Class 6

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 8 Basic Geometric Concepts Ex 8.2 Textbook Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 8th Lesson Basic Geometric Concepts Ex 8.2

Question 1.
Measure the lengths of the given line segments.

Solution:
i) \(\overline{\mathrm{AB}}\) = 2.4 cm
ii) \(\overline{\mathrm{PQ}}\) = 1.5 cm
iii) \(\overline{\mathrm{KL}}\) = 1cm, \(\overline{\mathrm{LM}}\) = 1 cm
\(\overline{\mathrm{KM}}=\overline{\mathrm{KL}}+\overline{\mathrm{LM}}\) =1 + 1 = 2 cm

Question 2.
Draw the following line segments.
i) AB = 6.3 centimeters ii) MN = 3.6 centimeters

Question 3.
Draw PQ = 4.6 cm and extend upto R such that PR = 6 cm.
Solution:

\(\overline{\mathrm{PR}}=\overline{\mathrm{PQ}}+\overline{\mathrm{QR}}\) = 4.6 + 1.4 = 6 cm

Question 4.
Draw a line segment \(\overline{\mathrm{OP}}\) with certain length and mark a point Q on it.
Check whether \(\overline{\mathrm{OP}}-\overline{\mathrm{PQ}}=\overline{\mathrm{OQ}}\)
Solution:

Given, \(\overline{\mathrm{OP}}\) = 8 cm; \(\overline{\mathrm{PQ}}\) = 3 cm
\(\overline{\mathrm{OP}}-\overline{\mathrm{PQ}}\) = 8cm – 3cm = 5 cm = \(\overline{\mathrm{OQ}}\)
∴ \(\overline{\mathrm{OP}}-\overline{\mathrm{PQ}}=\overline{\mathrm{OQ}}\) = 5cm

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 7 Introduction to Algebra InText Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 7th Lesson Introduction to Algebra InText Questions

Let’s Explore (Page No. 102)

Question 1.
Arrange 2 matchsticks to form the shape Continue the same shape for 2 times, 3 times and 4 times. Frame the rule for repeating the pattern.
Solution:
To make the given shape 2 matchsticks are needed.
To make the given 2 shapes 4 matchsticks are needed.
To make the given 3 shapes 6 matchsticks are needed.
To make the given 4 shapes 8 matchsticks are needed.
Continue and arrange the information in the following table.

Number of matchsticks required = 2 × Number of shapes to be formed
= 2 × x = 2x

Question 2.
Rita took matchsticks to form the shape
She repeated the pattern and gave a rule.
Number of matchsticks needed = 6.y, where y is the number of shapes to be formed. Is it correct ? Explain.
What is the number of sticks needed to form 5 such shapes ?
Solution:

To make the given shape 6 matchsticks are needed.
To make the given 2 shapes 12 matchsticks are needed.
To make the given 3 shapes 18 matchsticks are needed.
Continue and arrange the information in the following table.

Yes, it is correct.
Number of matchsticks required = 2 × Number of shapes to be formed
= 2 × y = 2y
Number of matchsticks needed to form 5 such shapes = 6 × 5 = 30

Let’s Explore (Page No. 103)

Question 1.
A line of shapes is constructed using matchsticks.

Shape-1 Shape-2 Shape-3 Shape-4
i) Find the rule that shows how many sticks are needed to make a line of such shapes ?
ii) How many matchsticks are needed to form shape -12 ?
Solution:

Number of matchsticks 3 5 7 9
i) Let us know the pattern
S₁ = 3 = 2 + 1 = (1 × 2) + 1
S₂ = 5 = 4 + 1 = (2 × 2) + 1
S₃ = 7 = 6 + 1 = (3 × 2) + 1
S₄ – 9 = 8 + 1 = (4 × 2) + 1
Now the rule for this pattern is number of matchsticks.

ii) Used to make ‘n’ number of shapes is Sn = (n × 2) + 1 = 2n + 1
Number of matchsticks needed to form shape – 12 is
S₁₂ = 2(12) + 1 = 24 + 1 = 25 sticks.

Check Your Progress (Page No. 105)

Question 1.
Fill the following table as instructed. One is shown for you.

S.No.	Expression	Verbal Form
1.	y + 3	Three more than y
2.	2x – 1
3.	5z
4.	\(<table border=”2″>\)

Solution:

S.No.	Expression	Verbal Form
1.	y + 3	Three more than y
2.	2x – 1	One less than the double of x
3.	5z	5 times of z
4.	\(<table border=”2″>\)	Half of the m

Let’s Explore ? (Page No. 106)

Question 1.
Find the general rule for the perimeter of a rectangle. Use variables T and ‘b’ for length and breadth of the rectangle respectively.
Solution:
Given length of rectangle = l
breadth of rectangle = b
We know that the perimeter of rectangle is twice the sum of its length and breadth.
Sum of length and breadth = l + b
Twice the sum of length and breadth = 2 × (l + b)
Rule for the perimeter of a rectangle = 2(l + b)

Question 2.
Find the general rule for the area of a square by using the variable ‘s’ for the side of a square.
Answer:
Given side of a square = s
We know that the area of a square is the product of side and side.
Area of a square = side × side
Rule for the area of a square = s.s

Side	Area
1	1 × 1
2	2 × 2
3	3 × 3
4	4 × 4
………..	…………
s	S × s

(Page No. 107)

Question 1.
Find the nth term in the following sequences.
0 3, 6, 9, 12, ii) 2, 5, 8, 11, iii) 1, 4, 9, 16,
Solution:
i) Given number pattern is 3, 6, 9, 12,……………..
To find the nth term in the given pattern, we put the sequence in a table.

First number = 3 × 1
Second number = 3 × 2
nth number = 3 × n = 3n
So, the nth term of the pattern 3, 6, 9, 12, is 3n.

ii) Given number pattern is 2, 5, 8, 11,
To find the nth term in the given pattern, we put the sequence in a table.

First number = 2 = 3 × 1 – 1
Second number = 5 = 3 × 2 – 1
Third number = 8 = 3 × 3 – 1
nth number = 3 × n – 1 = 3n – 1
So, the nth term of the pattern 2, 5, 8, 11 is 3n – 1.

iii) Given number pattern is 1, 4, 9, 16,
To find the nth term in the given pattern, we put the sequence in a table.

First number =1 = 1 × 1
Second number = 4 = 2 × 2
Third number =9 = 3 × 3
nth number = n × n = n²
So, the nth term of the pattern 1, 4, 9, 16 is n².

Check Your Progress (Page No. 108)

Question 1.
Complete the table and find the value of ‘p’ for the equation \(\frac{\mathbf{p}}{\mathbf{3}}\) = 4

p	\(\frac{\mathbf{p}}{3}\) = 4	Condition satisfied ? Yes/ No
3
6
9
12

Solution:

p	\(\frac{\mathbf{p}}{3}\) = 4	Condition satisfied ? Yes/ No
3	\(\frac{3}{3}\) ≠1 ≠ 4	No
6	\(\frac{6}{3}\) ≠2 ≠ 4	No
9	\(\frac{9}{3}\) ≠ 3 ≠ 4	No
12	\(\frac{12}{3}\) ≠ 4 ≠ 4	Yes

Question 2.
Write LHS and RHS of following simple equations.
i) 2x + 1 = 10
ii) 9 = y – 2
iii) 3p + 5 = 2p + 10
Solution;
i) 2x+ 1 = 10
Given equation is 2x + 1 = 10
L.H.S = 2x + 1
R.H.S = 10

ii) 9 = y – 2
Given equation is 9 = y – 2
LHS = 9
RHS = y – 2

iii) 3p + 5 = 2p + 10
Given equation is 3p + 5 = 2p + 10
LHS = 3p + 5
RHS = 2p + 10

Question 3.
Write any two simple equations and write their LHS and RHS.
Solution:
i) Consider 8x + 3 = 4 is a simple equation.
L.H.S = 8x + 3
RHS = 4

ii) Consider 5a + 6 = 8a – 3 is a simple equation.
LHS = 5a + 6
RHS = 8a – 3

Let’s Explore (Page No. 109)

Observe for what value of m, the equation 3m = 15 has both LHS and RHS become equal.
Solution:
Given equation is 3m = 15
If m = 1, then the value of 3m = 3(1) = 3≠15 ∴ LHS ≠RHS
If m = 2, then the value of 3m = 3(2) = 6 ≠ 15 ∴ LHS ≠ RHS
If m = 3, then the value of 3m = 3(3) = 9≠15 ∴ LHS ≠ RHS
If m = 4, then the value of 3m = 3(4) = 12 ≠ 15 ∴ LHS ≠ RHS
If m = 5, then the value of 3m = 3(5) = 15 = 15 ∴ LHS = RHS
From the above when m = 5 the both LHS and RHS are equal

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 6 Basic Arithmetic InText Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 6th Lesson Basic Arithmetic InText Questions

Check Your Progress (Page No. 84)

Question 1.
Express the terms 45 and 70 by using ratio symbol.
Solution:
Given terms are 45 and 70
Ratio = 45 : 70
It can be read as 45 is to 70.

