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Difficulty:
55%
(hard)
Question Stats:
64% (03:09) correct
36%
(03:19)
wrong
based on 67
sessions
History
Date
Time
Result
Not Attempted Yet
Alpha Optics and Beta Vision buy high-precision optical lenses from the same manufacturer. The price of a lens is inversely proportional to the cube of its diameter. Alpha Optics buys 30 lenses, each with a diameter of 6 mm. Beta Vision buys 90 lenses in total: 30 lenses with a diameter of 3 mm, 30 lenses with a diameter of 2 mm, and 30 lenses with a diameter of 1 mm. How many times as much does Beta Vision pay as Alpha Optics?
A. 80
B. 120
C. 180
D. 251
E. 300
A. 80
B. 120
C. 180
D. 251
E. 300
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19 Feb 2026, 06:08
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The key concept here is Inverse Proportionality — when price ∝ 1/d3, doubling the diameter makes a lens 8× cheaper, not 2×. That asymmetry is exactly what this problem is testing.
Step 1: Set up the price formula
Since price is inversely proportional to the cube of diameter:
Price = k/d3, where k is a constant.
Step 2: Calculate Alpha Optics' total spend
30 lenses, each with diameter 6 mm:
Total = 30 × k/63 = 30k/216
Step 3: Calculate Beta Vision's total spend
- 30 lenses at d = 3 mm: 30 × k/27 = 30k/27
- 30 lenses at d = 2 mm: 30 × k/8 = 30k/8
- 30 lenses at d = 1 mm: 30 × k/1 = 30k
Add them up using common denominator 216:
30k/27 = 240k/216
30k/8 = 810k/216
30k = 6480k/216
Beta Total = (240 + 810 + 6480)k/216 = 7530k/216
Step 4: Find the ratio
Beta/Alpha = (7530k/216) ÷ (30k/216) = 7530/30 = 251
Answer: D. 251
The common trap: Most students compute 30 × (1/33 + 1/23 + 1/13) but then divide by 1/63 without noticing the 30s cancel cleanly — leading to arithmetic errors mid-way. Just keep everything over 216 and it falls apart nicely.
Takeaway: When a problem says "inversely proportional to the cube," the small-diameter lenses are disproportionately expensive — always cube first, then compute totals.
Step 1: Set up the price formula
Since price is inversely proportional to the cube of diameter:
Price = k/d3, where k is a constant.
Step 2: Calculate Alpha Optics' total spend
30 lenses, each with diameter 6 mm:
Total = 30 × k/63 = 30k/216
Step 3: Calculate Beta Vision's total spend
- 30 lenses at d = 3 mm: 30 × k/27 = 30k/27
- 30 lenses at d = 2 mm: 30 × k/8 = 30k/8
- 30 lenses at d = 1 mm: 30 × k/1 = 30k
Add them up using common denominator 216:
30k/27 = 240k/216
30k/8 = 810k/216
30k = 6480k/216
Beta Total = (240 + 810 + 6480)k/216 = 7530k/216
Step 4: Find the ratio
Beta/Alpha = (7530k/216) ÷ (30k/216) = 7530/30 = 251
Answer: D. 251
The common trap: Most students compute 30 × (1/33 + 1/23 + 1/13) but then divide by 1/63 without noticing the 30s cancel cleanly — leading to arithmetic errors mid-way. Just keep everything over 216 and it falls apart nicely.
Takeaway: When a problem says "inversely proportional to the cube," the small-diameter lenses are disproportionately expensive — always cube first, then compute totals.
26 Feb 2026, 04:28
Kudos
Bookmarks
Price paid by AO - 30/(6^6^6) - 5/36
Price paid by BV - 30(3^3^3) + 30(2^2^2) + 30(1^1^1) - 1255/36
(1255/36) / (5/36) = 251
Price paid by BV - 30(3^3^3) + 30(2^2^2) + 30(1^1^1) - 1255/36
(1255/36) / (5/36) = 251
