Tag: average

Quizzes (2) Questions (95)

Questions Related to average

Multiple choice maths average arithmetic mean of ap introduction to averages means

State the following statement is True or False
Arithmetic mean of first five natural numbers is $3$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

First five natural numbers are $1 ,2 , 3 ,4, 5$

arithmetic mean = $\dfrac { sum\quad of\quad numbers }{ count\quad of\quad numbers } $
$ = \dfrac{1 + 2 + 3 + 4  + 5 }{ 5} $
$=\dfrac { 15 }{ 5 } =3$
Arithmetic mean of first five natural number is 3.
Hence the given statement is true.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean of the squares of the first $n$ natural numbers is

  1. $\dfrac { n\left( n+1 \right) \left( 2n+1 \right) }{ 6 } $
  2. $\dfrac { n\left( n+1 \right) \left( 2n+1 \right) }{ 2 } $
  3. $\dfrac { \left( n+1 \right) \left( 2n+1 \right) }{ 6 } $
  4. $\dfrac { \left( n+1 \right) \left( 2n+1 \right) }{ 3 } $
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Sum of squares of the first $n$ natural numbers $=\dfrac { n(n+1)(2n+1) }{ 6 } $


Their arithmetic mean is $\dfrac { Sum }{ n } =\dfrac { n(n+1)(2n+1) }{ 6n } =\dfrac { (n+1)(2n+1) }{ 6 } $
Hence, C is correct.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The middle terms , if four different numbers are in proportion are called ______ .

  1. Antecedents

  2. Means

  3. Extremes

  4. Consequents
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\Rightarrow$  The middle terms  of four different numbers are in proportion are called $Means.$

$\Rightarrow$  A proportion is simply a statement that two ratios are equal. It can be written in two ways:  $\dfrac{a}{b} = \dfrac{c}{d}$ or  $a:b = c:d.$
$\Rightarrow$  Here, outer terms $a$ and $d$ called extremes.
$\Rightarrow$  Middle terms $b$ and $c$ are called means.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean between $\cfrac { x+a }{ x } $ and $\cfrac { x-a }{ x } $ when $x\ne 0$, is (the symbol $\ne$ means "not equal to"):

  1. $2$, if $a\ne 0$
  2. $1$
  3. $1$, only if $a=0$
  4. $\dfrac {a}{x}$
  5. $x$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The arithmetic mean of two numbers a and b is (a + b) / 2. Here, ((x+a)/x + (x-a)/x) / 2 = (2x/x) / 2 = 2 / 2 = 1.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The arithmetic mean (average) of a set of $50$ numbers is $38$. If two numbers, namely, $45$ and $55$, are discarded, the mean of the remaining set of numbers is :

  1. $36.5$
  2. $37$
  3. $37.2$
  4. $37.5$
  5. $37.52$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

$Arithmetic\quad sum=\cfrac { sum\quad of\quad numbers }{ number\quad of\quad numbers } $
$\therefore S=50\times 38=1900;\quad \therefore x=\cfrac { 1900-45-55 }{ 48 } =37.5$

Multiple choice maths average arithmetic mean of ap introduction to averages means

If $A _1,A _2$ be two arithmetic means between $\dfrac{1}{3}$ and $\dfrac{1}{24}$, then their value are 

  1. $\dfrac{7}{72},\dfrac{5}{36}$
  2. $\dfrac{17}{72},\dfrac{5}{36}$
  3. $\dfrac{7}{36},\dfrac{5}{72}$
  4. $\dfrac{5}{72},\dfrac{17}{72}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

For two means A1, A2 between 1/3 and 1/24, the common difference d = (1/24 - 1/3) / (2 + 1) = (-7/24) / 3 = -7/72. A1 = 1/3 - 7/72 = 17/72. A2 = 17/72 - 7/72 = 10/72 = 5/36.

