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

The average weight of 45 students in a class is 25 kg. Five of them whose average weight is 48 kg leave the class and other 5 students whose average weight is 54 kg join the class. What is the new average weight (in kg) of the class?

  1. $52\frac{1}{3}$
  2. $52\frac{1}{2}$
  3. $52\frac{2}{3}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Multiple choice maths average arithmetic mean of ap introduction to averages means

If n A.M.'s are inserted between 3 and 17 such that the ratio of the last mean to the first mean is 3:1, then the value of n is

  1. 4

  2. 6

  3. 8

  4. 9
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When n arithmetic means are inserted between a and b, the common difference is d = (b - a) / (n + 1). The first mean is a + d and the last mean is b - d. Setting up the given ratio of last mean to first mean equal to 3:1 allows us to solve for n, yielding n = 6.

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

Four numbers are in proportion. The sum of the square of the four numbers is $50$ and the sum of the means is $5$. The ratio of first two terms is $1 : 3$. What is the average of the four numbers ?

  1. $2$
  2. $3$
  3. $5$
  4. $6$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let the four numbers in proportion be a, b, c, d, meaning ad = bc. The sum of the means (b + c) is 5, and the sum of their squares is a^2 + b^2 + c^2 + d^2 = 50. Using the ratio of the first two terms a : b = 1 : 3, we can find the values of the terms, leading to numbers whose sum is 12 and average is 12 / 4 = 3.

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

A line is such that the algebraic sum of the perpendicular on it from a number of point is zero. The line always passes through a fixed point that is 

  1. A. M of the given points

  2. GM of the given points

  3. HM of the given points

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The algebraic sum of perpendiculars from points (xi, yi) to a line ax + by + c = 0 being zero implies the line passes through the centroid (mean of x, mean of y).

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

Coefficient of variance of a distribution is 60% and the standard deviation is 25. The arithmetic mean of the distribution is

  1. $\cfrac {25}{3}$

  2. 35
  3. $\cfrac {125}{3}$
  4. $\cfrac {25}{6}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The coefficient of variation (CV) is given by (Standard Deviation / Mean) * 100. Rearranging the formula for the mean gives Mean = (Standard Deviation / CV) * 100 = (25 / 60) * 100 = 125 / 3.

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

If the arithmetic mean of the numbers $x _{1}, x _{2}, x _{3}.......,x _{3}$ is $\overline{X}$, then the arithmetic mean of numbers $ax _{1}+b, ax _{2}+b, ax _{3}+b,........, ax _{n}+b$, where  $a, b$ are two constants would be 

  1. $\overline{X}$
  2. $na\overline{X}+nb$
  3. $a\overline{X}$
  4. $a\overline{X}+b$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Arithmetic mean is affected by linear transformations in the same way. If each data point is multiplied by a constant a and has a constant b added to it, the new mean will be a times the old mean plus b, which is aX_bar + b.

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

If the total incomes of $M,N,O,P$ are in the rate a $2:3:4:5$ and the total income of $M$ is $Rs\ 8000$ , then find approximate average salary of all four?

  1. $Rs\ 14000$
  2. $Rs\ 7000$
  3. $Rs\ 9950$
  4. $Rs\ 4875$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Ratios 2:3:4:5. M = 2x = 8000, so x = 4000. Total = (2+3+4+5)x = 14x = 14 * 4000 = 56000. Average = 56000 / 4 = 14000.

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

The artimetic mean of $2 sin 2^o, 4 sin 4^o, 6sin6^o,...,180sin 180^o$ is equal to 

  1. $cosec1^o$
  2. $sec1^o$
  3. $cot1^o$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

This is a complex trigonometric series sum. The arithmetic mean of the given series is indeed cosec(1 degree).