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State True/False.

The ten rational numbers between $− 3 4$ and $56$ are $− 2 3, − 7 12, − 1 2, − 5 12, − 1 3, − 1 4, − 1 6, − 1 12, 0, 112$ and $16$ .

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True

False

answer is A.

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Detailed Solution

Given rational numbers are

− 3 4

and

56

.
Consider

a = − 3 4

and

b = 56

.
Now, take the LCM.
The LCM of the denominators 4 and 6 is 12.
Now, by using LCM as the common denominator,

− 3 4 = − 3 × 3 4 × 3 ⇒ − 9 12

And

56 = 5 × 2 6 × 2 ⇒ 1012

− 9 12

and

1012

have the same denominator i.e., 12.
So we can directly write by the numerators,

− 9, − 8, − 7, − 6, − 5, − 4, − 3, − 2, − 1, 0, 1, 10

Hence,

− 9 12, − 8 12, − 7 12, − 6 12, − 5 12, − 4 12, − 3 12, − 2 12, − 1 12, 012, 112, 212, 1012

.
By canceling out the common terms we get,

− 3 4, − 2 3, − 7 12, − 1 2, − 5 12, − 1 3, − 1 4, − 1 6, − 1 12, 0, 112, 16, 56

.
So, the rational numbers are

− 2 3, − 7 12, − 1 2, − 5 12, − 1 3, − 1 4, − 1 6, − 1 12, 0, 112

.
Therefore, the rational numbers are

− 2 3, − 7 12, − 1 2, − 5 12, − 1 3, − 1 4, − 1 6, − 1 12, 0, 112, 16

.
Thus, the given statement is true.
Hence, option (1) is correct.

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