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What is the range of a set consisting of the first \(\mathrm{100}\) multiples of \(\mathrm{7}\) that are greater than \(\mathrm{70}\)?
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
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.
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.
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:
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
Now we can calculate the range using our formula:
Range = Largest value - Smallest value
Range = \(770 - 77\)
Range = \(693\)
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.
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.
2. Confusing "first 100 multiples greater than 70" with "first 100 multiples of 7"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.
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 \(7 \times (11 + 99) = 7 \times 110 = 770\).
2. Arithmetic errors in basic multiplicationStudents 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.
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.