Question

The number of pets 40 students own is recorded

Number of pets: 2 , 4, 6, 8, 10
Number of students: x, 2, y, 6, 14

a) (i) show that x+y=18
(ii) If the mean of the distribution is 6.4, show that x +3y =30
(iii) Hence, find the value of x and of y
b) Using your answers in (a) (iii), find
(i) the median,
(ii) the mode,
Of the distribution

Answer 1

a.

i) & ii) -> See Explanation Below

a.

iii)
`y = 6` and
`x = 12`

b.

i)
`Median = 7`

ii)
`Mode = 10`

Explanation:

Given

Number of pets: 2 , 4, 6, 8, 10

Number of students: x, 2, y, 6, 14

Total students = 40

Required

a) (i) show that x+y=18

Given that total number of students is 40;

This implies that

`x + 2 + y + 6 + 14 = 40`

Collect like terms

`x + y = 40 - 14 - 6 - 2`

`x + y = 18`

a) (ii) If the mean of the distribution is 6.4, show that x +3y =30

The mean of a distribution is calculated as thus

`Mean = (\sum fx)/(\sum x)`

`\sum fx}` is gotten by multiplying number of pets by corresponding number of students

`\sum fx} = 2 * x + 4 * 2 + 6 * y + 8 * 6 + 10 * 14`

`\sum fx} = 2 x + 8 + 6y + 48 + 140`

`\sum fx} = 2 x + 6y + 48 + 140+ 8`

`\sum fx} = 2 x + 6y + 196`

`\sum x}` is the total number of students

`\sum x} = 40`

So;

`Mean = (\sum fx)/(\sum x)` becomes

`Mean = (2 x + 6y + 196)/(40)`

Substitute 6.4 for Mean

`6.4 = (2 x + 6y + 196)/(40)`

Multiply both sides by 40

`256 = 2 x + 6y + 196`

Subtract 196 from both sides

`256 -196= 2 x + 6y + 196-196`

`60= 2 x + 6y`

Divide both sides by 2

`(60)/(2)= (2 x)/(2) + (6y)/(2)`

`30 = (2 x)/(2) + (6y)/(2)`

`30 = x + 3y`

Reorder

`x + 3y = 30`

a) (iii) Hence, find the value of x and of y

Using

`x + y = 18` and

`x + 3y = 30`

Subtract both equations

`(x + y = 18) - (x + 3y = 30)`

`x - x + y - 3y = 18 - 30`

`y - 3y = 18 - 30`

`-2y = -12`

Divide both sides by -2

`(-2y)/(-2) = (-12)/(-2)`

`y = (-12)/(-2)`

`y = 6`

Substitute 6 for y in
`x + y = 18`

`x + y = 18` becomes

`x + 6 = 18`

Subtract 6 from both sides

`x + 6 - 6 = 18 - 6`

`x = 12`

b.

i) Calculate the Median

First, we need to tabulate the given data properly

Number of Pets ------ Number of students ------- Cumulative Frequency

2 ------------------------------12 -----------------------------------12

4 ------------------------------2 -----------------------------------14

6 ------------------------------6 -----------------------------------20

8 ------------------------------6 -----------------------------------26

10 ------------------------------14 -----------------------------------40

Since the number of pets is already arranged in ascending order,

the next step is to calculate the median element

Number of students = 40

`Median = (40)/(2)`

Median = 20

Given that number of students (40) is an even number,

The median is the average of the 20th and 21st element

From the table above; the median can be gotten from

6 ------------------------------6 -----------------------------------20

8 ------------------------------6 -----------------------------------26

The 20th element = 6

The 21st element = 8

`Median = (6 + 8)/(2)`

`Median = (14)/(2)`

`Median = 7`

ii) Mode

The mode is the corresponding data with the highest frequency

The highest frequency is 14;

The number of pets with frequency of 14 is 10

Hence,

`Mode = 10`