Examkin

Question

Difficulty: Very hardRatios and Proportions

In a military motor pool, the ratio of trucks to jeeps is 2:32:3, and the ratio of jeeps to motorcycles is 4:54:5. If there are exactly 140140 vehicles among these three types, how many more motorcycles are there than trucks?

  1. 28Answer
  2. B
    30
  3. C
    42
  4. D
    60

Answer

There are 28 more motorcycles than trucks.
To solve this, the two separate ratios must be unified. By finding the least common multiple for the overlapping term (jeeps, which are represented by 3 in the first ratio and 4 in the second), we scale the ratios to 8:12 and 12:15. This creates a combined ratio of 8:12:15 for trucks to jeeps to motorcycles. Adding these parts gives 35 total proportional units. Dividing the actual total (140 vehicles) by 35 units reveals that each unit equals 4 vehicles. The difference between motorcycles (15 units) and trucks (8 units) is 7 units. Multiplying 7 units by 4 vehicles per unit yields 28.

Step-by-Step Solution

1
Identify the common entity between the two given ratios to create a single, unified ratio.
Jeeps are the common entity. Trucks : Jeeps = 2:32:3, and Jeeps : Motorcycles = 4:54:5.
You cannot compare trucks and motorcycles directly or find the total parts until all three vehicles share the same proportional scale.
2
Find the least common multiple (LCM) for the 'Jeeps' portion of both ratios and scale them.
The LCM of 33 and 44 is 1212. Multiply the first ratio by 44 to get 8:128:12. Multiply the second ratio by 33 to get 12:1512:15. The combined ratio of Trucks : Jeeps : Motorcycles is 8:12:158:12:15.
Scaling ensures the common term is equal in both ratios, allowing them to be combined safely.
3
Calculate the value of a single ratio part based on the total number of vehicles.
Total parts = 8+12+15=358 + 12 + 15 = 35. Divide the total vehicles by total parts: 140÷35=4140 \div 35 = 4 vehicles per part.
This establishes the multiplier needed to convert ratio units into actual vehicle counts.
4
Find the difference in parts between motorcycles and trucks, then multiply by the part value.
Motorcycles (1515 parts) minus trucks (88 parts) equals 77 parts. 7×4=287 \times 4 = 28 vehicles.
Subtracting the parts before multiplying by the scale factor is more efficient than finding the total of each vehicle first, though both methods yield 28.

Key Concept

Combining multiple distinct ratios using a least common multiple to form a unified proportional relationship.
Rate this question

Topics

Ratios and Proportions

More questions from this topic