The given numbers are:
1, 3, 6, 10, 15, .
1, (1 + 2), (3 + 3), (6 + 4), (10 + 5), .....
∴ Next number in the sequence
= 15 + 6 = 21.

We have, 5.46 - 1.85
Now,
\(Ones\quad Tenths\quad Hundreaths\\\quad5\quad\quad\quad\quad4\quad\quad\quad\quad\quad6\\-1\quad\quad\quad\quad8\quad\quad\quad\quad\quad5\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\\\quad3\quad\quad\quad\quad6\quad\quad\quad\quad\quad1\\\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\)
\(\therefore\)    5.46 - 1.85 = 3.61

This figure has two lines of symmetry.

Given, area = 48 cm² and breadth (b) = 6 cm
We know that, Area = Length \(\times\)Breadth\(\Rightarrow Length=\frac { Area }{ Breadth } =\frac { 48 }{ 6 } =8cm\)
Hence, length of the cardboard is 8 cm.

Given, number is 97621.
Hence, place value of 7 is 7000 and face value of 7 is 7.

Suppose that initially (3x + 1) mangoes were pluck. [because when the collected mangoes were divided into three equal numbers, then there was one mango extra.]
Each equal part = x
Part got by first = x
Number of mangoes given to monkey = 1
∴ Number of rest mangoes after first division
=(3x+ 1)-(x+ 1)=2x
In second distribution
Number of mangoes given to monkey = 1
Rest mangoes = 2x - 1
On dividing into three equal parts, each part
= \(\frac { 2x-1 }{ 3 } \)
∴ Part obtained by second = \(\frac { 2x-1 }{ 3 } \)
Number of rest mangoes after second division
= \(\frac { 2x-1 }{ 3 } +\frac { 2x-1 }{ 3 } \)
= \(\frac { 2x-1+2x-1 }{ 3 } =\frac { 4x-2 }{ 3 } \)
In third distribution
Number of mangoes given to monkey = 1
∴ Rest mangoes = \(\frac { 4x-2 }{ 3 } -1=\frac { 4x-5 }{ 3 } \)
On dividing into three equal part, each part
=\(\frac { 1 }{ 3 } \left( \frac { 4x-1 }{ 3 } \right) \)
= \(\frac { 4x-5 }{ 9 } \)
∴ Part obtained by third =\(\frac { 4x-5 }{ 9 } \)
Number of rest mangoes afte r third division
= \(\frac { 4x-5 }{ 9 } +\frac { 4x-5 }{ 9 } \)
= \(\frac { 4x-5+4x-5 }{ 9 } =\frac { 8x-10 }{ 9 } \)
According to the question,
\(\frac { 8x-10 }{ 9 } \) =30
⇒ 8x-10 = 270
⇒ 8x= 280
⇒ x =\(\frac { 280 }{ 8 } \)
⇒ x= 35
∴ 3x + 1 = 3 x 35 + 1 = 105 + 1 = 106
Hence, 106 mangoes were plucked initially.

Three

Let the cost of 1 dozen be Rs x.
Then, cost of 3 dozen = Rs 3x
According to the question,
\(3x=54 \Rightarrow x=\frac{54}{3}=18\)
∴ The cost of one dozen = Rs 18
Hence, Rehman can buy 1 dozen of bananas.
(b) Humanity, Helpfulness, caring for others.

We have, 6895, 23787, 24569, 24659
Here, 6895 is a 4-digit number and other numbers are 5 digits, so it is clear that 6895 is the smallest number. Now, in remaining three numbers, the digit at ten thousands place is same in each number. So, we compare the thousands place digit.

∵ 4>3
Again, in two numbers, digit at thousands place is same. So, we compare the hundreds place digit.

∵ 6>5
∴ The greatest number is 24659 and smallest number is 6895.

Ratio of income to expenditure = 7 : 5
Let the income and expenditure per month be Rs 7x and 5x respectively.
Then, saving per month = Rs 7x - Rs 5x = Rs 2x
According to the question,
2x= 2000
⇒ x =\(\frac{2000}{2}\) =1000
⇒ 7x = 7 x 1000 = 7000
Hence, his income is Rs 7000 per month.

Vertical

We have, 1258
(i) Divisibility by 2
\(\because\)Unit digit of number = 8, so 1258 is divisible by 2.
(ii) Divisibility by 3
Sum of digits of given number = 1+ 2 + 5 + 8 = 16 is not divisible by 3, so 1258 is not divisible by 3. Now, we see that 1258 is divisible by 2 but not divisible by 3.
Hence, it is not divisible by 6.

According to the patterns of whole number,