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D
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Difficulty:
55%
(hard)
Question Stats:
61% (02:32) correct
39%
(03:03)
wrong
based on 38
sessions
History
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Not Attempted Yet
A health watchdog in Turlandia calculates an indicator I for each city in the country. The indicator is proportional to the cube root of the number of inhabitants of the city and inversely proportional to the square root of the number of health districts in the city. Albor has 20 health districts and 125,000 inhabitants. The capital city of Brondon has 8,000,000 inhabitants.
If the value of the indicator I is the same for both cities, how many health districts are in Brondon?
(A) 20
(B) 80
(C) 160
(D) 320
(E) 1,280
If the value of the indicator I is the same for both cities, how many health districts are in Brondon?
(A) 20
(B) 80
(C) 160
(D) 320
(E) 1,280
09 Mar 2026, 03:17
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kevincan
Indicator I is proportional to the cube root of population, and inversely proportional to the square root of Health districts.
I (Albor) and I ( Brondon) has the same value.
Dividing both, we need to get 1.
I(Albor) / I(Brondon)
= x / y
x = Cube root of (Albor population) / square root of (Albor heath districts).
y = = Cube root of (Brondon population) / square root of (Brandon heath districts).
We need to find: Number of Brandon heath districts ?
Cube root of Albor population = Cube root ( 125*10^3) = 50
Square root of Albor health districts = Square root (20) = 2* square root (5).
x = 50/ ( 2* square root (5) ).
Cube root of Brondon Population = cube root of (8*10^6) = 2*10^2
Square root of Brondon health district = a
x/y = [50 / 2*square root (5)] * {a/ (2*10^2)}
= 50*a / 2* square root (5) * 2 * 10^2
= a / 2* square root (5) * 2 * 2
= a / (8*square root (5))
Therefore, for ( x/y) =1, a should be equal to 8* square root (5)
a = 8 * Square root (5)
= 64*5
= 320
Option D
09 Mar 2026, 04:52
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Answer: (D) 320
Key Concept: Proportionality — translating a word relationship into a formula, then applying it as a ratio.
The first instinct many students have here is to set up a table or guess-and-check with numbers. The trap is spending time computing exact values of I for Albor, then trying to match Brondon. You don't need I's actual value at all — you only need the condition that the two I values are equal.
Step 1: Write the proportionality formula.
"I is proportional to the cube root of population and inversely proportional to the square root of health districts" translates to:
I = k × ∛(population) / √(districts)
where k is a constant that is the same for every city in Turlandia.
Step 2: Set up the equal-indicator condition.
Since I is the same for Albor and Brondon:
k × ∛(125,000) / √(20) = k × ∛(8,000,000) / √(d)
The k cancels immediately:
∛(125,000) / √(20) = ∛(8,000,000) / √(d)
Step 3: Compute the cube roots.
∛(125,000) = ∛(125 × 1,000) = 5 × 10 = 50
∛(8,000,000) = ∛(8 × 1,000,000) = 2 × 100 = 200
So the equation becomes:
50 / √(20) = 200 / √(d)
Step 4: Solve for √(d), then d.
Cross-multiply:
50 × √(d) = 200 × √(20)
√(d) = 4 × √(20)
d = 16 × 20 = 320
Answer: (D)
Common trap: Students who compute ∛(8,000,000) as 20 (confusing cube root with the square root of 400, since √400 = 20) end up with d = 80, which is answer choice (B). If you are getting 80, check your cube root.
Takeaway: When a GMAT word problem gives you a "proportional to / inversely proportional to" relationship, immediately write I = k × (formula), set up the ratio condition, and let k cancel — you almost never need to find k's actual value.
Key Concept: Proportionality — translating a word relationship into a formula, then applying it as a ratio.
The first instinct many students have here is to set up a table or guess-and-check with numbers. The trap is spending time computing exact values of I for Albor, then trying to match Brondon. You don't need I's actual value at all — you only need the condition that the two I values are equal.
Step 1: Write the proportionality formula.
"I is proportional to the cube root of population and inversely proportional to the square root of health districts" translates to:
I = k × ∛(population) / √(districts)
where k is a constant that is the same for every city in Turlandia.
Step 2: Set up the equal-indicator condition.
Since I is the same for Albor and Brondon:
k × ∛(125,000) / √(20) = k × ∛(8,000,000) / √(d)
The k cancels immediately:
∛(125,000) / √(20) = ∛(8,000,000) / √(d)
Step 3: Compute the cube roots.
∛(125,000) = ∛(125 × 1,000) = 5 × 10 = 50
∛(8,000,000) = ∛(8 × 1,000,000) = 2 × 100 = 200
So the equation becomes:
50 / √(20) = 200 / √(d)
Step 4: Solve for √(d), then d.
Cross-multiply:
50 × √(d) = 200 × √(20)
√(d) = 4 × √(20)
d = 16 × 20 = 320
Answer: (D)
Common trap: Students who compute ∛(8,000,000) as 20 (confusing cube root with the square root of 400, since √400 = 20) end up with d = 80, which is answer choice (B). If you are getting 80, check your cube root.
Takeaway: When a GMAT word problem gives you a "proportional to / inversely proportional to" relationship, immediately write I = k × (formula), set up the ratio condition, and let k cancel — you almost never need to find k's actual value.
