Question 5 Miss Jones decided to divide her class of 24 students into groups according to the colour of their hair: brown, black, blonde and red. She counted three times as many brown haired students as those with black hair. There were six less blonde haired students than brown haired students and the number of red haired students was one third of those with blonde hair.

To solve this problem, we can set up a system of equations based on the given information and then solve for the number of students with each hair color. Let's denote: - \( b \) as the number of brown-haired students - \( k \) as the number of black-haired students - \( l \) as the number of blonde-haired students - \( r \) as the number of red-haired students From the given information, we can form the following equations: 1. \( b = 3k \) (three times as many brown-haired students as black-haired students) 2. \( l = b - 6 \) (six less blonde-haired students than brown-haired students) 3. \( r = \frac{1}{3}l \) (the number of red-haired students is one third of those with blonde hair) 4. \( b + k + l + r = 24 \) (total number of students is 24) Now, we can solve this system of equations to find the number of students with each hair color. Solve the system of equations by following steps: - step0: Solve using the substitution method: \(\left\{ \begin{array}{l}b=3k\\l=b-6\\r=\frac{1}{3}l\\b+k+l+r=24\end{array}\right.\) - step1: Substitute the value of \(b:\) \(\left\{ \begin{array}{l}l=3k-6\\r=\frac{1}{3}l\\3k+k+l+r=24\end{array}\right.\) - step2: Simplify: \(\left\{ \begin{array}{l}l=3k-6\\r=\frac{1}{3}l\\4k+l+r=24\end{array}\right.\) - step3: Substitute the value of \(l:\) \(\left\{ \begin{array}{l}r=\frac{1}{3}\left(3k-6\right)\\4k+3k-6+r=24\end{array}\right.\) - step4: Simplify: \(\left\{ \begin{array}{l}r=k-2\\7k-6+r=24\end{array}\right.\) - step5: Substitute the value of \(r:\) \(7k-6+k-2=24\) - step6: Simplify: \(8k-8=24\) - step7: Move the constant to the right side: \(8k=24+8\) - step8: Add the numbers: \(8k=32\) - step9: Divide both sides: \(\frac{8k}{8}=\frac{32}{8}\) - step10: Divide the numbers: \(k=4\) - step11: Substitute the value of \(k:\) \(r=4-2\) - step12: Simplify: \(r=2\) - step13: Substitute the value of \(k:\) \(l=3\times 4-6\) - step14: Simplify: \(l=6\) - step15: Substitute the value of \(k:\) \(b=3\times 4\) - step16: Simplify: \(b=12\) - step17: Calculate: \(\left\{ \begin{array}{l}b=12\\k=4\\l=6\\r=2\end{array}\right.\) - step18: Check the solution: \(\left\{ \begin{array}{l}b=12\\k=4\\l=6\\r=2\end{array}\right.\) - step19: Rewrite: \(\left(b,k,l,r\right) = \left(12,4,6,2\right)\) The solution to the system of equations is: - \( b = 12 \) (number of brown-haired students) - \( k = 4 \) (number of black-haired students) - \( l = 6 \) (number of blonde-haired students) - \( r = 2 \) (number of red-haired students) Therefore, Miss Jones has 12 brown-haired students, 4 black-haired students, 6 blonde-haired students, and 2 red-haired students in her class.