Question 2.
Write antecedent in the ratio 7:15.
Solution:
Given ratio 7 : 15
In the ratio first term is called antecedent.
In 7 : 15 antecedent is 7.

Question 3.
Write the consequent in the ratio 8 : 13.
Solution:
Given ratio 8 : 13
In the ratio second term is called consequent.
In 8 : 13 consequent is 13.

Question 4.
Express the ratio 35 : 55 in the simplest form.
Solution:
Given ratio 35 : 55 (or)
To write the ratio in the simplest form we have to divide by the common factor of two terms 35 and 55.
Common factor is 5.
Now divide by 5,
\(\frac{35}{55}=\frac{35 \div 5}{55 \div 5}=\frac{7}{11}\)
Simplest form of \(\) is \(\frac{7}{11}\)

Question 5.
In the given figure, find the ratio of
i) Shaded part to unshaded parts.
ii) Shaded part to total parts,
iii) Unshaded parts to total parts.

Solution:
i) In the given figur.e,
Number of shaded parts = 1
Number of unshaded parts = 3
Ratio = shaded parts : unshaded parts = 1:3

ii) Number of shaded parts = 1
Total parts = 4
Ratio = shaded parts : total parts = 1:4

iii) Number of unshaded parts = 3
Total parts = 4
Ratio = unshaded parts : total parts = 3:4

Question 6.
Express the following in the form of ratio.
a) The length of a rectangle is triple its breadth. ‘
b) In a school, the workload of teaching 19 sections has been assigned to 38 teachers.
Solution:
a) Let breadth of rectangle = x or one part = 1 part
length of rectangle = triple the breadth
= 3 x x = 3x = 3 parts
Ratio = l : b = x : 3x =\(\frac{1 x}{3 x}=\frac{1}{3}\) = 1:3

b) Given number of sections = 19
Number of teachers = 38
Ratio = 19 : 38 = \(\frac{19}{38}=\frac{1}{2}\) = 1 : 2

(Page No. 88)

Question 1.
Which ratio is larger in the following pairs ?
(a) 5 : 4 or 9 : 8
(b) 12 : 14 or 16 : 18
(c) 8: 20 or 12: 15
(d)4:7 or 7:11
Solution:
a) 5 : 4 or 9 : 8
Write the given ratios as fractions, we have 5 : 4 = \(\frac{5}{4}\) and 9 : 8 = \(\frac{9}{8}\)
Now find the LCM of the denominators of 4 and 8 is 8.
Make the denominator of the each fraction equal to 8.
We have \(\frac{5}{4} \times \frac{2}{2}=\frac{10}{8}\) and \(\frac{9}{8} \times \frac{1}{1}=\frac{9}{8}\)
Clearly we know that 10 > 9
∴ \(\frac{10}{8}>\frac{9}{8}\) (or) 10 : 8 > 9 : 8
10 : 8 is equal to 5 : 4
Therefore the larger ratio is 5 : 4.

b) 12 : 14 or 16:18
12 : 14 = \(\frac{12}{14}=\frac{6}{7}\) and 16 : 18 = \(\frac{16}{18}=\frac{8}{9}\)
Now find the LCM of the denominators of 7 and 9 is 63.
Make the denominator of the each fraction equal to 63.
we have \(\frac{6}{7} \times \frac{9}{9}=\frac{54}{63}\) and \(\frac{8}{9} \times \frac{7}{7}=\frac{56}{63}\)
Clearly, we know that 54 < 56
∴ \(\frac{54}{63}<\frac{56}{63}\) (or) 54:63 < 56:63
56 : 63 is equal to 16 : 18 (or) 8 : 9
∴ The larger ratio is 16 : 18.

c) 8 : 20 or 12 : 15
Write the given ratios as fractions we have
8:20 = \(\frac{8}{20}=\frac{2}{5}\) and 12:15 = \(\frac{12}{15}=\frac{4}{5}\)
\(\frac{2}{5}\) and \(\frac{4}{5}\)

Clearly \(\frac{2}{5}\) < \(\frac{4}{5}\)
i.e., 2:5 < 4 : 5 (or) 8: 20 < 12: 15
Therefore the larger ratio is 12 : 15.

d) 4: 7 or 7: 11
Write the given ratios as fractions, we have 4 7
4 : 7 = \(\frac{4}{7}\) and 7:11 = \(\frac{7}{11}\) .
Now find the LCM of the denominators of 7 and 11 is 77.
Make the denominators of the each fraction equal to 77.
We have \(\frac{4}{7} \times \frac{11}{11}=\frac{44}{77}\) and \(\frac{7}{11} \times \frac{7}{7}=\frac{49}{77}\)
\(\frac{44}{7}\) and \(\frac{49}{77}\)
Clearly we know that 44 < 49
∴ \(\frac{44}{77}<\frac{49}{77}\) (or) 44 : 77 < 49 : 77
i.e.,4: 7 < 7 : 11
Therefore the larger ratio is 7 : 11

Question 2.
Find three equivalent ratios of 12 : 16.
Solution:
Given ratio is 12 : 16
Write the given ratio as fraction we have 12:16= \(\frac{12}{16}=\frac{3}{4}\)
Now, write equivalent fractions of \(\frac{3}{4}\)

i. e., 6 : 8 = 9 : 12 = 12 : 16 = 15 : 20 = 18 : 24
∴ Equivalent ratios of 12 : 16 are 6 : 8, 9 : 12, 12 : 16, 15 : 20 and 18 : 24.

(Page No. 90)

Question 1.
Check whether the following terms are in proportion ?
1) 5,6,7,8
2) 3,5,6,10
3) 4,8,7,14
4) 2,12,3,18
Solution:
1) Given, 5, 6, 7, 8
If a, b, c, d are in proportion i.e., a : b :: c : d
If 5, 6, 7, 8 are in proportion i.e., 5 : 6 : : 7 : 8
We know that, product of extremes = Product of means [a x d : b x c]
5 x 8 = 6 x 7
40 ≠ 42
So, 5, 6, 7, 8 are not in proportion.

2) Given, 3, 5, 6, 10
If a, b, c, d are in proportion i.e., a : b :: c : d
If 3, 5, 6, 10 are in proportion i.e., 3 : 5 :: 6 : 10
We know that, product of extremes = Product of means a x d = b x c
3 x 10 = 5 x 6
30 = 30
So, 3, 5, 6, 10 are in proportion.

3) Given, 4, 8, 7, 14.
If a, b, c, d are in proportion i.e., a : b : : c : d
If 4, 8, 7, 14 are in proportion i.e., 4 : 8 : : 7 : 14
We know that, product of extremes = Product of means a x d = b x c
4 x 14 = 8 x 7
56 = 56
So, 4, 8, 7, 14 are in proportion.

