Question

If $x_1,x_2,x_3,\dots ,x_n$ are n-values of a variable x such that ${\sum }_{i=1}^n{u}_i=90$ and ${\sum }_{i=1}^n{V}_i=150$ where $u_i=2x_i–3$ and $V_i=3x_i–4$, for i = 1, 2, 3, 4, …, n, then the value of n will be

• A. 30
• B. 40
• C. 50
• D. 60

Answer 1

The correct option is A 30

Given: ${\sum }_{i=1}^n{u}_i=90$ and ${\sum }_{i=1}^n{V}_i=150$
where, $u_i=2x_i–3$ and $V_i=3x_i–4$; i = 1, 2, 3, 4, …, n,
Now,${\sum }_{i=1}^n{u}_i=90$
$\Rightarrow {\sum }_{i=1}^n(2x_i-3)=90$
$\Rightarrow 2{\sum }_{i=1}^n{x}_i-3n=90$
$\Rightarrow 2(x_1+x_2+.....+x_n)-3n=90$
$\Rightarrow x_1+x_2+.....+x_n=\frac{90+3n}{2}$.....(i)
Similarly, ${\sum }_{i=1}^n{V}_i=150$
$\Rightarrow {\sum }_{i=1}^n(3x_i-4)=150$
$\Rightarrow 3{\sum }_{i=1}^n{x}_i-4n=150$
$\Rightarrow 3(x_1+x_2+.....+x_n)-4n=150$
$\Rightarrow x_1+x_2+.....+x_n=\frac{150+4n}{3}$.....(ii)
From (i) and (ii), we get
$\frac{90+3n}{2}=\frac{150+4n}{3}$
$\Rightarrow 270+9n=300+8n$
$\Rightarrow n=30$
Hence, the correct answer is option (a).