On an automobile assembly line, every third car is green. Every fourth car is a convertible.

Question

How many cars out of the first 100 will be green convertibles?
Which number of car is the first green convertible?
Explain how writing common multiples help to solve this problem

Answer 1

To determine how many cars out of the first 100 will be green convertibles, we need to find the common multiples of the two sequences: every third car (which is green) and every fourth car (which is a convertible).

Step 1: Identify the sequences

1. Green Cars: The green cars are the 3rd, 6th, 9th, … car, which can be expressed mathematically as the multiples of 3:

   • Green cars: 3, 6, 9, 12, 15, ..., up to 99.

2. Convertible Cars: The convertible cars are the 4th, 8th, 12th, … car, which can be expressed mathematically as the multiples of 4:

   • Convertible cars: 4, 8, 12, 16, 20, ..., up to 100.

Step 2: Finding common multiples

To find the green convertibles, we look for the least common multiples (LCMs) of 3 and 4.

• The LCM of 3 and 4 is 12.

This means that every 12th car will be both green and a convertible.

Step 3: Counting common multiples up to 100

Next, we need to count how many multiples of 12 fit within the first 100 cars.

The multiples of 12 are:

• 12, 24, 36, 48, 60, 72, 84, 96.

Step 4: Count these cars

To find out how many multiples of 12 are there up to 100, we can find the largest integer \(n\) such that \(12n \leq 100\).

Calculating:

• \(12 \times 1 = 12\)
• \(12 \times 2 = 24\)
• \(12 \times 3 = 36\)
• \(12 \times 4 = 48\)
• \(12 \times 5 = 60\)
• \(12 \times 6 = 72\)
• \(12 \times 7 = 84\)
• \(12 \times 8 = 96\)

The multiples we found are: \(12, 24, 36, 48, 60, 72, 84, 96\) — totaling 8 green convertibles.

Step 5: Identify the first green convertible

The first green convertible is simply the first multiple of 12, which is:

• First green convertible is car number 12.

Explanation of Common Multiples

Writing common multiples helps to simplify the problem by reducing it to a matter of finding the least common multiple of two criteria (3 and 4). By identifying these common multiples, we can efficiently count how many cars meet both conditions across the total range (in this case, the first 100 cars). Thus, finding and counting the multiples of the least common multiple (12) gives us the exact number of green convertibles.