4) Given, 2, 12, 3, 18
If a, b, c, d are in proportion i.e., a : b :: c : d
If 2, 12, 3, 18 are in proportion i.e., 2 : 12 : : 3 : 18
We know that, product of extremes = Product of means [ a x d = b x c ]
2 x 18 = 12 x 3
36 = 36
So, 2, 12, 3, 18 are in proportion.

Let’s Explore (Page No. 92)

Question 1.
Read the table and fill in the boxes.

Prepare two similar problems and ask your friend to solve them
Solution:

Weight	Cost of Tomato	Cost of Potato
5 kg	₹ 75	₹ 60
1 kg	₹15	₹ 12
3 kg	₹ 45	₹ 36

(Page. No. 94)

Question 1.
Represent the following in other forms.

Solution:

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 5 Fractions and Decimals InText Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 5th Lesson Fractions and Decimals InText Questions

(Page No. 63)

Question 1.
Is it true to say that 3 × \(\frac{1}{5}=\frac{1}{5}\) x 3?
Solution:
3 × \(\frac{1}{5}=\frac{1}{5}\) × 3. Yes, it is true.
By using commutative property over multiplication a × b = b × a
3 × \(\frac{1}{5}=\frac{1}{5}\) × 3 = \(\frac{3}{5}\)

Check Your Progress (Page No. 63)

Find :
i) \(5 \times 3 \frac{2}{7}\)
ii) \(2 \frac{5}{9} \times 3\)
iii) \(2 \frac{4}{7} \times 3\)
iv) \(3 \times 1 \frac{3}{4}\)
Solution:

Let’s Explore (Page No. 64)

Question 1.
Observe the products of fractions.
Have you observed the products of any two fractions is always lesser or greater than each of its fraction, write conclusion.
\(\frac{1}{5} \times \frac{2}{3}=\frac{2}{15}\) (Product of two proper fractions)
Solution:
Product of any two proper fractions is always less than each of its fraction.
i. e., \(\frac{2}{15}<\frac{1}{5} \text { and } \frac{2}{15}<\frac{2}{3}\)

ii) \(\frac{3}{2} \times \frac{5}{4}=\frac{15}{8}\) (Product of two improper fractions) •
Solution:
The product of any two improper fractions is always greater than each of its fraction.
i.e, \(\frac{3}{2}<\frac{15}{8} \text { and } \frac{5}{4}<\frac{15}{8}\)

iii) \(\frac{2}{3} \times \frac{5}{3}=\frac{10}{9}\) (Product of proper and improper fractions)
Solution:
The product of a proper fraction and an improper fraction is always greater than its proper fraction and less than its improper fraction.
i.e., \(\frac{2}{3}<\frac{10}{9} \text { and } \frac{5}{3}>\frac{10}{9}\)

(Pg. No. 66)

Question 1.
i) 4 ÷ \(\frac{1}{8}\)
ii) 9 ÷ \(\frac{3}{4}\)
iii) 7 ÷ \(\frac{2}{3}\)
iv) 35 ÷ \(\frac{7}{3}\)
v) 4 ÷ \(\frac{15}{8}\)
Solution:

(Pg. No. 67)

Question 1.
Observation these products and fill in the blanks.

Solution:

Check Your Progress (Page No. 68)

Question 1.
Write the reciprocal of fractions in the given table.

Solution:

(Reciprocal of a fraction \(\frac{\mathrm{a}}{\mathrm{b}}\) is \(\frac{\mathrm{b}}{\mathrm{a}}\))

(Page No. 69)

Question 1.
Find
i) \(\frac{7}{9}\) ÷ 4
ii) \(\frac{3}{4}\) ÷ 9
iii) 4\(\frac{1}{2}\) ÷ 6
iv) \(\frac{1}{5}\) ÷ 3
Solution:

Check Your Progress (Page No. 73, 74 & 75)

Question 1.
Fill in the blanks.

Fraction	Decimal Number	” Read as
\(\frac{6}{10}\)	0.6	Zero point six
\(\frac{37}{100}\)	0.37	Zero point three seven
	0.721	Zero point seven two one
		Seventeen point two

Solution:

Fraction	Decimal Number	Read as
\(\frac{6}{10}\)	0.6	Zero point six
\(\frac{37}{100}\)	0.37	Zero point three seven
\(\frac{721}{1000}\)	0.721	Zero point seven two one
\(\frac{172}{10}\)	17.2	Seventeen point two

Question 2.
Write the place value of the circled digits.

Solution:

Question 3.
a) 700 + 40 + 2 + \(\frac{1}{10}+\frac{3}{100}+\frac{6}{1000}\)
Solution:
700 + 40 + 2 + 0.1 + 0.03 + 0.006 = 742.136

b) 9000 + 800 + 3 + 0.2 + 0.05 + 0.007
Solution:
9000 + 800 + 3 + 0.2 + 0.05 + 0.007 = 983.257

c) 6000 + 400 + 20 + 1 + \(\frac{2}{10}+\frac{5}{100}+\frac{9}{1000}\)
Solution:
6000 + 400 + 20 + 1 + 0.2 + 0.05 + 0.009 = 6421. 259

d) 400 + 5+ \(\frac{1}{10}+\frac{8}{100}\)
Solution:
400 + 5 + 0.1 +0.08 = 405.18

Question 4.
Expand the following into decimals and fractional forms,
a) 164.238
b) 968.054
Solution:
a) 164.238 = 100 + 60 + 4 + 0.2 + 0.03 + 0.008
= 100 + 60 + 4 + \(\frac{2}{10}+\frac{3}{100}+\frac{8}{1000}\)

b) 968.054
Solution:
= 900 + 60 + 8 + 0.0 + 0.05 + 0.004
= 900 + 60 + 8 + 0 + \(\frac{5}{100}+\frac{4}{1000}\)
= 900 + 60 + 8 + \(\frac{5}{100}+\frac{4}{1000}\)

Question 5.
Write fractions as decimals.
1. \(\frac{23}{10}\) = ………..
2. \(\frac{6}{100}\) = ………..
3. \(\frac{3}{8}\) = ………..
4. \(\frac{2}{25}\) = ………..
Solution:
1. \(\frac{23}{10}\) = 2.3
2. \(\frac{6}{100}\) = 0.06
3. \(\frac{3}{8}\) = 0.375
4. \(\frac{2}{25}\) = 0.08

Question 6.
Write decimals as fractions in simplest form.
1. 0.2 = ……………
2. 0.38 = ……………
3. 1.62 = ……………
4. 8.1 = ……………
Solution:
1. 0.2 = \(\frac{2}{10}\)
2. 0.38 = \(\frac{38}{100}\)
3. 1.62 = \(\frac{162}{100}\)
4. 8.1 = \(\frac{81}{10}\)

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 4 Integers InText Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 4th Lesson Integers InText Questions

Check Your Progress?(Page No. 47)

Question 1.
Write any five positive integers.
Solution:
1, 2, 3, 4, 5, 6, 7,

Question 2.
Write any five negative integers.
Solution:
-1, -2, -3, -4, -5, -6,

Question 3.
Which number is neither positive nor negative?
Solution:
0 (zero)

Question 4.
Represent the following situations with integers,
(i) A gain of ₹ 500 ( )
(ii) Temperature is below 5°C ( )
Solution:
i) + 7 500
ii) – 5° C

Question 5.
Represent the following using either positive or negative numbers.
a) A bird is flying at a height of 25 meters above the sea level and a fish at a depth of 2 meters.
b) A helicopter is flying at a height of 60m above the sea level and a submarine is at 400m below sea level.
Solution:
a) Height of the flying bird 25 meters from th$ sea level = + 25 meters
Depth of the fish 2 meters from the sea level = – 2 meters
b) Height of the flying helicopter 60 meters from the sea level = + 60 m
Depth of the submarine 400 m from the sea level = – 400 m

Check Your Progress (Page No. 49)

Question 1.
Draw a vertical number line and represent -5,4,0,-6, 2 and 1 on it.
Solution:

Question 2.
Represent opposite integers of – 200 and + 400 on integer number line.
Solution:
Opposite integers means, additive inverse.
∴ Opposite integer (additive inverse) of – 200 is 200.
Opposite integer (additive inverse) of +400 is – 400.

