Class 9 Maths Chapter 1 Question Answers - Number System

Q1. Find a rational number between 1 and 2.

Sol: Express 1 and 2 as rational numbers with the same denominator:
1 = 1010, 2 = 2010
Now, the rational numbers between 1010 and 2010 are:
1110, 1210, 1310, 1410, 1510, 1610, 1710, 1810, 1910
Out of these, select any five:

1110, 1210, 1310, 1410, 1510
Answer: The five rational numbers between 1 and 2 are:
1110, 1210, 1310, 1410, 1510$\backslash frac\{11\}\{10\},\; \backslash frac\{12\}\{10\},\; \backslash frac\{13\}\{10\},\; \backslash frac\{14\}\{10\},\; \backslash frac\{15\}\{10\}$

Q2. Find two rational numbers between 0.1 and 0.3.

Sol:  Express $0.10.1$ and $0.30.3$ as rational numbers with the same denominator:

$0.1=\frac{1}{10}, 0.3=\frac{3}{10}$

Now, the rational numbers between $\frac{1}{10}\backslash frac\{1\}\{10\}$ and $\frac{3}{10}\backslash frac\{3\}\{10\}$ can be written with a larger denominator to find numbers in between.
Let us express them with a denominator of $100100$100:

$0.1=\frac{10}{\mathrm{100 }},0.3=\frac{30}{\mathrm{100 }}\mathrm{ .}$

The rational numbers between $\frac{10}{100}\backslash frac\{10\}\{100\}$ and $\frac{30}{100}\backslash frac\{30\}\{100\}$ are:

$\frac{11}{100},\frac{12}{100},\frac{13}{100},\dots ,\frac{29}{100}.\backslash frac\{11\}\{100\},\; \backslash frac\{12\}\{100\},\; \backslash frac\{13\}\{100\},\; \backslash ldots,\; \backslash frac\{29\}\{100\}$

any two:

$\frac{12}{100},\frac{25}{100}.\backslash frac\{12\}\{100\},\; \backslash frac\{25\}\{100\}.$$<br\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">$

Q3. Express  in the form of a decimal.

We have,
Now, dividing 25 by 8,
 

Since, the remainder is 0.
∴ The process of division terminates.

Q4. Express  as a rational number.

Multiplying (1) by 100, we have 100x = 100 x 0.3333…
⇒ 100x = 33.3333              …(2)
Subtracting (2) from (1), we have
100x – x = 33.3333… – 0.3333…
⇒ 99x = 33

Q5. Simplify: (4+ √3) (4 −√3)

∵ (a + b)(a – b) = a² – b²
(4 +√3) (4 −√3) = (4)² – ( √3)² = 16 – 3 = 13
Thus, (4 +√3) (4 −√3) = 13

Q6. Simplify: (√3 +√2)²

∵ (a + b)² = a² + 2ab + b²
(√3 +√2)² = (√3)² + √2 ( √3 ×2) + (√2)² = 3 + 2 √6 + 2 = 5 + 2 √6
Thus, (√3 +√2)² = 5 + 2√6

Q7. Rationalise the denominator of  1√6 - √5

Multiply and divide the given number by √6 + √5

 1√6 - √5

= √6 + √5(√6 - √5) (√6 + √5)

= √6 + √56 - 5

= √6 + √5

Q8. Find (64)-1/3

As, 64 = 4 x 4 x 4 = 4³

∴ (64)-1/3 = 1(64)1/3 = 1(43)1/3
= 14^3×1/3 = 14
Thus, (64)-1/3 = 14

Q9. Find a rational number lying between 

Rational numbers between 1/2 and 1/5 are infinite. Some of them are 3/10 , 4/10 , 45/100 , 35/100.Step-by-step explanation:As per the question, We need to find drational numbers lying between 1/5 and 1/2 As we know,

• Rational Numbers are numbers that can be expressed in the form of p/q where q is not equal to zero.
• Now, we know that  1/5 = 0.2 and 1/2 = 0.5
• So, numbers between 0.2 and 0.5 are infinite. Some of them are 0.3,0.4,0.45,,0.35 etc.
• And these may be written as 3/10,4/10,45/100,35/100 tec.

Hence, Rational numbers between 1/2 and 1/5 are infinite. Some of them are 3/10 , 4/10 , 45/100 , 35/100. 

Q10. Express  as a fraction in the simplest form.

Let X =  = 0.24545 ... (1)
Then, multiplying (1) by 10,
We have 10X = 10 x 0.24545...
⇒ 10x = 2.4545                      ... (2)
Again multiplying (1) by 1000,
we get 1000 x X = 0.24545... x 1000
⇒ 1000X = 245.4545                ... (3)
Subtracting (2) from (3),
1000X – 10X = 245.4545... – 2.4545...

Thus