Question

The weight of coffee (in gms) in $70$ packets is given in the following table.

Weighs in grams	$200 - 201$	$201 - 202$	$202 - 203$	$203 - 204$	$204 - 205$	$205 - 206$
No. of packets	$12$	$26$	$20$	$9$	$2$	$1$

Hence determine the modal weight of the coffee packet.

Answer 1

Hint: To find the modal weight of the given data set, first we need to find the highest number of the no. of packets and the weight as noted. The width is the difference between the weight. The highest weight is added to the difference between the no. of packets and the multiplication of width. The highest no. of packets is taken first.
Formula:
Modal weight of given data,
$Mode = I + \left( {\dfrac{{{f_m} - {f_1}}}{{2{f_m} - {f_1} - {f_2}}}} \right) \times h$
I is the highest weight.
${f_m}$ is the highest number of packets
${f_1}$ is the lowest number of packets
$h$is the width
${f_2}$ is the second highest number of packets.
The width is the difference between the weights which have the highest number of packets.

Complete answer:
Given,

Weighs in grams	$200 - 201$	$201 - 202$	$202 - 203$	$203 - 204$	$204 - 205$	$205 - 206$
No. of packets	$12$	$26$	$20$	$9$	$2$	$1$

By considering the above equation,
The highest number of packets is $26$
The second number of packets is $20$.
The weight which has the highest number of packets is $201 - 202$.
The width is $1$.
The lowest number of packets is $12$.
The lowest weight is $201$
Substituting $I = 201$, ${f_m} = 26$, ${f_1} = 12$, ${f_2} = 20$and $h = 1$,
Calculate $Mode = I + \left( {\dfrac{{{f_m} - {f_1}}}{{2{f_m} - {f_1} - {f_2}}}} \right) \times h$
$Mode = 201 + \left( {\dfrac{{26 - 12}}{{2 \times 26 - 12 - 20}}} \right) \times 1$
According to BODMAS rule, the values inside the solved first,
$Mode = 201 + \left( {\dfrac{{14}}{{2 \times 26 - 32}}} \right) \times 1$
Multiply the above term,
$Mode = 201 + \left( {\dfrac{{14}}{{52 - 32}}} \right) \times 1$
Multiply the above term,
$Mode = 201 + \left( {\dfrac{{14}}{{52 - 32}}} \right)$
Subtract the terms in denominator
$Mode = 201 + \left( {\dfrac{{14}}{{20}}} \right)$
Divide the numerator and the denominator
$Mode = 201 + 0.7$
Add the above terms to get the modal weight value,
$Mode = 201.7$
The modal weight of the given data set is $201.7$.

Note:
The value should be correctly substituted. If the highest value is taken wrong the total value will be wrong and while solving the values always use the BODMAS rule which basically starts from Bracket, of, multiplication, addition and subtraction. We need to be so precise while finding the highest value, second highest value and corresponding terms.