By using Extra Questions for Class 9 Maths and Ganita Manjari Class 9 Maths Chapter 3 The World of Numbers Extra Questions, students can improve their problem-solving skills.
Class 9 The World of Numbers Extra Questions
Extra Questions on The World of Numbers Class 9
Class 9 Ganita Manjari Chapter 3 Extra Questions
Question 1.
Find a rational number between \(\frac{1}{4}\) and \(\frac{3}{4}\).
Solution:
We know that
1 < 2 < 3
⇒ \(\frac{1}{4}<\frac{2}{4}<\frac{3}{4}\)
⇒ \(\frac{1}{4}<\frac{1}{2}<\frac{3}{4}\)
Hence, one rational number between \(\frac{1}{4}\) and \(\frac{3}{4}\) is \(\frac{1}{2}\).
Aliter:
One rational number between \(\frac{1}{4}\) and \(\frac{3}{4}\)
= \(\frac{\frac{1}{4}+\frac{3}{4}}{2}=\frac{1}{2}\)
Question 2.
Find three irrational numbers between \(\frac{5}{7}\) and \(\frac{9}{11}\)
Solution:
\(\frac{5}{7}\) = 0.714285………..
\(\frac{9}{11}\) = 0.818181 ………….
Hence, three irrational numbers between \(\frac{5}{7}\) and \(\frac{9}{11}\) can be taken as
0.72 01 001 00001 ……….
0.720 200 2000 20000 2 ……….
and 0.72 03 003 0003 0000 3 ………….
Question 3.
Write one rational number and one irrational number between 2 and √5.
Solution:
√5 = 2.23606…………….
So, one rational number between 2 and √5 is 2.1 i.e., \(\frac{21}{10}\)
Here, p = 21, q = 10 (≠ 0)
Also, one irrational number between 2 and √5 is 2.10 100 1000 1…….
Question 4.
Express 3\(\frac{4}{5}\) in the decimal form and state the kind of decimal expansion.
Solution:
3\(\frac{4}{5}\) = 3 + \(\frac{4}{5}\)
= 3 + \(\frac{4 \times 2}{5 \times 2}\)
= 3 + \(\frac{8}{10}\)
= 3 + 0.8
= 3.8
The decimal expansion is terminating.
The World of Numbers Class 9 Very Short Question Answer
Question 5.
Is the number (3 – √7) (3 + √7) rational or irrational ?
Solution:
(3 – √7) (3 + √7)
= (3)2 – (√7)2
= 9 – 7 = 2
Question 6.
Find two rational numbers between \(\frac{1}{2}\) and \(\frac{2}{3}\).
Solution:
Question 7.
Find the rational number of the form \(\frac{p}{q}\) corresponding to the decimal representation 0.222 , where p and q are integers and q ≠ 0.
Solution:
Let x = 0.222 …(1)
Multiplying both sides by 10, we get
10x = 2.222 …(2)
Subtracting (1) from (2), we get
9x = 2 ⇒ x = \(\frac{2}{9}\)
⇒ 0.222… = \(\frac{2}{9}\) which is a rational number of the form \(\frac{p}{q}\), where p = 2 and q = 9
(≠ 0) are integers.
Question 8.
Express \(0 . \overline{3178}\) in the form of where p and q are integers and q ≠ 0.
Solution:
Let x = \(0 . \overline{3178}\)
Then, x = 0.3178 3178 3178… …(1)
Multiplying both sides by 10000 (since
four digits are repeating), we get
10000x = 3178.3178 3178 3178 …(2)
Subtracting (1) from (2), we get
9999x = 3178
⇒ x = \(\frac{3178}{9999}\)
⇒ \(0 . \overline{3178}\) = \(\frac{3178}{9999}\)
which is in the form of \(\frac{p}{q}\)
where p = 3178 and q = 9999(≠ 0) integers.
Question 9.
Express \(3 . \overline{651}\) in the form of f, where p and q are integers and q ≠ 0.
Solution:
Let x = \(3 . \overline{651}\)
Then, x = 3.6 51 51 51 …………………
Multiplying both sides by 10, we get
10x = 36.51 51 51 …………… (1)
Multiplying both sides by 100 (since digits are repeating), we get
1000 x = 3651.515151 ……. (2)
Subtracting (1) from (2), we get
990x = 3615
⇒ 330x = 1205
⇒ x = \(\frac{1205}{330}\)
Thus, \(3 . \overline{651}\) = \(\frac{1205}{330}\)
Here, p = 1205
and q = 330 (≠ 0) are integers.