Let’s Think (Page No. 50)

Question 1.
For any two integers, say 3 and 4, we know that 3 < 4.
Is it true to say -3 < -4? Give reason.
Solution:
On the number line, the value of a number increases as we move to right and decreases as we move to the left. As -3 lies right to -4 on the number line.
So, -3 < -4 is not true.

(Pg. No. 52)

Question 1.
What is additive inverse of 7 ?
Solution:
Additive inverse of 7 is -7.

Question 2.
What is additive inverse of -8 ?
Solution:
Additive inverse of -8 is 8.

Let’s Explore (Page.No. 52)

Question 1.
Find the value of the following using a number line.
i) (-3) + 5 ii) (-5) + 3
Make two questions on your own and solve them using the number line.
Solution:
i) (-3) + 5

On the number line, we first move 3 steps to the left of 0 to reach -3.
Then, we move 5 steps to the right of -3 and reach +2. So, (-3) + 5 = 2

ii) (-5) + 3

On the number line, we first move 5 steps to the left of 0 to reach -5. Then, we move 3 steps to the right of -5 and reach -2. So, (-5) + 3 = – 2

iii)(+6) + (-3)

On the number line, we first move 6 steps to the right of 0 to reach +6.
Then, we move 3 steps to the left of 6 and reach +3. So, (+6) + (-3) = 3

iv) (-4) + (-3)

On the number line, we first move 4 steps to the left of 0 to reach -4. Then, we move 3 steps to the left of -4 and reach -7.
So (-4) + (-3) = -7.

Question 2.
Find the solution of the following:
i) (+5) + (-5) (ii) (+6) + (-7) (iii) (-8) + (+2)
Ask your friend five such questions and solve them.
Solution:
i) (+5) + (-5) = 0

On the number line, we first move 5 steps to the right of 0 to reach +5.
Then, we move 5 steps to the left of +5 and reach 0.

ii) (+6) + (-7) = -1

On the number line, we first move 6 steps to the right of 0 to reach +6.
Then, we move 7 steps to the left of +6 and reach -1.

(iii) (-8) + (+2)

On the number line, we first move 8 steps to the left of 0 to reach -8. Then, we move 2 steps to the right of -8 and reach -6.

iv)(-4) + (+8) = +4

On the number line, we first move 4 steps to the left of 0 to reach -4. Then, we move 8 steps to the right of -4 and reach +4.

v) (+3) + (-4) = -1

On the number line, we first move 3 steps to the right of 0 to reach +3. Then, we move 4 steps to the left of +3 and reach -1.

vi) (+5) + (-6) = -1

On the number line, we first move 5 steps to the right of 0 to reach +5. Then, we move 6 steps to the left of +5 and reach -1.

vii) (+4) + (-4) = 0

On the number line, we first move 4 steps to the right of 0 to reach +4. Then, we move 4 steps to the left of +4 and reach 0.

viii) (-6) +(+4) =-2

On the number line, we first move 6 steps to the left of 0 to reach -6. Then, we move 4 steps to the right of -6 and reach -2.
So, (-6) + (+4) = -2

Let’s Explore (Page No. 55)

Question 1.
Take any two integers a and b. Check whether a+b is also an integer.
Case (i) : Consider two integers 3 and -2 (Positive and negative)
Sum = 3 + (-2) = +1 + 2 – 2 = +1 + 0 = +1 is also an integer.
Case (ii) : Consider two ihtegers 5 and 6 (Both are positive)
Sum = 5 +6 = + 11 is also an integer
Case (iii) : Consider two integers -4 and -6 (Both are negative)
Sum = -4 + (-6) = -4 -6 = -10 is also an integer
Case (iv) : Consider two integers -5 and 4 (Negative and positive)
Sum = -5 + 4 = -1 -4 + 4 = -1 + 0 = -1 is also an integer.
So, if a and b are integers, then their sum a + b is also an integer. Integers are closed under addition.

Question 2.
Check the following Properties on integers, a, b, c are any integers.
i) Closure Property under subtraction ‘
ii) Commutative Property under addition and subtraction (a + b = b + a ?, a – b = b – a?)
iii) Associative Property under addition and subtractioji.
(a + b) + c = a + (b + c) ? (a – b).- c = a – (b – c)?
Solution:
i) Closure Property under subtraction :
Case (i) : Consider two integers 4, -5 (positive and negative)
Then, difference a – b = 4 – (-5) = 4 + 5 = + 9is also an integer.
Case (ii) : Consider two integers 3, 8 (Both are positive)
Then, difference a – b = 3 – (+8) = 3-8
= +3 – 3 – 5 = 0 – 5 = -5is also an integer.
Case (iii) : Consider two integers -2, -6 (Both are negative) .
Then, difference a – b = -2 – (-6) = -2 + 6
= -2 + 2 + 4 = 0 + 4 = +4 is also an integer.

Case (iv) : Consider two integers -3, 2 (Negative and positive)
Then, difference a – b = -3 – (+2) = -3 -2 = -5 is also an integer.
So, if a and b are any two integers, then their difference a – b is also an integer. Integers are closed under subtraction.

ii) Commutative Property under addition and subtraction :
(a + b = b + a, a-b = b-a)
(A) Case (Q : Consider two integers-3 and 5 (Negative and positive)
Then, a + b = -3 + (+5)
= -3 + 5 = – 3 + 3 + 2 = 0 + 2 = + 2
b + a = +5 + (-3) = +2 + 3- 3 = +2 + 0 = + 2
∴ a + b = b + a

Case (ii) : Consider two integers +4 and +2 (Both are positive)
Then, a + b = +4 + (+2) = +4 + 2 = + 6 ‘b + a = +2 + (+4) = +2 + 4 = + 6
∴ a + b = b + a

Case (iii) Consider two integers -5 and -3 (Both are negative) Then, a + b = -5 + (-3) = -5 – 3 = -8 b + a = -3 + (-5) = – 3 – 5 = -8
∴ a + b = b + a

Case (iv) : Consider two integers +4 and -1 (Positive and negative)
Then, a + b = +4 + (*1) = +4 -1 = +3 + 1 -1 = +3 + 0 = +3 b + a = -1 + (+4) =-l + 4 = -l + l+ 3 = 0 + 3 = + 3
∴ a + b = b + a
So, integers are commutative under addition.

(B) Consider two integers -4 and +6 (Negative and positive)
Then, a – b = -4 – (+6) = -4 – 6 = -10 b-a = + 6-(-4) = +6 +4 = +10 -10*10
a – b ≠ b – a
So, integers are not commutative under subtraction.

iiO Associative property under addition and subtraction :
(a + b) + c = a + (b + c) ; (a – b) – c = a – (b – c)
(A) Case (i) : Consider any three integers 2, 4, -5
(a + b) + c = (2 + 4) + (-5) = 6 – 5 = +1 + 5- 5 = +1 + 0 = +1
a + (b + c) = 2 + (4 + (-5)) = 2 + (4 – 5) = 2 + (4 – 4 -1)
= 2 + (0-1) = + 2 – 1 = + 1 + 1 – 1 = +1 + 0 = +1
∴ (a + b) + c = a + (b + c)

Case (ii) : Consider any three integers +2, -5, +3
Then, (a + b) + c = [2 + (-5)] + 3 = [+2 -5] + 3 = +2 -2 -3 + 3 = 0 + 0 = 0
a + (b + c) = +2 + [( – 5)+3] = +2 + [-2 – 3 + 3] = +2 + (-2 + 0) = +2 -2 = 0
∴ (a + b) + c = a + (b + c)

Case (iii) : Consider any three integers 3, 4, 6
Then, (a + b) + c = [2 + (-5)] + 3 [+2-5] + 3
a + (b + c) = 3 + (4 + 6) = 3 + 10 = + 13
∴ (a + b) + c = a + (b + c)

