The numbers $3,5,7,4 $ have frequencies $x,x+4,x-3,x+8.$ If their arithmetic mean is $4 $, the value of $x$ is
- $\displaystyle \frac{5}{3}$
- $\displaystyle \frac{7}{4}$
- $\displaystyle \frac{2}{3}$
-
none of these
Mean = (Sum of (value * frequency)) / (Sum of frequencies) = 4. (3x + 5(x+4) + 7(x-3) + 4(x+8)) / (x + x+4 + x-3 + x+8) = 4. (3x + 5x + 20 + 7x - 21 + 4x + 32) / (4x + 9) = 4. (19x + 31) / (4x + 9) = 4. 19x + 31 = 16x + 36. 3x = 5. x = 5/3.
The formula for the arithmetic mean is the sum of the observations multiplied by their frequencies divided by the total frequency. The numerator becomes 3x plus 5(x+4) plus 7(x-3) plus 4(x+8), which simplifies to 3x plus 5x plus 20 plus 7x minus 21 plus 4x plus 32, equaling 19x plus 31. The denominator is x plus (x+4) plus (x-3) plus (x+8), which simplifies to 4x plus 9. Setting the mean to 4 gives the equation (19x + 31)/(4x + 9) = 4; solving this yields 19x plus 31 equals 16x plus 36, so 3x equals 5 and x equals 5/3.