Question 10.
Which is greater? (\(\frac{1}{2}\))1/2 or (\(\frac{1}{2}\))1/3
Solution:
Hence, (\(\frac{1}{2}\))1/2 is greater.
The World of Numbers Class 9 Short Question Answer
Question 1.
Find live rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\)
Solution:
\(\frac{3}{5}=\frac{3 \times 10}{5 \times 10}=\frac{30}{50}\)
We know that
30 < 31 < 32 < 33 < 34 < 35 < 40
⇒ \(\frac{30}{50}<\frac{31}{50}<\frac{32}{50}<\frac{33}{50}<\frac{34}{50}<\frac{35}{50}<\frac{40}{50}\)
⇒ \(\frac{3}{5}<\frac{31}{50}<\frac{16}{25}<\frac{33}{50}<\frac{17}{25}<\frac{7}{10}<\frac{4}{5}\)
Hence, five rational numbers between \(\frac{3}{5}\) and \(\frac{4}{5}\) can be taken as \(\frac{31}{50}, \frac{16}{25}, \frac{33}{50}, \frac{17}{25}\) and \(\frac{7}{10}\)
Question 2.
Express 3.651̄ in the form of \(\frac{p}{q}\) where p and q are integers and q≠ 0.
Solution:
Let x = 3.651̄
Then, x = 3.65111… …(1)
Multiplying both sides of (1) by 100, we get 100x = 365.1111… …(2)
Multiplying both sides of (2) by 10, we get 1000x = 3651.1111… …(3)
Subtracting (2) from (3), we get
900x = 3286
⇒ x = \(\frac{1}{2}\) ⇒ x = \(\frac{1643}{450}\)
⇒ 3.651̄ = \(\frac{1643}{450}\)
which is in the form of \(\frac{p}{q}\) where p = 1643 and q = 450 (≠ 0) are integers.
Question 3.
Show that 0.4777777… can be expressed in the form of \(\frac{p}{q}\), where p and q are integers and q ≠ 0.
Solution:
Let x = 0.477777… …(1)
Multiplying both sides of (1) by 10, we get
10x = 4.777777… …(2)
Multiplying both sides of (2) by 10, we get
100x = 47.777777…
Subtracting (2) from (3), we get
90x = 43
⇒ x = \(\frac{43}{90}\)
0. 4777777… = \(\frac{43}{90}\)
which is in the form of \(\frac{p}{q}\) where p = 43 and q = 90 (≠ 0) are integers.
Question 4.
Find ‘a’ and ‘b’ if \(\frac{1}{2}\) = a + b√3
Solution:
⇒ 11 – 6√3 = a + b√3
⇒ a = 11 and b = -6
Question 5.
If \(\frac{\sqrt{3}-1}{\sqrt{3}+1}\) = a – b√3 determine the rational numbers a and b.
Solution:
⇒ 2 – √3 = a – b√3
⇒ a = 2 and b = 1
The World of Numbers Class 9 Long Question Answer
Question 1.
Represent –\(\frac{7}{5}\) on the number line.
Solution:
\(\frac{-7}{5}=\frac{(-5)+(-2)}{5}=\frac{(-5)}{5}+\frac{(-2)}{5}\)
= -1 + \(\frac{(-2)}{5}\)
Take a number line 1 with O as zero. Mark points A and B on it to the left of O such that OA = AB = 1 unit.
Now divide the line segment AB into five equal parts. Let P, Q, R and S be the points of division. Then,
AP = PQ = QR = RS = SB
Clearly, AQ = \(\frac{2}{5}\) AB.
So, Q represents the rational number \(\frac{-7}{5}\) on the number line.
Question 2.
Locate √7 on the number line.
Solution:
Mark the distance 7 from a fixed point A on a given line l to obtain a point B such that AB = 7 units. From B mark a distance of 1 unit and mark the new point as C. Find the mid-point of AC and mark that point as O. Draw a semi-circle with centre O and radius OC. Draw a line perpendicular to AC passing through B and intersecting the semi-circle at D. Then, BD = √7
Now, let us treat BC as the number line with B as zero, C as 1 and so on. Draw an arc with centre B and radius BD, which
intersects the number line in E.
Then, E represents √7 on the number line.
Question 3.
Represent \(\sqrt{13}\) on number line geometrically. Verify using mathematically.
Or
Represent \(\sqrt{13}\) on a number line.
Solution:
Take a number line l with O as zero. Mark a point A on it such that OA = 3 units. Draw a line AB = 2 units passing through A and perpendicular to line /. Join OB.
