To identify a rational number between 5 and 5 1/4, we first need to convert 5 1/4 into an improper fraction:
5 1/4 = 5 + 1/4 = 20/4 + 1/4 = 21/4.
Now, we are looking for rational numbers between 5 (which is 20/4) and 5 1/4 (which is 21/4).
Now, let's evaluate the given options:
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√26: This is an irrational number, so it does not meet the criteria.
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5 1/8: Convert it to an improper fraction: 5 1/8 = 5 + 1/8 = 40/8 + 1/8 = 41/8. This is between 20/4 (which is 40/8) and 21/4 (which is also 42/8). So this option is valid.
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51 eighth: This is 51/8, which is greater than 5 1/4 (21/4 or 42/8), so this option is not valid.
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5 2/5: Convert it to an improper fraction: 5 2/5 = 5 + 2/5 = 25/5 + 2/5 = 27/5. Now, compare 27/5 with 20/4 (which is 27/5 = 5.4) and 21/4 (which is 5.25). This option is greater than 5 1/4, so it's not valid.
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5 1/3: Convert it to an improper fraction: 5 1/3 = 5 + 1/3 = 15/3 + 1/3 = 16/3. To compare 16/3 with 20/4 and 21/4, convert to a common denominator (12):
- 20/4 = 60/12,
- 21/4 = 63/12,
- 16/3 = 64/12. Since 64/12 is greater than both, this option is also not valid.
The only rational number between 5 and 5 1/4 from the given options is 5 1/8.