Case (iv) : Consider any three integers -4, -2, +5
Then, (a + b) + c = [-4 + (-2)] + 5 = [-4 -2] + 5 = -6 + 5
= -1-5 + 5 = – 1 + 0 = – 1
a + (b + c) = -4 + [(-2) + 5] = -4 + [-2 + 5]
= _4 + [_2 + 2 +3] = -4 + 0 + 3 =-1-3+ 3
= -1 + 0 = -1
∴ (a + b) + c = a + (b + c)

Case (v) : Consider any three integers-3, 4, 1
Then, (a + b) + c = (-3 + 4) + 1 = (-3 + 3 + 1) + 1 = 0 + 1 + 1 = + 2
a + (b + c) = -3 + (4 + 1) = -3 + 5 = -3 + 3 + 2 = 0 + 2 = +2
∴ (a + b) + c = a + (b + c)

Case (vi) : Consider any three integers -2, 6, -7
Then, (a + b) + c = (-2 + 6) + (-7) = (-2 + 2 + 4) + (-7) = 0 + 4 – 7
= + 4 – 4 – 3 = 0 – 3 = -3
a + (b + c) = -2 + [6 + (-7)] = -2 + (6 – 7) = -2 + [+6 – 6 -1]
= -2 + (-1) = -2 -1 = -3
∴ (a + b) + c = a + (b + c)

Case (vii) : Consider any three integers +6, -3, -1
Then, (a + b) +c = [+6 + (-3)] + (-1) = (6 – 3) – 1 = (+3 +3 -3) -1
=+3-1 =+2 + 1 – 1 =+2 + 0 = + 2
a + (b + c) = +6 + [-3 + (-1)] = 6 + [-3 – 1] = 6 + (-4) .
= +2 + 4 – 4 = 2 + 0 = + 2
∴ (a + b) + c = a + (b + c)

Case(viii)
Consider any three integers -4, -1, -7
Then, (a + b) + c = [-4 + (-1)] + (-7) = (-4 -1) – 7 = -5 -7 = -12
a + (b + c) = -4 + [(-1) + (-7)] = -4 + [-1 -7] = -4 + (-8)
= -4 – 8 = -12
∴ (a + b) + c = a + (b + c)
From all the above cases we conclude that, integers are associative under addition.

(B) Consider any three integers +5, -4, 1
Then, (a – b) – c = (+5 – (-4)) – (+1) = (5 + 4) – 1 .
= + 9 – 1 = + 8 + 1 – 1 = + 8 + 0 = + 8
a – (b – c) = +5 – [- 4 – (+1)] = + 5 – [-4 – 1]
= + 5 – [ -5] =+ 5 + 5 = + 10
+ 8 ≠ +10
∴ (a – b) – c ≠ a – (b – c)
So, integers are not associative under subtraction.

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 3 HCF and LCM InText Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 3rd Lesson HCF and LCM InText Questions

(Page No. 35)

Question 1.
How does the Sieve of Eratosthenes work ?
Solution:
The following example illustrates how the Sieve of Eratosthenes, can be used to find all the prime numbers that are less than 100. .
Step 1: Write the numbers from 1. to 100 in ten rows as shown below.
Step 2: Cross out 1 as 1 is neither a prime nor a composite number.
Step 3: Circle 2 and cross out all the multiples of 2. (2, 4, 6, 8, 10, 12, ………… )
Step 4: Circle 3 and cross out all the multiples of 3. (3, 6, 9, 12, 15, 18,………….)
Step 5: Circle 5 and cross out all the multiples of 5. (5, 10. 15, 20, 25……………… )
Step 6: Circle 7 and cross out all multiples of 7. (7, 14, 21. 28, 35, ………………….. )

Circle all the numbers that are not crossed out and they are the required prime numbers less than 100.

First arrange the numbers ffom 1 to 100 in a table as shown above.
Enter 6 numbers in each row until the last number 100 is reached.
First we select a number and we strike off all the multiples of it.
Start with 2 which is greater than 1.
Round off number 2 and strike off entire column until the end.
Similarly strike off 4th column and 6th column as they are divisible by 2.
Now round off next number 3 and strike off entire column until end.
The number 4 is already gone.

Now round off next number 5 and strike off numbers in inclined fashion as shown in the figure (they are all divisible by 5). When striking off ends in some row, start again striking off with number in another end which is divisible by 5. New striking off line should be parallel to previous strike off line as. shown in the figure.
The number 6 is already gone.
Now round off number 7 and strike off numbers as we did in case of number 5.
8,9,10 are also gone. .
Stop at this point.
Count all remaining numbers. Answer will be 25.

Prime numbers :
There are 25 prime numbers less than 100.
These are:

What if we go above 100 ? Around 400 BC the Greek mathematician. Euclid, proved that there are infinitely many prime numbers.

Co-primes: Two numbers are said to be co-prime if they have no factors in common. Example: (2, 9), (25, 28)
Any two consecutive numbers always form a pair of co-prime numbers.

Example:
Co-prime numbers are also called relatively prime number to one another.
Example: 3, 5, 8, 47 are relatively prime to one another/co-prime to each other.

Twin primes: Two prime numbers are said to be twin primes, if they differ by 2.
Example: (3, 5), (5, 7), (11, 13), …etc.

Prime factorization: The process of expressing the given number as the product of prime numbers is called prime factorization.
Example: Prime factorization of 24 is
24 = 2 x 12
= 2 x 2 x 6
= 2 x 2 x 2 x 3, this way is unique.

Every number can be expressed as product of primes in a unique manner. We can factorize a given number in to product of primes in two methods. They are
a) Division method
b) Factor tree method

Common factors: The set of all factors which divides all the given numbers are called their common factors.
Example: Common factors to 24, 36 & 48 are 1, 2, 3, 4, 6 & 12
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 & 24
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 & 36
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 & 48
Common factors to 24, 36 & 48 are 1, 2, 3, 4, 6 & 12
We can see that among their common factors 12 is the highest common factor. It is called H.C.F. of the given numbers. So H.C.F. of 24, 36 & 48 is 12.

H.C.F./G.C.D : The highest common factor or the greatest common divisor of given numbers is the greatest of their common factors.
H.C.F. of given two or more numbers can be found in two ways.
a) By prime factorization
b) By continued division
H.C.F. of any two consecutive numbers is always 1.
H.C.F. of relatively prime/co-prime numbers is always 1.
H.C.F. of any two consecutive even numbers is always 2.
H.C.F. of any two consecutive odd numbers is always 1.

Common multiples:
Multiples of 8: 8, 16, 24, 32, 40, 48,
Multiples of 12: 12, 24, 36, 48, … .
Multiples.common to 8 & 12: 24, 48; 72, 96, …
Least among the common multiple is 24. This is called L.C.M. of 8 & 12. The number of common multiples of given two or more numbers is infinite, as such greatest common multiple cannot be determined.

L.C.M.: The least common multiple of two or more numbers is the smallest natural number among their common multiples.
L.C.M. of given numbers can be found by the
a) Method of prime factorization.
b) Division method.
L.C.M. of any two consecutive numbers is always equal to their product.
L.C.M. of 8 &9 is 8 x 9 = 72
L.C.M. of co-prime numbers is always equal to their product.
L.C.M. of 8 & 15 is 8 x 15 = 120

Relation between the L.C.M. & H.C.F:
For a given two numbers Nj & N2 , the product of the numbers is equal to the product of their L.C.M.(L) & H.C.F.(H)
N₁ x N₂ = L x H

Check Your Progress (Page No. 29)

Question 1.
Are the numbers 900, 452, 9534, 788 divisible by 2? Why?
Solution:
Yes. Because these numbers have 0, 2, 4 and 8 in their ones place. The numbers having 0, 2, 4, 6 and 8 in their ones place are divisible by 2.

