Let one part be $x$.

Then, other part = $28-x$

Given that
$\frac{6}{5}$ of one part = $\frac{2}{3}$ of the other
$\Rightarrow \frac{6}{5}x=\frac{2}{3}(28-x)$
$\Rightarrow \left(\frac{6\times 3}{2}\right)x=5(28-x)$
$\Rightarrow 9x=5(28-x)$
$\Rightarrow 9x=140-5x$
$\Rightarrow 14x=140$
$\Rightarrow x=10$

The other part would be 28 - 10 = 18
Hence, the smaller part is 10.

$\text{Given, the sum of the digits of a two-digit number is 7.}$
$\text{Let the units digit of the}\text{original number be}x.$
$\text{Then, the tens digit of the}\text{original number is}7-x.$
$\text{The two-digit number}=10(7-x)+(1\times x)$
$=70-10x+x=70-9x$

$\text{When the digits are interchanged,}\text{the new number is}10\left(x\right)+1(7-x)=10x-x+7=9x+7$
$\text{New number = Original number - 27}$

$9x+7=70-9x-2718x=36x=2$

$\text{The tens digit}=7-x=7-2=5$

$\therefore \text{The original number is 52}$.

In the given equation, x is the variable (whose value has to be found)
2x + 3 = 3x + 2

3x - 2x = 3 - 2

x = 1

Let the numerator of a rational number be $x$.
Then the denominator of a rational number would be $(x+3)$.
When the numerator is increased by $7$, it becomes $(x+7)$ 
When the denominator is decreased by 1, it becomes
$\Rightarrow x+3-1=x+2$
The new number formed $=\frac{3}{2}$
$\text{So,}\frac{(x+7)}{(x+2)}=\frac{3}{2}2x+14=3x+63x-2x=14-6x=8$

$\therefore$ Numerator, $x=8$
Denominator, $x+3=8+3=11$
Hence, the original rational number is $\frac{8}{11}$

Let us assume the first multiple of 4 as 4x. Hence, the next two multiples are 4x + 4 and 4x + 8

According to the question,

$4x+(4x+4)+(4x+8)=444$

$\Rightarrow 12x+12=444$
$\Rightarrow 4(3x+3)=4\left(111\right)$

$\Rightarrow 3x+3=111$

$\Rightarrow 3x=108$

$\Rightarrow$ x = 36

$⟹4x=4\times 36=144$,
$4x+4=144+4=148$ and
$4x+8=144+8=152$

Therefore, the multiples are 144, 148 and 152.

Given, 
$\frac{x+1}{x+4}=\frac{2}{3}$
On cross multiplication, we get,
$3\times (x+1)=2\times (x+4)\Rightarrow 3x+3=2x+8$
Transposing $2x$ from RHS to LHS and 3 from LHS to RHS, we get,
$3x-2x=8-3x=5$

If the degree of an equation is one, then that equation is called a linear equation. In all the equations given, the degree of each equation is one as the highest exponent of variables in a term is one.
Equations which have only one variable or which can be reduced to equations having only one variable are known as linear equations in one variable.
Since, 4a + 3b = 3b + 6 can be reduced to a linear equation in variable "a" by subtracting "3b" from both sides, this is a linear equation in one variable.
Also, 2x = 3x + 4 is a linear equation in one variable as it has only one variable, x. 
The other equations are also linear but they have two variables, so they are not linear equations in one variable.

Total number of gifts = 150

Let the number of gifts worth ₹50 be x

Then the number of gifts worth ₹100 would be (150 - x)

Amount spent on x gifts of ₹ 50 = ₹50x

Amount spent on (150 - x) gifts of ₹100 = ₹100(150 - x)

So, 50x + 100 (150 - x) = 10000

50x + 15000 - 100x = 10000

50x = 15000 - 10000 = 5000

x = 100

The total number of ₹50 gifts = x = 100 and hence the given statement is False.

Let the present age of Pratik be 2x and Arjun be 3x.

4 years from now, Pratik 's age = 2x + 4 and Arjun 's age = 3x + 4

Given that, $\frac{2x+4}{3x+4}=\frac{4}{5}$

$⟹10x+20=12x+16$

Thus, $2x=4⟹x=2$

Pratik's present age is $2x=4$.

The equation is, 7x - 2 = 2x + 8

or 7x-2x = 8+2

or 5x = 10

x = 2