Question

Matt gets a \[\ $ 1,000\] commission on a big sale. This commission alone raises his average commission by \[\ $ 150\] . If Matt's new average commission is \[\ $ 400\] , how many sales has Matt made?
(A) \[6\]
(B) \[5\]
(C) \[4\]
(D) \[7\]

Answer 1

Hint: In order to find the number of sales made by Matt, first it is needed to find the new average commission of Matt on his sale and after that put it equals to the given new average commission \[\ $ 400\] and hence solve it.

Complete step-by-step solution:
Let average commission be \[x\] .
Let the number of items sold be \[y\] .
So the total commission is \[xy\] .
As given Matt got a \[\ $ 1,000\] commission on a big sale, so the new commission is given as below.
New commission is \[xy + 1000\] .
As given that after getting \[\ $ 1000\] commission by Matt, Matt’s average commission raised by \[\ $ 150\] .
As after getting \[\ $ 1000\] commission the new average commission is \[x + 150\] .
The new average commission is \[\ $ 400\] .
So equate the above new expression for the new average commission as below.
 \[
  x + 150 = 400 \\
\Rightarrow x = 250
 \] ………………(1)
So the new average commission can be written as shown below.
 $ \Rightarrow $ New average $ = \left( {xy + 1000} \right)/\left( {y + 1} \right) $
So, equate the above two expressions in order to find the number of sales Matt that is written as shown below.
 $ \Rightarrow $ $ \left( {xy + 1000} \right)/\left( {y + 1} \right) = x + 150 $
Multiply both sides by \[\left( {y + 1} \right)\] in the above expression.
 $ \Rightarrow $ \[\left( {xy + 1000} \right) = \left( {x + 150} \right)\left( {y + 1} \right)\]
Simplify the expression that shows in the right-hand side in the above equation.
 $ \Rightarrow $ \[xy + 1000 = xy + x + 150y + 150\]
Subtract \[xy\] from both the sides of the above equation, we get the equation in terms of \[x\& y\] as below.
 $ \Rightarrow $ \[1000 = x + 150y + 150\]
Subtract $ 150 $ of the both sides of the above equation as shown below.
 $ \Rightarrow $ \[x + 150y = 850\] .................(2)
Solve equation \[\left( 1 \right)\& \left( 2 \right)\] as shown below.
 \[
  250 + 150y = 850 \\
\Rightarrow 150y = 600 \\
\Rightarrow y = 4
 \]
So, the number of sales made by Matt is calculated as:
 \[
  y + 1 = 4 + 1 \\
   = 5
 \]
Therefore, the number of sales made by Matt is \[5\] .

Hence, the correct answer is (B).

Note: As we know that to solve the unknown variables by using the equations we need the same number of equations as the unknown, that is if the number of unknown are two then we need at least two equations and as the number of unknown increases, the number of equations increases.