Question 2.
Are the numbers 953, 457, 781, 325, 269 divisible by 2? Why?
Solution:
No. Because, these numbers have 3, 7, 1, 5 and 9 in their ones place. The numbers having 0, 2,
4, 6 and 8 in their ones place are only divisible by 2. So, these are not divisible by 2.

Question 3.
Are the numbers 452, 673, 259, 356 divisible by 2? Verify.
Solution:
452 and 356 have 2 and 6 in their ones place respectively.
So, they are divisible by 2.
673 and 259 have 3 and 9 in their ones place respectively.
So, they are not divisible by 2.

Question 4.
Check whether the following numbers are divisible by 3 (using rule). Verify by actual division.
(i) 12345 (ii) 61392 (iii)8747
Solution:
i)12345
1 + 2 + 3 + 4 + 5=15 is a multiple of 3.
If the sum of the digits of a number is the multiple of 3, then the number is divisible by 3.
So, 12345 is divisible by 3.

ii) 61392
6 + 1 + 3 + 9 + 2 = 21is a multiple,of 3.
If the sum of the digits of a number is a multiple of 3, then the number is divisible
by 3.

So, 61392 is divisible by 3.

iii) 8747
8 + 7 + 4 + 7 = 26is not a multiple of 3.
If the sum of the digits of a number is a multiple of 3, then the numbei is divisible by 3. So, 8747 is not divisible by 3.

So, 8747 is not divisible by 3.

Let’s Explore (Page No. 29)

Question 1.
Is 8430 divisible by 6? Why?
Given number is 8430.
The given number has zero in the ones place.
So, 8430 is divisible by 2. –
And the unit sum is8 + 4 + 3 + 0 = 15 is a multiple of 3.
So, 8430 is divisible by 3.
If a number is divisible by both 2 and 3, then only it is divisible by 6.
8430 is divisible by both 2 and 3.
Therefore 8430 is divisible by 6.

Question 2.
Take any three 4 digit numbers and check whether they are divisible by 6.
Solution:
Consider: i) 5632, ii) 6855, iii) 9600 are three 4 digit numbers.
i) 5632 has 2 in its ones place. So, 5632 is divisible by 2.
The unit sum is
5 + 6 + 3 + 2 = 16 is not a multiple of 3. So, 5632 is not divisible by 3.
If a number is divisible by both 2 and 3, then only it is divisible by 6.
5632 is qnly divisible by 2, but not divisible by 3.
So, 5632 is not divisible by 6.

ii) 6855 has 5 in the ones place. So, 6855 is not divisible by 2.
6 + 8 + 5 + 5 = 24 is a multiple of 3.
So, 6855 is divisible by 3.
If a number is divisible by both 2 and 3, then only it is divisible by 6.
6855 is not divisible by 2, but it is divisible by 3.
So, 6855 is not divisible by 6.

iii) 9600 has ‘0’ in the ones place. So, 9600 is divisible by 2.
9 + 6 + 0 + 0 = 15 is a multiple of 3.
So, 9600 is divisible by 3. .
If a number is divisible by both 2 and 3, then only it is divisible by 6, 9600 is divisible by 1 both 2 and 3.
So, 9600 is divisible by 6.

Question 3.
Can you give an example of a number which is divisible by 6 but not by 2 and 3? Why?
Solution:
No. We can’t give any example, because if any number is divisible by both 2 and 3, then only it is divisible by 6. Otherwise it is not possible.

Check Your Progress (Page No. 30 & 31))

Question 1.
Test whether 6669 is divisible by 9. ,
Solution:
Given number is 6669.
Sum of the digits = 6 + 6 + 6 + 9 = 27 is divisible by 9
If the sum of the digits of a number is divisible by 9 then, it is divisible by 9. ” 27 is divisible by 9. So, 6669 is divisible by 9.

Question 2.
Without actual division, find whether 8989794 is divisible by 9.
Solution:
Given number is 8989794.
Sum of the digits =8+9+8+9+7+9+4=54 – 1
If the sum of the digits of a number is divisible by 9.
Then, it is divisible by 9.
54 is divisible by 9. So, 8989794 is divisible by 9.

Question 3.
Are the numbers 28570, 90875 divisible by 5? Verify by actual division also.
Solution:
a) Given number be 28570.
The numbers with zero or five at ones place are divisible by 5
28570 has zero in its ones place. So, 28570 is divisible by 5.
Actual division :

So, 28570 is completely divisible by 5.

b) Given number is 90875
In 90875, ones place digit is 5. So, 90875 is divisible by 5.
Actual division :

In the given number 90875 the digit in the units place is 5,
‘ So, it is divisible by 5.
So, 90875 is divisible by 5.

Question 4.
Check whether the number 598, 864, 4782 and 8976 are divisible by 4. Use divisibility rule and verify by actual division.
Solution:
a) Given number is 598.
The number formed by the digits in tens and ones places of 598 is 98.
If the number formed by last two digits (Ones and Tens) of the number is divisible
by 4, then the number is divisible by 4.
98 is not divisible by 4. So, 598 is not divisible by 4.

b) Given number is 864.
The number formed by tens and ones places of 864 is 64.
If the number formed by last two digits (ones and tens)
of the number is divisible by 4, then the number is divisible by 4.
64 is divisible by 4. So, 864 is divisible by 4.

c) Given number is 4782.
The number formed by ones and tens places of 4782 is 82.
If the number formed by last two digits (ones and tens) of the number is divisible by 4, then the number is divisible by 4. 82 is not divisible by 4.

d) Given number is 8976.
The number formed by the digits in tens and ones places of 8976 is 76.
If the number formed by last two digits
(Tens and ones) of the number is divisible by 4. Then the number is divisible by 4.
76 is divisible by 4. So, 8976 is divisible by 4.

Question 1.
Fill the blanks and complete the table. (Page No. 32)
Solution:

Lets Explore (Page No.33)

Question 1.
1221 is a polindrome number, which on reversing its digits gives the same number. Thus, every polindrome number with even number of digits is always divisible by 11. Write polindrome number of 6 – digits.
Solution:

There are some polindrome number of 6 – difits.

Check Your Progress (Page No.34)

Question 1.
Find the factors of 60.
Solution:
60 = 1 x 60
60 = 2 x 30
60 = 3 x 20
60 = 4 x 15
60 = 5 x 12
60 = 6 x 10
∴ The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.

Question 2.
Do all the factors of a given number divide the number exactly? Find the factors of 30 and verify by division.
30 = 1 x 30
30 = 3 x 10
30 = 2 x 15
30 = 5 x 6
The factors of 30 are 1, 2, 3, 5, 6,10, 15 and 30.

Yes, the factors of a given numbers are divide the number exactly.

Question 3.
3 is a factor of 15 and 24. Is 3 a factor of their difference also?
Solution:
Difference = 24 – 15 = 9 is the multiple of 3.
Yes, 3 is a factor of difference of 15 and 24.

Let’s Explore (Page No. 35)

Question 1.
What is the smallest prime number?
Solution:
2

Question 2.
What is the smallest composite number?
Solution:
4

Question 3.
What is the smallest odd prime number?
Solution:
3

Question 4.
What is the smallest odd composite number?
Solution:
9

Question 5.
Write 10 odd and 10 even composite numbers.
Solution:
Odd composite numbers are 9, 15, 21, 25, 27, 33, 35, 39, 45, 49.
Even composite numbers are 4, 6, 8, 10, 12, 14, 16, 18, 20, 22.
Except 2, every even number is a composite number.

Let’s Explore (Page No. 36)

Question 1.
Can you guess a prime number which when on reversing its digits, gives another prime number? (Hint a 2 digit prime number)
Solution:
13 and 31; 17 and 71, 37 and 73 79 and 97.

