1. Most boys grow a lot between the ages of 9 and 14.

2. Most boys grow a lot between the age of 9 and the age of 14.

2-1. Most boys grow a lot between at the age of 9 and at the age of 14.

2-2. Most boys grow a lot between at the age of 9 and the age of 14.

3. Most boys grow a lot between at 9 and 14.

4. Most boys grow a lot between at 9 and at 14.

(Are they all the same?)

5. He grew 7 cm last year.

6. He grew 7 centimeters last year.

7. He grew by 7 cm last year.

(Are the three sentences grammatical?)

#2.1, #2.2, and #3, #4 = for both of these you can not use both "between" and "at" and because of 2 years, between is best.

The last set is OK. (5, 6, 7)

Sra

1. Most boys grow a lot between the ages of 9 and 14.

To determine if this statement is correct, you would need to observe the growth patterns of boys within the specified age range. This can be done by conducting research, analyzing data, or referencing studies or records that track growth rates of boys.

2. Most boys grow a lot between the age of 9 and the age of 14.
This statement is similar to the first one, but it emphasizes that growth occurs specifically between the ages of 9 and 14. The same methods as mentioned earlier can be used to verify this statement.

2-1. Most boys grow a lot between at the age of 9 and at the age of 14.
This sentence includes the preposition "at" before both age values, but it is not necessary and can be omitted. The statement is still clear without it.

2-2. Most boys grow a lot between at the age of 9 and the age of 14.
This sentence is grammatically incorrect because it includes "at" before the second age value. The correct phrasing would be "between the age of 9 and the age of 14" or "between 9 and 14".

3. Most boys grow a lot between at 9 and 14.
This sentence is also grammatically incorrect as it is missing the prepositions "the" before both age values. The correct phrasing would be "between 9 and 14".

4. Most boys grow a lot between at 9 and at 14.
Similar to sentence 3, this sentence is grammatically incorrect because it includes "at" before both age values. The correct phrasing would be "between 9 and 14".

(Are they all the same?)
No, sentences 2-1, 2-2, 3, and 4 have grammatical errors while sentence 1 is correct. The statement itself is the same, but the incorrect grammar in the other sentences makes them different from the first one.

5. He grew 7 cm last year.
This sentence is grammatically correct. It states that someone grew 7 centimeters within the duration of the past year.

6. He grew 7 centimeters last year.
This sentence is also grammatically correct. It uses the same information as sentence 5 but expresses the measurement unit, centimeters, explicitly.

7. He grew by 7 cm last year.
This sentence is grammatically correct as well. It indicates that someone's growth increased by a specific amount, in this case, 7 centimeters, during the previous year.

(Are the three sentences grammatical?)
Yes, all three sentences are grammatically correct. They differ in structure and precise wording, but each conveys the same basic meaning that someone experienced 7 centimeters of growth in the past year.