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Difficulty: HardBasic Ratios

An engine transmission system uses three interconnected gears: Gear A, Gear B, and Gear C. The ratio of the number of teeth on Gear A to the number of teeth on Gear B is 4:34:3. The ratio of the number of teeth on Gear B to the number of teeth on Gear C is 5:25:2. If Gear C has exactly 1818 teeth, what is the total number of teeth on all three gears combined?

Answer: 123 teeth

Answer

123
The correct answer is found by linking the two separate proportions using their common element, Gear B. By converting the A:B ratio (4:34:3) and the B:C ratio (5:25:2) to share a common value for B (1515), we get a unified, continuous ratio for A:B:C of 20:15:620:15:6. The problem states that Gear C has 1818 teeth. Since Gear C represents 66 parts in our combined ratio, each part must equal 33 teeth (18÷6=318 \div 6 = 3). The total number of parts across all three gears is 4141 (20+15+620 + 15 + 6). Multiplying the 4141 total parts by 33 teeth per part results in a total of 123123 teeth for the entire system.

Step-by-Step Solution

1
Align the separate ratios based on the common element.
Gear A : Gear B = 4:34:3 and Gear B : Gear C = 5:25:2.
To compare all three gears simultaneously, Gear B must have the same value in both ratios.
2
Find a common multiple for the ratio numbers representing Gear B (33 and 55).
The least common multiple of 33 and 55 is 1515.
Creating a common denominator allows us to link the two independent proportions.
3
Scale both ratios so Gear B is represented by 1515 parts.
Ratio A:B becomes (4×5):(3×5)=20:15(4 \times 5):(3 \times 5) = 20:15. Ratio B:C becomes (5×3):(2×3)=15:6(5 \times 3):(2 \times 3) = 15:6.
Multiplying both sides of a ratio by the same number maintains the proportion.
4
Combine into a single three-part ratio and determine total parts.
Combined ratio A:B:C is 20:15:620:15:6. Total ratio parts = 20+15+6=4120 + 15 + 6 = 41.
This single ratio precisely maps the relationships between all three gears in the system.
5
Calculate the multiplier using Gear C's known teeth count.
66 parts = 1818 teeth. Therefore, 11 part = 18/6=318 / 6 = 3 teeth.
This establishes the real-world value of a single ratio unit in this specific transmission system.
6
Calculate the total number of teeth on all gears combined.
4141 total parts ×3\times 3 teeth per part = 123123 teeth.
Multiplying the total number of parts by the value of a single part gives the final aggregate quantity.

Key Concept

Combining overlapping ratios by calculating the least common multiple of the shared term, then scaling the combined ratio to determine unknown totals.
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Topics

Basic Ratios

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