Then, OB = \(\sqrt{9+4}=\sqrt{13}\) units.
Draw an arc with centre O and radius OB, which intersects the number line l at C. Then, C represents \(\sqrt{13}\) on the number line.
Verification
In right triangle OAB,
OB2 = OA2 + AB2 |By Pythagoras Theorem
= 32 + 22 = 9 + 4 = 13
⇒ OB = \(\sqrt{13}\)
⇒ OC = \(\sqrt{13}\)
Hence verified.
Question 4.
Find two rational numbers between:
(a) \(\frac{1}{4}\) and \(\frac{4}{5}\)
Solution:
\(\frac{3}{10}, \frac{4}{10}, \frac{5}{10} ;\)
(b) –\(\frac{4}{7}\) and –\(\frac{1}{4}\)
Solution:
\(\frac{-13}{28}, \frac{-11}{28}, \frac{-9}{28}\)
Question 5.
If \(\frac{1}{13}\) = 0.076923, determine the decimal expansions of without actual long division?
Solution:
\(0 . \overline{846153}\)
Question 6.
Express \(\frac{1}{2}\) as rational numbers in \(\frac{p}{q}\) form.
Solution:
\(\frac{707}{333}\)
Question 7.
Express 0.2315 as rational numbers in \(\frac{p}{q}\) form.
Solution:
\(\frac{463}{2000}\)
Question 8.
What can the maximum number of digits be in the repeating block of digits in the decimal expansion of \(\frac{11}{23}\)?
Solution:
22
Question 9.
Find three irrational numbers between the rational numbers:
(a) 2 and 3
(b) 0.030030003… and 0.131131113…
Solution:
(a) 2.010010001…, 2.0200020002…, 2.030030003…
(b) 0.04004…, 0.050050005…, 0.060060006…
Question 10.
Find three rational numbers between -2.18 and -2.17.
Solution:
-2.179, –2.177, -2.175
The World of Numbers Class 9 Case Based Questions
Question 1.
w = \(\frac{1}{2}\), x = \(\frac{1}{2}\) and z = \(\frac{1}{2}\)
Based on the above information, answer the following questions:
(i) Which of the following are rational numbers?
(a) w only
(b) w and x only
(c) y and z only
(d) z only
Solution:
(b) w and x only
(ii) Which of the following are irrational numbers?
(a) w only
(b) w and x only
(c) y and z only
(d) z only
Solution:
(c) y and z only
(iii) Which of the following can be represented as a terminating decimal ?
(a) w
(b) x
(c) y
(d) z
Solution:
(b) x
(iv) Which of the following can be represented as a non-terminating recurring decimal ?
(a) w
(b) x
(c) y
(d) z
Solution:
(a) w
The World of Numbers Extra Questions for Practice
Very Short Answer Type Questions
Question 1.
Give an example of two irrational numbers whose:
(i) difference is a rational number.
(ii) sum is a rational number.
(iii) product is a rational number.
(iv) division is a rational number. Justify also.
Solution:
(i) 1 + √3 ; 2 + √3
(ii) 1 + √3; 1 – √3
(iii) √3 + 1; √3 – 1
(iv) 2√3 , √3
Question 2.
Identify an irrational number among the following numbers:
0. 13, \(\frac{1}{2}\), 0.3013001300013…
Solution:
0.301 3001 300013…
Question 3.
Which numbers have their decimal expansions non-terminating and non-repeating?
Solution:
Irrational numbers
Question 4.
What type of decimal expansion \(\frac{1}{7}\) has?
[Hint: \(\frac{1}{7}\) = \(0 . \overline{142857}\)]
Solution:
non-terminating recurring
Question 5.
Find two rational numbers between – 2 and 5.
Solution:
0, 1
Question 6.
Find two rational numbers between 2.1 and 2.2.
Solution:
\(\frac{211}{100}, \frac{53}{25}\)
Question 7.
Find two rational numbers between \(\frac{2}{3}\) and \(\frac{5}{3}\).
Solution:
1, \(\frac{4}{3}\)
Question 8.
Write a rational number between \(\frac{2}{3}\) and \(\frac{3}{4}\).
Solution:
\(\frac{17}{24}\)
Question 9.
Find two irrational numbers between 2 and 3.
Solution:
2.101 001 0001 …
2.201 001 0001 …
Question 10.
Find two irrational numbers between 786 and 787.
Solution:
786.101 001 0001 ……….