Question 2.
311 is a prime number. Can you find the other two prime numbers just by rearrang-ing the digits?
Solution:
113, 131

Check Your Progress (Page No. 36)

Question 1.
From the following numbers identify different pairs of co-primes. 2, 3,4, 5,6,7, 8, 9 and 10.
Solution:
The numbers which have only 1 as the common factor are called co-primes, (or) Numbers having no common factors, other than 1 are called co-primes.
2,3; 2, 5; 2, 7; 2, 9; 3,4; 3, 5; 3, 7; 3, 8; 3,10; 4,5; 4, 7; 4, 9; 5, 6; 5, 7; 5,8; 5, 9; 6, 7; 8, 9 and 9,10. These are the different pairs of co-primes with 2, 3, 4, 5, 6, 7, 8, 9 and 10.
1) Any two primes always forms a pair of co-primes.
2) Any two consecutive numbers always form a pair of co-primes.
3) Any two primes cilways form a pair of co-primes. .

Question 2.
Write the pairs of twin primes less than 50.
Solution:
Two prime numbers are said to be twin primes, if they differ each other by 2.
Twin primes less than 50 are (3, 5); (5, 7); (11,13); (17, 19); (29, 31) and (41, 43).

(Page No. 36)

Question 1.
Find the HCF of 12, 16 and 28
Solution:

Thus 12 = 2 x 2 x 3
16 = 2 x 2 x 2 x 2
28 = 2 x 2 x 7
The common factor of 12, 16 and 28 ¡s 2 x 2 = 4.
Hence, H.C.F of 12, 16 and 28 is 4.

Let’s Explore (Page No. 40)

What is the HCF of any two
i) Consecutive numbers ?
ii) Consecutive even numbers ?
iii) Consecutive odd numbers? What do you observe? Discuss with your Mends.
Solution:
Consider the two consecutive number are 5, 6 and 9, 10.

We observed that HCF of any two consecutive numbers is always 1.

ii) Consider two consecutive even numbers are 8, 10 and 20, 22.

We observed that HCF of any two consecutive even numbers is always 2.

iii)Consider the consecutive odd numbers are 7, 9 and 13, 15.

We observed that HCF of any two consecutive odd numbers is always 1.

(Page No. 42)

Question 1.
Find LCM of (i) 3, 4 (ii) 10, 11 (iii) 10, 30 (iv) 12, 24 (v) 3, 12 by prime factorization method.
Solution:
i) Given numbers are 3, 4
Factors of 3 = 1 x 3
Factors of 4 = 2 x 2
LCM of 3, 4 = 1 x 3 x 2 x 2 = 12

ii) Given numbers are 10, 11
Factors of 10 = 2 x 5
Factors of 11 = 1 x 11
LCM of 10, 11 = 2 x 5 x 11 = 110

iii) Given numbers are 10, 30
Factors of 10 = 2×5
Factors of 30 = 2x3x5
LCM of 10, 30 = 2 x 3 x 5 = 30

iv) Given numbers are 12, 24
Factors of 12 = 2x2x3 ,
Factors of 24 =. 2 x 2 x 2 x 3
LCM of 12, 24 = 2 x 2 x 2 x 3 = 24

v) Given numbers are 3,12 Factors of 3 = 1×3
Factors of 12 =2 x 2 x 3,
LCM of 3, 12 = 3 x 2 x 2 = 12

(Page No. 43)

Question 1.
What is the LCM and HCF of twin prime numbers ?
Solution:
LCM = Product of the taken twin primes and HCF = 1

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 5 Fractions and Decimals Ex 5.1 Textbook Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 5th Lesson Fractions and Decimals Ex 5.1

Question 1.
Classify the fractions as proper, improper and mixed.
\(\frac{3}{4}\), \(\frac{6}{5}\), \(\frac{3}{2}\),\(\frac{4}{1}\), \(\frac{2}{3}\), \(\frac{1}{4}\), \(\frac{18}{13}\), 1\(\frac{5}{7}\),\(\frac{1}{3}\), 11\(\frac{1}{2}\)
Answer:
If in a fraction numerator is less than the denominator, then it is called a proper fraction.
Proper fractions are \(\frac{3}{4}\), \(\frac{2}{3}\), \(\frac{1}{4}\), \(\frac{1}{3}\)
If in a fraction numerator is greater than the denominator, then it is called an improper fraction.
Improper fractions are \(\frac{6}{5}\), \(\frac{3}{2}\), \(\frac{4}{1}\), \(\frac{18}{13}\)
A combination of a whole number and a proper fraction is called a mixed fraction.
Mixed fractions are 1\(\frac{5}{7}\), 11\(\frac{1}{2}\)

Question 2.
Write the following fractions in an ascending order.
i) \(\frac{3}{4}\), \(\frac{3}{2}\), \(\frac{2}{3}\), \(\frac{1}{5}\), \(\frac{18}{7}\)
ii) \(\frac{2}{7}\), \(\frac{3}{8}\), \(\frac{3}{4}\), \(\frac{5}{7}\), \(\frac{4}{9}\)
Answer:
Given fractions are \(\frac{3}{4}\), \(\frac{3}{2}\), \(\frac{2}{3}\), \(\frac{1}{5}\), \(\frac{18}{7}\)
These are unlike fractions.
To arrange the unlike fractions in ascending order / descending order first we have to convert them into equivalent fractions with LCM of their denominators, then compare the like fractions (i.e.,) we convert them to like fractions.
LCM of Denominators = 2 × 2 × 3 × 5 × 7 = 420

\(\frac{315}{420}\), \(\frac{630}{420}\), \(\frac{280}{420}\), \(\frac{84}{420}\), \(\frac{1080}{420}\)
These are like fractions. Now, we can compare them.
\(\frac{84}{420}\)< \(\frac{280}{420}\) < \(\frac{315}{420}\) < \(\frac{630}{420}\) < \(\frac{1080}{420}\)
i.e, \(\frac{1}{5}\) < \(\frac{2}{3}\) < \(\frac{3}{4}\) < \(\frac{3}{2}\) < \(\frac{18}{7}\)
∴ Ascending order: \(\frac{1}{5}\), \(\frac{2}{3}\), \(\frac{3}{4}\), \(\frac{3}{2}\), \(\frac{18}{7}\)
Descending order: \(\frac{18}{7}\), \(\frac{3}{2}\), \(\frac{3}{4}\), \(\frac{2}{3}\), \(\frac{1}{5}\)

ii) \(\frac{2}{7}\), \(\frac{3}{8}\), \(\frac{3}{4}\), \(\frac{5}{7}\), \(\frac{4}{9}\)
These are unlike fractions.
To arrange the unlike fractions in ascending order / descending order first we have to convert them into equivalent fractions with LCM of their denominators. Then compare the like fractions (i.e.,) we convert them to like fractions.
LCM of denominators = 2 × 2 × 2 × 3 × 3 × 7 = 504

\(\frac{144}{504}\), \(\frac{189}{504}\), \(\frac{378}{504}\), \(\frac{360}{504}\), \(\frac{224}{504}\)
These are like fractions. Now, we can compare them.
\(\frac{144}{504}\) < \(\frac{189}{504}\) < \(\frac{224}{504}\) < \(\frac{360}{504}\) < \(\frac{1080}{420}\)
i.e, \(\frac{2}{7}\) < \(\frac{3}{8}\) < \(\frac{4}{9}\) < \(\frac{5}{7}\) < \(\frac{3}{4}\)
∴ Ascending order: \(\frac{2}{7}\), \(\frac{3}{8}\), \(\frac{4}{9}\), \(\frac{5}{7}\), \(\frac{3}{4}\)
Descending order: \(\frac{3}{4}\), \(\frac{5}{7}\), \(\frac{4}{9}\), \(\frac{3}{8}\), \(\frac{2}{7}\)

Question 3.
Without doing calculation, find the result \(\frac{2}{3}\) + 1\(\frac{3}{4}\) + \(\frac{1}{3}\) – \(\frac{1}{4}\)
Answer:
Given \(\frac{2}{3}\) + 1\(\frac{3}{4}\) + \(\frac{1}{3}\) – \(\frac{1}{4}\)

