Question

the mean is 42 find the missing frequency of X and y if the total frequency is 100.
class interval 0-10, 10-20 ,20-30, 30-40 ,40-50 ,50-60 ,60-70, 70-80
frequency 7, 10, X ,13, y, 10, 14, 9​

Answer 1

$\sf\large\undeline{Given:}$

$\sf{\implies Mean=42}$

$\sf{\implies Total frequency=100}$

$\sf\large\undeline{To\:Find:}$

• The value of missing frequency=?]

$\sf\large\undeline{Solution:}$

• To calculate the missing frequency at first we have to make a table after that we have to calculate the value of x and y to calculate the value of x and y , we have to set up equation then solve.]

$\begin{array}{|c|c|c|c|} \cline{1-4} \sf CI & \sf f & \sf x & \sf fx \\ \cline{1-4} 0-10 & 7 & 5 & 35 \\ \cline{1-4} 10-20 & 10 & 15 & 150 \\ \cline{1-4} 20-30 & \sf X & 25 & \sf 25X \\ \cline{1-4} 30-40 & 13 & 35 & 455 \\ \cline{1-4} 40-50 & \sf Y & 45 & \sf 45Y \\ \cline{1-4} 50-60 & 10 & 55 & 550 \\ \cline{1-4} 60-70 & 14 & 65 & 910 \\ \cline{1-4} 70 - 80 & 9 & 75 & 675 \\ \cline{1-4} & \sf \sum\:f=63 + X + Y & & \sf \sum\:f\:x=2775 + 25X + 45Y \\ \cline{1-4} \end{array}$

By calculating we get here:]

$\sf{\implies \sum\:f\:x=2775+25x+45y}$

$\sf{\implies \sum\:f=63+x+y}$

$\sf{\implies setting\:up\: first\: equation:}$

$\sf{\implies Total\: frequency=63+x+y}$

$\sf{\implies 100=63+x+y}$

$\sf{\implies x+y=100-63}$

$\sf{\implies x+y=37-------(i)}$

$\sf{\implies setting\:up\:2nd\: equation:}$

• To calculate 2nd equation we have to apply the formula of mean:]

$\tt{\implies Mean=\dfrac{\sum\:f\:x}{\sum\:f}}$

$\tt{\implies 42=\dfrac{2775+25x+45y}{63+x+y}}$

$\tt{\implies 2646+42x+42y=2775+25x+45y}$

$\tt{\implies 42x-25x+42y-45y=2775-2646}$

$\tt{\implies 17x-3y=129----(ii)}$

Now solving here , equation:]

• In eq (I) multiplying by 17 the subtracting from eq (ii)

$\tt{\implies 17x+17y=629}$

$\tt{\implies 17x-3y=129}$

• By solving we get , here]

$\tt{\implies 20y=500\implies y=25}$

• Putting the value of y=25 in eq (i)]

$\sf{\implies x+y=37}$

$\sf{\implies x+25=37}$

$\sf{\implies x=37-25\implies x=12}$

$\sf\large{Hence,}$

$\sf\purple{\implies The\: missing\: frequency=25\:and\:12}$

Answer 2

Answer:

$\boxed{\begin{array}{ccccc}\sf Class\: interval&\sf Frequency&\sf Class\:marks&\sf f_i\:x_i\\\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}&\frac{\qquad \qquad \qquad \qquad\qquad}{}\\\sf 0-10&\sf 7&\sf5&\sf 35 \\\\\sf 10-20 &\sf 10&\sf 15 &\sf 150 \\\\\sf 20-30 &\sf x &\sf 25 &\sf 25x\\\\\sf 30-40&\sf 13&\sf 35&\sf 455\\\\\sf 40-50 &\sf y &\sf 45&\sf 45y \\\\\sf 50-60 &\sf 10 &\sf 55 &\sf 550\\\\\sf 60-70 &\sf 14 &\sf 65&\sf 910 \\\\\sf 70-80 &\sf 9 &\sf 75&\sf 675\\\frac{\qquad\qquad \qquad \qquad}{}&\frac{\qquad \qquad \qquad\qquad \qquad}{}&\frac{\qquad \qquad \qquad}{}&\frac{\qquad \qquad\qquad \qquad \qquad \qquad \qquad\qquad}{}\\\\\sf Total &\sf \sum\:f_i=63+x+y &\sf &\sf \sum f_i\:x_i=2775+25x+45y\end{array}}$

$\rule{130}{1}$

Sum of Frequencies = 100

$:\implies\sf \sum i \ fi= 100\\\\\\:\implies\sf 63+x+y =100\\\\\\:\implies\sf x+y = 100-63\\\\\\:\implies\sf x+y= 37\\\\\\:\implies\sf x= 37-y\:\:\:\Bigg\lgroup\bf{Equation\:(1)}\Bigg\rgroup$

$\rule{170}{2}$

Now, the mean of the given data is given by,

$\dashrightarrow\sf\:\:x = \dfrac{\sum\limits_i\:fi\:xi}{\sum\limits_i\:fi}\\\\\\\dashrightarrow\sf\:\:42 = \dfrac{2775+25x+45y}{100}\\\\\\\dashrightarrow\sf\:\:4200-2775= 25x + 45y\\\\\\\dashrightarrow\sf\:\:1425= 25(37-y) +45y\:\:\:\Bigg\lgroup\bf{From\: Equation\:(1)}\Bigg\rgroup\\\\\\\dashrightarrow\sf\:\:1425 = 925 - 25y + 45y\\\\\\\dashrightarrow\sf\:\:1425 - 925 = -25y + 45y\\\\\\\dashrightarrow\sf\:\:500= 20y\\\\\\\ \dashrightarrow\sf\:\:y = \dfrac{550}{20}\\\\\\\dashrightarrow\:\:\underline{\boxed{\sf y = 25}}$

$\rule{130}{1}$

$\begin{lgathered}\bullet\:\:\textsf{y = \textbf{25}}\\\\\bullet\:\:\textsf{x = 37 - y = 37 - 25 = \textbf{12}}\end{lgathered}$

$\therefore\:\underline{\textsf{The value of x and y is \textbf{12 and 25 respectively}}}.$