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Between 5 and 10 on a number line, exclusive, a dark gray tick mark is
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13 Jan 2023, 06:11
13 Jan 2023, 06:11
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81% (01:47) correct
18% (03:05) wrong based on 11 sessions
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Between 5 and 10 on a number line, exclusive, a dark gray tick mark is placed at every multiple of 1/3 and a light gray tick mark at every multiple of 1/9. At how many places will a dark gray tick mark and a light gray tick mark overlap?
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14
Re: Between 5 and 10 on a number line, exclusive, a dark gray tick mark is
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11 Feb 2023, 02:16
11 Feb 2023, 02:16
Expert Reply
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Note that each third will always overlap a ninth. Therefore, the question can be rephrased as “how many dark gray tick marks are there?” Since the line segment does not include the endpoints (5< x < 10), there are dark gray tick marks at 4 integers (6, 7, 8, and 9) plus two dark gray tick marks between consecutive integers. This results in an additional 2 in each of 5 spans between adjacent integers, or an additional 10. Thus, there are 14 dark gray tick marks
Note that each third will always overlap a ninth. Therefore, the question can be rephrased as “how many dark gray tick marks are there?” Since the line segment does not include the endpoints (5< x < 10), there are dark gray tick marks at 4 integers (6, 7, 8, and 9) plus two dark gray tick marks between consecutive integers. This results in an additional 2 in each of 5 spans between adjacent integers, or an additional 10. Thus, there are 14 dark gray tick marks
Between 5 and 10 on a number line, exclusive, a dark gray tick mark is
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27 Dec 2023, 12:17
27 Dec 2023, 12:17
A basic note not explained above that I didn't know before this question. What are multiples of fractions? Recall, x is a multiple of number n if you can divide x by n, and are left with an integer. So multiples of 1/3 are any number you can divide by 1/3 and get an integer out.
So, multiples of 1/3 are 1/3, 2/3, 3/3, 4/3, 5/3, ... etc.
Also another way to do the question. Once you get to: "Therefore, the question can be rephrased as 'how many dark gray tick marks are there?'"
You want to find the number of 1/3 multiples between 5 and 10. So use the equation for finding the number of terms. The multiples of 1/3 between 5 and 10 are 16/3, 17/3, 18/3, ... 29/3.*
This is equivalent to finding the number of terms between 16 and 29. Formula for that is [(last term - first term)/interval] +1. These are just consecutive integers, so interval between terms is 1. So it's 29 - 16 +1 = 14.
* To find these, note 5 expressed as 1/3 is 15/3, so we wold start with that if it was inclusive of 5, but since we don't include 5, we start with the first multiple of 3 past 5, which is 16/3. Same deal for finding one multiple less than 10 (aka 30/3).
So, multiples of 1/3 are 1/3, 2/3, 3/3, 4/3, 5/3, ... etc.
Also another way to do the question. Once you get to: "Therefore, the question can be rephrased as 'how many dark gray tick marks are there?'"
You want to find the number of 1/3 multiples between 5 and 10. So use the equation for finding the number of terms. The multiples of 1/3 between 5 and 10 are 16/3, 17/3, 18/3, ... 29/3.*
This is equivalent to finding the number of terms between 16 and 29. Formula for that is [(last term - first term)/interval] +1. These are just consecutive integers, so interval between terms is 1. So it's 29 - 16 +1 = 14.
* To find these, note 5 expressed as 1/3 is 15/3, so we wold start with that if it was inclusive of 5, but since we don't include 5, we start with the first multiple of 3 past 5, which is 16/3. Same deal for finding one multiple less than 10 (aka 30/3).
