CBSE Class 7 Mathematics Chapter 9: Rational Numbers NCERT Solutions

To find five rational numbers between -1 and 0, we first express them with a common denominator. Let's choose a denominator that allows us to easily find five numbers in between. We can write -1 and 0 as fractions with a denominator of, for example, 10:

$$-1\; =\; \backslash frac\{-10\}\{10\}$$

$$0\; =\; \backslash frac\{0\}\{10\}$$

Now, we can list five rational numbers between $\backslash frac\{-10\}\{10\}$ and $\backslash frac\{0\}\{10\}$. These numbers will have the same denominator (10) and numerators between -10 and 0.

Simplifying the fractions where possible, we get:

Thus, five rational numbers between -1 and 0 are $\backslash frac\{-9\}\{10\},\; \backslash frac\{-4\}\{5\},\; \backslash frac\{-7\}\{10\},\; \backslash frac\{-3\}\{5\},\; \backslash frac\{-1\}\{2\}$.

To find five rational numbers between -2 and -1, we first express them with a common denominator. Let's use a denominator of 10:

$$-2\; =\; \backslash frac\{-20\}\{10\}$$

$$-1\; =\; \backslash frac\{-10\}\{10\}$$

Now, we can identify five rational numbers between $\backslash frac\{-20\}\{10\}$ and $\backslash frac\{-10\}\{10\}$ by choosing numerators between -20 and -10.

Simplifying the fractions where possible:

Therefore, five rational numbers between -2 and -1 are $\backslash frac\{-19\}\{10\},\; \backslash frac\{-9\}\{5\},\; \backslash frac\{-17\}\{10\},\; \backslash frac\{-8\}\{5\},\; \backslash frac\{-3\}\{2\}$.

To find five rational numbers between $\backslash frac\{-4\}\{5\}$ and $\backslash frac\{-2\}\{3\}$, we first need to find a common denominator. The least common multiple (LCM) of 5 and 3 is 15.

Convert the fractions to equivalent fractions with a denominator of 15:

$$\backslash frac\{-4\}\{5\}\; =\; \backslash frac\{-4\; \backslash times\; 3\}\{5\; \backslash times\; 3\}\; =\; \backslash frac\{-12\}\{15\}$$

$$\backslash frac\{-2\}\{3\}\; =\; \backslash frac\{-2\; \backslash times\; 5\}\{3\; \backslash times\; 5\}\; =\; \backslash frac\{-10\}\{15\}$$

Now we have $\backslash frac\{-12\}\{15\}$ and $\backslash frac\{-10\}\{15\}$. We need to find five rational numbers between these. Since there is only one integer between -12 and -10 (which is -11), we need to increase the denominator to find more numbers. Let's use a denominator of 30 (which is 15 x 2):

$$\backslash frac\{-12\}\{15\}\; =\; \backslash frac\{-12\; \backslash times\; 2\}\{15\; \backslash times\; 2\}\; =\; \backslash frac\{-24\}\{30\}$$

$$\backslash frac\{-10\}\{15\}\; =\; \backslash frac\{-10\; \backslash times\; 2\}\{15\; \backslash times\; 2\}\; =\; \backslash frac\{-20\}\{30\}$$

Now we can find five rational numbers between $\backslash frac\{-24\}\{30\}$ and $\backslash frac\{-20\}\{30\}$:

Let's try a larger denominator, say 45 (LCM of 5 and 3 is 15, and 15*3 = 45):

$$\backslash frac\{-4\}\{5\}\; =\; \backslash frac\{-4\; \backslash times\; 9\}\{5\; \backslash times\; 9\}\; =\; \backslash frac\{-36\}\{45\}$$

$$\backslash frac\{-2\}\{3\}\; =\; \backslash frac\{-2\; \backslash times\; 15\}\{3\; \backslash times\; 15\}\; =\; \backslash frac\{-30\}\{45\}$$

Now we can find five rational numbers between $\backslash frac\{-36\}\{45\}$ and $\backslash frac\{-30\}\{45\}$:

Simplifying these fractions:

$$\backslash frac\{-35\}\{45\}\; =\; \backslash frac\{-7\}\{9\}$$

$$\backslash frac\{-33\}\{45\}\; =\; \backslash frac\{-11\}\{15\}$$

So, five rational numbers between $\backslash frac\{-4\}\{5\}$ and $\backslash frac\{-2\}\{3\}$ are $\backslash frac\{-7\}\{9\},\; \backslash frac\{-34\}\{45\},\; \backslash frac\{-11\}\{15\},\; \backslash frac\{-32\}\{45\},\; \backslash frac\{-31\}\{45\}$.

To find five rational numbers between $\backslash frac\{-1\}\{2\}$ and $\backslash frac\{2\}\{3\}$, we first find a common denominator. The LCM of 2 and 3 is 6.

Convert the fractions to equivalent fractions with a denominator of 6:

$$\backslash frac\{-1\}\{2\}\; =\; \backslash frac\{-1\; \backslash times\; 3\}\{2\; \backslash times\; 3\}\; =\; \backslash frac\{-3\}\{6\}$$

$$\backslash frac\{2\}\{3\}\; =\; \backslash frac\{2\; \backslash times\; 2\}\{3\; \backslash times\; 2\}\; =\; \backslash frac\{4\}\{6\}$$

Now we have $\backslash frac\{-3\}\{6\}$ and $\backslash frac\{4\}\{6\}$. We can list rational numbers between these two fractions. The integers between -3 and 4 are -2, -1, 0, 1, 2, 3. We can use these as numerators with the denominator 6.

Simplifying these fractions:

$$\backslash frac\{-2\}\{6\}\; =\; \backslash frac\{-1\}\{3\}$$

$$\backslash frac\{0\}\{6\}\; =\; 0$$

$$\backslash frac\{1\}\{6\}$$

$$\backslash frac\{2\}\{6\}\; =\; \backslash frac\{1\}\{3\}$$

$$\backslash frac\{3\}\{6\}\; =\; \backslash frac\{1\}\{2\}$$

So, five rational numbers between $\backslash frac\{-1\}\{2\}$ and $\backslash frac\{2\}\{3\}$ are $\backslash frac\{-1\}\{3\},\; \backslash frac\{-1\}\{6\},\; 0,\; \backslash frac\{1\}\{6\},\; \backslash frac\{1\}\{3\}$.