Question

5. Of the four numbers the average of the first three is 30 and of the last three is 32. If the last
number is 38, find the first number.
​

Answer 1

Given:

the average of the first three is 30 and of the last three is 32. If the last number is 38

To find:

The first number

Explanation:

Let the numbers be a,b,c,38

Average of first three is 30

$\boxed{\sf average=\dfrac{sum \: of \: observations}{total\: no. \: of \:observations}}$

$\longrightarrow \: \sf \dfrac{a + b + c}{3} = 30$

$\longrightarrow \: \sf \: a + b + c = 90.........(1)$

Average of last three is 32

$\longrightarrow \: \sf \dfrac{b + c + 38}{3} = 32$

$\longrightarrow \: \sf \: b + c + 38 = 96$

$\longrightarrow \: \sf \: b + c = 58$

substituting the value of b+c in (1)

$\longrightarrow \: \sf \: a + 58 = 90$

$\longrightarrow \: \sf \: a = 90 - 58$

$\longrightarrow \: \boxed{ \sf \: a = 32}$

Therefore,the first number is 32

Answer 2

$\bold{\underline{\underline{\huge{\sf{AnsWer:}}}}}$

First Number = 32

$\bold{\underline{\underline{\large{\sf{StEp\:by\:stEp\:explanation}}}}}$

GiVeN :

• Of the four numbers the average of the first three is 30
• Average of last three number is 32
• Last number is 38

To FiNd :

• The First Number

SoLuTiOn :

Let the first three numbers be,

• x
• y
• z

We have the last number = 38

Using the formula for average

FoRmUlA :

$\bold{\large{\boxed{\rm{\pink{Average\:=\:{\dfrac{Sum\:of\:observations}{Total\:Number\:of\:observations}}}}}}}$

Block in the values,

For First Three Numbers :

$\longrightarrow$$\bold{\tt{30\:=\:{\dfrac{x+y+z}{3}}}}$

$\longrightarrow$$\bold{\tt{30\:\times\:3\:=\:x+y+z}}$

$\longrightarrow$$\bold{\tt{90\:=\:x+y+z}}$

$\longrightarrow$$\bold{\tt{x+y+z=90}}$ ---> (1)

For Last Three Numbers :

Here we are calculating the average of the last three number. Since in the question we have even number of observation i.e 4, we exclude the first score (observation) and perform the calculation with the remaining scores.

Last number = 38

Average = 32

Block in the values in the formula for average,

$\longrightarrow$$\bold{\tt{32\:=\:{\dfrac{y+z\:+\:38}{3}}}}$

$\longrightarrow$$\bold{\tt{32\:\times\:3\:=\:y+z+38}}$

$\longrightarrow$$\bold{\tt{96\:=\:y+z+38}}$

$\longrightarrow$$\bold{\tt{96-38\:=\:y+z}}$

$\longrightarrow$$\bold{\tt{58\:=\:y+z}}$

$\longrightarrow$$\bold{\tt{y+z=58}}$ ---> (2)

Substitute y + z = 58 in equation 1,

$\longrightarrow$$\bold{\tt{x+y+z=90}}$ ---> (1)

$\longrightarrow$$\bold{\tt{x+58\:=\:90}}$

$\longrightarrow$$\bold{\tt{x=90-58}}$

$\longrightarrow$$\bold{\tt{x=32}}$

•°• First Number = x = 32