CBSE Class 7 Maths Chapter 8 Rational Numbers NCERT Solutions

The correct option is (d) $q\; \backslash neq\; 0$.

By definition, a rational number is any number that can be written as a fraction $\backslash frac\{p\}\{q\}$, where 'p' (the numerator) and 'q' (the denominator) are both integers. A fundamental condition for a number to be rational is that the denominator 'q' must not be equal to zero, as division by zero is undefined.

The correct option is (c) $\backslash frac\{-3\}\{-4\}$.

A rational number is considered positive if both its numerator and denominator have the same sign. This means either both 'p' and 'q' are positive, or both 'p' and 'q' are negative. In option (c), both -3 and -4 are negative, so their ratio $\backslash frac\{-3\}\{-4\}$ results in a positive value.

The correct option is (d) $\backslash frac\{3\}\{-7\}$.

A rational number is considered negative if its numerator and denominator have opposite signs. This means one is positive and the other is negative. In option (d), the numerator 3 is positive and the denominator -7 is negative, making the rational number $\backslash frac\{3\}\{-7\}$ negative.

The correct option is (b) 1.

A rational number is said to be in its standard form (or simplest form) if its denominator is a positive integer and the greatest common divisor (GCD) of the numerator and the denominator is 1. This means they share no common factors other than 1.

The correct option is (a) 1.

The reciprocal of a number is 1 divided by that number. For a rational number $\backslash frac\{p\}\{q\}$, its reciprocal is $\backslash frac\{q\}\{p\}$. We are looking for a number that is equal to its reciprocal. Let the number be 'x'. Then $x\; =\; \backslash frac\{1\}\{x\}$. Multiplying both sides by x gives $x^2\; =\; 1$. The integer solutions for this equation are x = 1 and x = -1. However, among the given options, only 1 is present.

The correct option is (b) 2.

The reciprocal of a fraction is obtained by swapping its numerator and denominator. For the rational number $\backslash frac\{1\}\{2\}$, the numerator is 1 and the denominator is 2. Swapping them gives $\backslash frac\{2\}\{1\}$, which is equal to 2.

The correct option is (c) $\backslash frac\{-4\}\{5\}$.

To express a rational number in its standard form, we need to divide both the numerator and the denominator by their greatest common divisor (GCD) and ensure the denominator is positive. The GCD of 48 and 60 is 12. Dividing both the numerator and the denominator by 12:

$$\backslash frac\{-48\}\{60\}\; =\; \backslash frac\{-48\; \backslash div\; 12\}\{60\; \backslash div\; 12\}\; =\; \backslash frac\{-4\}\{5\}$$

Since the denominator 5 is positive and the GCD of -4 and 5 is 1, $\backslash frac\{-4\}\{5\}$ is the standard form.

The correct option is (c) 16/20.

Equivalent rational numbers represent the same value. We can check each option:

(a) 5/4 is the reciprocal, not equivalent.

(b) 16/25: The GCD of 16 and 25 is 1. This fraction cannot be simplified to 4/5.

(c) 16/20: We can simplify this fraction by dividing both the numerator and the denominator by their GCD, which is 4. $\backslash frac\{16\; \backslash div\; 4\}\{20\; \backslash div\; 4\}\; =\; \backslash frac\{4\}\{5\}$. Thus, 16/20 is equivalent to 4/5.

(d) 15/25: The GCD of 15 and 25 is 5. $\backslash frac\{15\; \backslash div\; 5\}\{25\; \backslash div\; 5\}\; =\; \backslash frac\{3\}\{5\}$. This is not equivalent to 4/5.