Multiple choice maths average arithmetic mean of ap introduction to averages means

Sum of $4$ numbers in GP is $60$. And the AM of first and last no. is $18$ find the first term and common difference of the GP

  1. $a=4, r=2$
  2. $a=32, r=\dfrac {1}{2}$
  3. $a=3, r=1$
  4. $a=6, r=3$
Reveal answer Fill a bubble to check yourself
A,B Correct answer
Explanation

Let 4 numbers in G.P. be $a,ar,{ ar }^{ 2 }{ ,ar }^{ 3 }\ a(1+r+{ r }^{ 2 }+{ r }^{ 3 })=60\ \cfrac { a+{ ar }^{ 3 } }{ 2 } =18=>a+{ ar }^{ 3 }=36\ a(1+r)(1+{ r }^{ 2 })=60\ a(1+r)(1+{ r }^{ 2 }-r)=36\ \cfrac { 1+{ r }^{ 2 }-r }{ 1+{ r }^{ 2 } } =\cfrac { 36 }{ 60 } \ 5+5{ r }^{ 2 }-5r=3+3{ r }^{ 2 }\ 2{ r }^{ 2 }-5r+2=0\ r=2,\cfrac { 1 }{ 2 } \ if\quad r=2,a(1+8)=36=>a=4\ if\quad r=\cfrac { 1 }{ 2 } ,a(1+\cfrac { 1 }{ 8 } )=36=>a=32\ a=4,r=2\quad (or)\quad a=32,r=\cfrac { 1 }{ 2 } $

Multiple choice maths average arithmetic mean of ap introduction to averages means

The A.M. of the observations $1.3.5, 3.5.7, 5.7.9,...,(2n-1)(2n+1)(2n+3)$ is $(\forall n\in N)$

  1. $2n^3+6n^2+7n-2$
  2. $n^3+8n^2+7n-2$
  3. $2n^3+5n^2+6n-1$
  4. $2n^3+8n^2+7n-2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$1.3.5, 3.5.7, 5.7.9, ........, (2n-1)(2n+1)(2n+3) $
 $x _1$       $x _2$       $x _3$                                   $x _n=n \,  terms$

$AM=\displaystyle\sum _{i=1}^{n}\dfrac{x _{i}}{n}$

$\displaystyle\sum _{i=1}^{n}x _{i}=\sum _{r=1}^{n}(2r-1)(2r+1)(2r+3)$

$=\displaystyle\sum _{r=1}^{n} 8r^{3}-2r+12r^{2}-3$

$=\displaystyle 8\sum _{r=1}^{n}+12\sum _{r=1}^{n}r^{2}-2\sum _{r=1}^{n}-3\sum _{r=1}^{n}(1) ....... (1)$

$1^{2}+2^{2}+.........+x^{2}=\dfrac{n(2n+1)(n+1)}{6}=\displaystyle\sum _{r=1}^{n}r^{2}$

$\displaystyle\sum _{r=1}^{n}r^{3}=1^{3}+2^{3}+......+ n^{3}=\left(\dfrac{n(n+1)}{2}\right)^{2}$

$\displaystyle\sum _{r=1}^{n}r=1+2+3+.......+ n=\dfrac{n(n+1)}{2}$

from $(1)$

$8\left[\dfrac{n(n+1)}{2}\right]^{2}+12\left[\dfrac{n(2n+1)(n+1)}{6}\right]-\dfrac{2n(n+1)}{2}-3n$

$=2(n(n+1))^{2}+2n(2n+1)(n+1)-n(n+1)-3n$

$=n(n+1)[2n^{2}+2n+4n+2-1]-3n$

$\displaystyle\sum =n(n+1)[2x^{2}+6x+1]-3n$

$A.M=\dfrac{n(n+1)(2n^{2}+6n+1)-3n}{n}$

$=(n+1)(2n^{2}6n+1)-3$

$=2n^{3}+6n^{2}+n-2n^{2}+6n+1-3=2n^{3}+8n^{2}+7n-2$
Multiple choice maths average arithmetic mean of ap introduction to averages means

If $n\ AM's$ are inserted between $1$ and $31$ and ratio of ${7}^{th}$ and $(n-1)^{th}$ $A.M.$ is $5:9$ then $n$ equals ?

  1. $12$
  2. $13$
  3. $14$
  4. $None$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using the formula for the k-th arithmetic mean between a and b, A_k = a + k(b-a)/(n+1). Setting up the ratio A_7 / A_(n-1) = 5/9 and solving for n yields n = 14.