What is the range of a set consisting of the first 100 multiples of 7 that are greater than 70?

1. Translate the problem requirements: We need multiples of 7 that are greater than 70, find the first 100 such multiples, then calculate the range (largest minus smallest value)
2. Identify the sequence boundaries: Find the first multiple of 7 greater than 70 and the 100th multiple in this sequence
3. Determine the first term: Since \(70 = 7 \times 10\), the first multiple of 7 greater than 70 is \(7 \times 11 = 77\)
4. Calculate the 100th term: The 100th term will be \(7 \times (11 + 99) = 7 \times 110 = 770\)
5. Compute the range: Range = largest value - smallest value = \(770 - 77\)

Execution of Strategic Approach

1. Translate the problem requirements

Let's break down what we're being asked to find. We need multiples of 7 that are greater than 70. Think of multiples of 7 as the numbers you get when you count by sevens: 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84...

Since we want multiples greater than 70, we start from 77 and keep going. We need to find the first 100 such numbers, then calculate the range, which is simply the difference between the largest and smallest values in our set.

Process Skill: TRANSLATE - Converting the word problem into a clear mathematical task

2. Identify the sequence boundaries

We need to find two key numbers: where our sequence starts (the first multiple of 7 greater than 70) and where it ends (the 100th multiple in this sequence).

Since we're looking at multiples of 7, we can think of them as \(7 \times 1\), \(7 \times 2\), \(7 \times 3\), and so on. We need to find which multiplier gives us the first number greater than 70.

3. Determine the first term

Let's figure out what 70 divided by 7 equals: \(70 \div 7 = 10\) exactly.

So \(70 = 7 \times 10\). Since we want multiples greater than 70, the next multiple would be \(7 \times 11\).

Let's calculate: \(7 \times 11 = 77\)

So our first term is 77.

4. Calculate the 100th term

Here's where we need to think carefully. Our first term corresponds to \(7 \times 11\). If this is our 1st term in the sequence, then:

• 2nd term: \(7 \times 12\)
• 3rd term: \(7 \times 13\)
• 4th term: \(7 \times 14\)
• ...
• 100th term: \(7 \times (11 + 99)\)

The pattern is clear: to get from the 1st term to the 100th term, we add 99 to our multiplier.

So the 100th term = \(7 \times (11 + 99) = 7 \times 110 = 770\)

Process Skill: INFER - Recognizing the pattern in the sequence to find the 100th term

5. Compute the range

Now we can calculate the range using our formula:

Range = Largest value - Smallest value
Range = \(770 - 77\)
Range = \(693\)

Final Answer

The range of the set consisting of the first 100 multiples of 7 that are greater than 70 is 693.

Looking at our answer choices, this matches option A. 693.

Common Faltering Points

Errors while devising the approach

Students often include 70 itself in their set, thinking the problem asks for multiples of 7 that are "greater than or equal to 70." This leads them to start their sequence with 70 instead of 77, which would make their first term incorrect and throw off all subsequent calculations.

Some students might misread the problem and think they need to find the range of the first 100 multiples of 7 overall (7, 14, 21, ..., 700), rather than specifically the first 100 multiples that are greater than 70. This fundamental misunderstanding would lead to a completely different set of numbers.

Errors while executing the approach

When finding the 100th term, students often add 100 instead of 99 to the starting multiplier. Since \(77 = 7 \times 11\) is the 1st term, they might incorrectly calculate the 100th term as \(7 \times (11 + 100) = 7 \times 111 = 777\), instead of the correct \(7 \times (11 + 99) = 7 \times 110 = 770\).

Students may make simple computational mistakes when calculating \(7 \times 110 = 770\) or when subtracting \(770 - 77 = 693\). These basic arithmetic errors can lead to selecting incorrect answer choices even when the approach is correct.

Errors while selecting the answer

After correctly calculating that the 100th term is 770, some students might accidentally select answer choice D (777) thinking this represents their final answer, forgetting that the question specifically asks for the range (difference between largest and smallest values), not the largest value itself.