Chapter 6.3, Problem 6CB

To determine

To find time taken by the hairdresser to give a haircut and dye customer’s hair using augmented matrix.

Answer to Problem 6CB

Time for hair cut $=45$ min

Time for hair dye $=90$ min

Explanation of Solution

Given:

A hairdresser can give 3 haircuts and 2 hair dyes in 315 min.

He can give 2 haircuts and 4 hair dyes in 450min.

Calculation:

Let $x$ be the time given by hairdresser for one haircut.

Let $y$ be the time given by the hairdresser for one hair dye.

Now, according to the question, the equations will be,

  $3x+2y=315$ ….(i).

  $2x+4y=450$

  $x+2y=225$ ….(ii).

Now writing equation (i) and equation (ii) in a matrix and row reducing it,

  $\Rightarrow \left[\begin{array}{ccc}3& 2& 315\\ 1& 2& 225\end{array}\right]$

  $R_1\to R_1-R_2$

  $\Rightarrow \left[\begin{array}{ccc}3& 2& 315\\ 1& 2& 225\end{array}\right]$

  $R_1\to R_1-R_2$

  $\Rightarrow \left[\begin{array}{ccc}2& 0& 90\\ 1& 2& 225\end{array}\right]$

  $R_1\to \frac{1}{2}{R}_1$

  $\Rightarrow \left[\begin{array}{ccc}1& 0& 45\\ 1& 2& 225\end{array}\right]$

  $R_2\to R_2\to R_1$

  $\Rightarrow \left[\begin{array}{ccc}1& 0& 45\\ 0& 2& 180\end{array}\right]$

  $R_2\to \frac{1}{2}{R}_2$

  $\Rightarrow \left[\begin{array}{ccc}1& 0& 45\\ 0& 1& 90\end{array}\right]$

Hence by the matrix,

  $x=45$

  $y=90$

Hence, time taken by the hairdresser to give one haircut is 45 min.

Time taken by the hairdresser to dye a customer’s hair is 90min.

Conclusion:

Therefore, the hairdresser takes 45 min to give one haircut and 90 min to dye a customer’s hair.