Question

The mean of the following distribution is 50.
X 10 30 50 70 90
F 17 5a+3 32 7a-11 19
Find the value of a and hence the frequencies of 30 and 70​

Answer 1

Question

The mean of the following distribution is 50.

$\begin{array}{|c|c|c|c|c|c|}\cline{1-6}X_i & 10 & 30 & 50 & 70 & 90 \\ \cline{1-6} F_i & 17 & 5a+3 & 32 & 7a-11 & 19 \\ \cline{1-6}\end{array}$

Solution

$\begin{tabular}{|c|c|c|}\cline{1-3} X_i & F_i & F_iX_i \\ \cline{1-3}10 & 17 & 170 \\ \cline{1-3}30 & 5a+3 & 150a+90 \\ \cline{1-3}50 & 32 & 1600 \\ \cline{1-3} 70 & 7a-11 & 490a-770 \\ \cline{1-3} 90 & 19 & 1710 \\ \cline{1-3}Total & \Sigma\:F_i = 60 + 12a & \Sigma X_iF_i = 2800 + 640a \\ \cline{1-3}\end{tabular}$

Mean = Sum of all observations/Total number of observations

Given, mean is 50.

$\implies\:\sf{50\:=\:\dfrac{\sum X_i F_i}{\sum X_i}}$

$\Rightarrow\:\sf{50\:=\:\dfrac{2800+650a}{60+12a}}$

$\Rightarrow\:\sf{50(60+12a)\:=\:2800+650a}$

$\Rightarrow\:\sf{3000+600a\:=\:2800+650a}$

$\Rightarrow\:\sf{3000\:-\:2800\:=\:650a\:-\:600a}$

$\Rightarrow\:\sf{200\:=\:40a}$

$\Rightarrow\:\sf{\cancel{\frac{200}{40}}\:=\:a}$

$\Rightarrow\:\sf{\boxed{\red{a\:=\:5}}}$

Therefore, the frequency of 30 = 5a + 3

⇒ 5(5) + 3

⇒ 25 + 3

⇒ 28

Frequency of 70 = 7a - 11

⇒ 7(5) - 11

⇒ 35 - 11

⇒ 24

Answer 2

Given :---

• X = 10 30 50 70 90
• F = 17 (5a+3) 32 (7a-11) 19
• Mean of Distribution = 50 .

To Find :--

• The value of a and hence the frequencies of 30 and 70..

______________________________

❁❁ Refer To Image First .. ❁❁

|| ★★ FORMULA USED ★★ ||

→ Mean of distribution is given by , Sum of all observation divided by Total Number of observation .

or,

→ Mean = (sum of FiXi /(sum of Fi)

|| ✰✰ ANSWER ✰✰ ||

From image we can see that,

→ Sum of Fixi we get = (640a + 2800) [ By mulitplying each Xi with Fi and than adding all of them . ]

→ Sum of Fi (Frequency) = (12a + 60)

And , we have given Mean = 50.

__________________________

Putting all values in Mean formula now, we get,

→ 50 = (640a + 2800) / (12a+60)

Cross - Multiplying ,

→ 50(12a + 60) = 640a + 2800

→ 600a + 3000 = 640a + 2800

→ 640a - 600a = 3000 - 2800

→ 40a = 200

Dividing both sides by 40,

→ a = 5 (Ans).

__________________________

Now, putting value of a ,

→ Frequency of 30 = (5a + 3) = 5*5 + 3 = 25 + 3 = 28 .

→ Frequency of 70 = (7a -11) = 7*5 - 11 = 35 - 11 = 24 .