\)
\(1.44\mathrm{a} + 1.53\mathrm{b} = 35(1.49)\)