Question 4.
Neha bought a cake. She ate \(\frac{7}{15}\) th of the cake immediately add in the afternoon she ate the remaining part. How much part die ate in the afternoon ?
Answer:
Whole cake = 1 = \(\frac{15}{15}\)
Neha divided the cake into 15 parts.
Part of a cake eaten by Neha = \(\frac{7}{15}\)
Remaining part of cake = Whole – eaten part immediately
= \(\frac{1}{1}\) – \(\frac{7}{15}\)
= \(\frac{15}{15}\) – \(\frac{7}{15}\)
= \(\frac{15-7}{15}\)
= \(\frac{8}{15}\)
∴ Part of a cake eaten by Neha in the afternoon = \(\frac{8}{15}\)

Question 5.
Simplify:
i) \(\frac{2}{5}\) + \(\frac{1}{3}\)
ii) \(\frac{5}{7}\) + \(\frac{2}{3}\)
iii) \(\frac{3}{5}\) – \(\frac{7}{20}\)
iv) \(\frac{17}{20}\) – \(\frac{13}{25}\)
Answer:
i) \(\frac{2}{5}\) + \(\frac{1}{3}\)
LCM of denominators = 3 × 5 = 15

ii) \(\frac{5}{7}\) + \(\frac{2}{3}\)
LCM of denominators = 7 × 3 = 21

iii) \(\frac{3}{5}\) – \(\frac{7}{20}\)
LCM of denominators = 2 × 2 × 5 = 20

iv) \(\frac{17}{20}\) – \(\frac{13}{25}\)
LCM of denominators = 2 × 2 × 5 × 5 = 100

Question 6.
Represent \(\frac{16}{5}\) pictorially
Answer:
Given fraction is \(\frac{16}{5}\) (Improper fraction)
\(\frac{16}{5}\) = Mixed fraction is 3\(\frac{1}{5}\)

3 + \(\frac{1}{5}\) = 3\(\frac{1}{5}\)

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 1 Numbers All Around us Ex 1.3 Textbook Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 1st Lesson Numbers All Around us Exercise 1.3

Question 1.
Write each of the following numbers in digits by using International place value chart. Also write them in expanded form.
i) Nine million seven hundred thousand and six hundred five.
Answer:
9,700,605: 9,000,000 + 700,000 + 600 + 5

ii) Seven hundred million eight hundred seventy two thousand and four hundred seven.
Answer:
700,872,407: 700,000,000 + 800, 000 + 70,000 + 2000 + 400 + 7

Question 2.
Rewrite each of the following numerals with proper commas in the International system of numeration and write the numbers in word form.
i) 717858
ii) 3250672
iii) 75623562
iv) 956237676
Answer:
i) 717,858: Seven hundred seventeen thousand eight hundred and fifty eight.
ii) 3,250,672: Three million two hundred fifty thousand six hundred and seventy two.
iii) 75,623,562: Seventy five million six hundred twenty three thousand five hundred and sixty two.
iv) 956,237,676: Nine hundred fifty six million two hundred thirty seven thousand six hundred and seventy six.

Question 3.
Write the following number names in both Indian and International systems.
i) 6756327
ii) 45607087
iii) 8560707236
Answer:
i) 6756327
Indian system:
67,56,327: Sixty seven lakh fifty six thousand three hundred and twenty seven.
International system:
6,756,327: Six million seven hundred fifty six thousand three hundred and twenty seven.

ii) 45607087 Indian system:
4,56,07,087: Four crores fifty six lakhs seven thousand eighty seven.
International system:
45,607,087: Forty five million six hundred seven thousand and eighty seven.

iii) 8560707236 Indian system:
856,07,07,236: Eight hundred fifty six crores seven lakh seven thousand two hundred and thirty six.
International system:
8,560,707,236: Eight billion five hundred sixty million seven hundred seven thousand two hundred and thirty six.

Question 4.
Express the following numbers in other system.

S.No.	Indian	International
1.	42,56,876
2.		800,000,000
3.	956,76,72,345
4.		6,303,448,433

Answer:

S.No.	Indian	International
1.	42,56,876	4,256,876
2.	80,00,00,000	800,000,000
3.	956,76,72,345	9,567,672,345
4.	630,34,48,433	6,303,448,433

Question 5.
Write the following numbers in International system (Word Form).
i) Twenty Nine crore thirty five lakh forty six thousand seven hundred and fifty three.
Answer:
293,546,753
Word form: Two hundred ninety three million five hundred forty six thousand seven hundred and fifty three.

ii) Thousand crore ninety nine lakh and forty three.
Answer:
10,009,900,043
Word form: Ten billions nine million nine hundred thousand and forty three.

Question 6.
Write following numbers in Indian system (Word Form).
i) Nine billion twenty four million fifty thousand and seventy two.
Answer:
902,40,50,072
Word form: Nine hundred two crores forty lakhs fifty thousand and seventy two.

ii) Seven hundred billions six millions four thousand seven hundred and five.
Answer:
70000,60,04,705
Word form: Seventy thousand crores sixty lakhs four thousand seven hundred and five.

AP State Syllabus AP Board 6th Class Maths Solutions Chapter 1 Numbers All Around us Ex 1.2 Textbook Questions and Answers.

AP State Syllabus 6th Class Maths Solutions 1st Lesson Numbers All Around us Exercise 1.2

Question 1.
Write each of the following in numeral form.
i) Sixty crores seventy five lakhs ninety two thousands five hundred and two.
Answer:
60, 75, 92, 502

ii) Nine hundred forty four crores six lakhs fifty five thousand four hundred and eighty six.
Answer:
944, 06, 55, 486

iii) Ten crores ten thousand and ten.
Answer:
10,00,10,010

Question 2.
Insert commas in the correct positions to separate periods and write the following numbers in words.
i) 57657560
ii) 70560762
iii) 97256775613
Answer:
i) 5,76,57,560: Five crores seventy six lakhs fifty seven thousands five hundred and sixty.
ii) 7,05,60,762: Seven crores five lakhs sixty thousands seven hundred and sixty two.
iii) 9725,67,75,613: Nine thousand seven hundred and twenty five crores sixty seven lakhs seventy five thousand six hundred and thirteen.

Question 3.
Write the following in expanded form.
i) 756723
ii) 60567234
iii) 8500756762
Answer:
i) 756723
Expanded form: 7 × 1,00,000 + 5 × 10,000 + 6 × 1,000 + 7 × 100 + 2 × 10 + 3 × 1
: 7 lakhs + 5 ten thousands + 6 thousands + 7 hundreds + 2 tens + 3 ones
Word form: Seven lakh fifty six thousand seven hundred and twenty three.

ii) 60567234
Expanded form: 6 × 1,00,00,000 + 5 × 1,00,000 + 6 × 10,000 + 7 × 1,000 + 2 × 100 + 3 × 10 + 4 × 1
: 6 crores + 5 lakhs + 6 ten thousands + 7 thousands + 2 hundreds + 3 tens + 4 ones
Word form: Six crore five lakh sixty seven thousand two hundred and thirty four.

iii) 8500756762
Expanded form: 8 × 1,00,00,00,000 + 5 × 10,00,00,000 + 7 × 1,00,000 + 5 × 10,000 + 6 × 1000 + 7 × 100 + 6 × 10 + 2 × 1
: 8 hundred crores + 5 ten crores + 7 lakhs & 5 ten thousands + 6 thousands + 7 hundreds + 6 tens + 2 ones
Word form: Eight hundred and fifty crore seven lakh fifty six thousand seven hundred and sixty two.

Question 4.
Determine the difference between the place value and the face value of 6 in 86456792.
Answer:
Given number is 86456792. By putting commas to separate periods the given number can be written as 8,64,56,792.
i) Place value of ‘6’ in thousand place = 6 x 1000 = 6,000
Face value of 6 = 6
Difference = 6,000 – 6 = 5,994
ii) Place value of ‘6’ in ten lakhs palce = 6 x 10,00,000 = 60,00,000
Face value of 6 = 6
Difference = 60,00,000 – 6 = 59,99,994