786.201 001 0001 ………..
Question 11.
Find a rational number between √2 and √3.
Solution:
\(\frac{3}{2}\)
Question 12.
Write a rational number not lying between – \(\frac{1}{5}\) and \(\frac{-2}{5}\)
Solution:
0
Short Answer Type Questions
Question 1.
Find two rational numbers between \(\frac{2}{3}\) and \(\frac{4}{5}\).
Solution:
\(\frac{101}{150}\) and \(\frac{17}{25}\)
Question 2.
Express 2.3̄ in the form of \(\frac{p}{q}\), where p and q are integers and q ≠ 0.
Solution:
\(\frac{7}{3}\)
Question 3.
Express \(1 . \overline{27}\) in the form of \(\frac{p}{q}\), where p and q are integers and q ≠ 0.
Or
Show that 1.27 27 27 ……. = \(1 . \overline{27}\) can be expressed in the form \(\frac{p}{q}\) where p and q are integers and q ≠ 0.
Solution:
\(\frac{14}{11}\)
Question 4.
Express \(0 . \overline{437}\) in the form of \(\frac{p}{q}\).
Solution:
\(\frac{437}{999}\)
Question 5.
Express 0. 516̄ in \(\frac{p}{q}\) form.
Solution:
\(\frac{31}{60}\)
Question 6.
Express in the form of \(\frac{p}{q}\):
\(0 . \overline{38}+1.2 \overline{7}\)
Solution:
\(\frac{329}{198}\)
Question 7.
Represent √5 on the number line.
Solution:
Do yourself
Long Answer Type Questions
Question 1.
Locate √2 on the number line.
Or
Represent √2 on the number line.
Solution:
Do yourself
Question 2.
Rationalize the denominator of \(\frac{4 \sqrt{3}+3 \sqrt{2}}{\sqrt{48}+\sqrt{18}}\)
Solution:
1
Question 3.
Calculate the following using Brahmagupta’s laws:
(a) (-14) × 3
(b) (-8) × (-3)
(c) 0 – (-25)
(d) (-42) + 7
Solution:
(a) -42,
(b) 24,
(c) 25
(d) -6
Question 4.
Represent the rational numbers \(-\frac{2}{5}, \frac{3}{4}\) and 1\(\frac{1}{2}\) on a single number line.
Solution:
Do yourself
Question 5.
Find three distinct rational numbers that lie \(\left(-\frac{1}{4}\right)\) and \(\frac{3}{5}\)
Solution:
0, \(\frac{1}{4}, \frac{1}{2}\)
Question 6.
Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: \(\frac{9}{20}, \frac{7}{15}\) and \(\frac{23}{250}\)
Solution:
(a) terminating,
(b) non-terminating recurring,
(c) terminating
Question 7.
Classify the following numbers as rational or irrational:
(a) \(\sqrt{56}\)
(b) \(\sqrt{289}\)
(c) 0.474747…
(d) 1.01221222122221…
(e) 0.123456789
Solution:
(a) irrational,
(b) rational,
(c) rational,
(d) irrational,
(e) rational
Question 8.
Convert the following rational numbers in the form of a terminating decimal or non- terminating and repeating decimal, whichever the case may be, by the process of long division:
(a) \(\frac{7}{16}\)
(b) \(\frac{15}{35}\)
Solution:
(a) 0.4375,
(b) 0.42871
Question 9.
Prove that √7 is an irrational number.
Solution:
Do yourself
Question 10.
Locate the following rational numbers on the number line:
(a) 3.6 (b) -2.35
Solution:
Do yourself
Question 11.
A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 3rd decimal place. Show that such a number can be written in the form \(\frac{p}{10^3}\) where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by 23 or 53? Give reasons.
Solution:
Yes
Question 12.
Without performing division, determine whether the decimal expansion of \(\frac{51}{375}\) is terminating or non-terminating. If it terminates, state the number of decimal places.
Solution:
Terminating; 3 places of decimal
Case Study Based Questions
Ms. Leena, a Mathematics teacher conducts a test at the end of every chapter to check the performance of students.
Take a look at the following questions she designed for her students and answer them.
Question 1.
Show \(\sqrt{30.25}\) is a rational number between 5 and 6.
Solution:
5.5
Question 2.
If \(0.04 \overline{37}\) is written in \(\frac{p}{q}\) form.
Solution:
\(\frac{433}{9900}\)
Question 3.
If the decimal for figure A is \(0 . \overline{285714}\) then find the decimal for figure B.
Solution:
0.714285