The mean height of a group of students is 171 cm. In the group, there are 4 more females than males. The mean height of the females is 162 cm. The mean height of the males is 183 c... The mean height of a group of students is 171 cm. In the group, there are 4 more females than males. The mean height of the females is 162 cm. The mean height of the males is 183 cm. Calculate the total number of students in the group. Read more

Steps to Solve

1. Define Variables Let $m$ represent the number of males in the group. Then, the number of females is $m + 4$.

2. Set Up the Mean Height Formula The overall mean height is given as 171 cm. We can set up the equation using the formula for mean height: [ \text{Mean Height} = \frac{(\text{Number of females} \times \text{Mean height of females}) + (\text{Number of males} \times \text{Mean height of males})}{\text{Total number of students}} ]

3. Substitute Known Values Now we substitute the values we have: [ 171 = \frac{((m + 4) \times 162) + (m \times 183)}{(m + (m + 4))} ]

4. Simplify the Equation The total number of students can be simplified to $2m + 4$. Now, rewriting the equation: [ 171(2m + 4) = (m + 4) \times 162 + m \times 183 ]

5. Expand Both Sides Expanding both sides gives us: [ 342m + 684 = 162m + 648 + 183m ]

6. Combine Like Terms This reduces to: [ 342m + 684 = 345m + 648 ]

7. Rearrange and Solve for m Rearranging gives: [ 342m - 345m = 648 - 684 ] [ -3m = -36 \implies m = 12 ]

8. Find the Number of Females Using $m = 12$, the number of females is: [ m + 4 = 12 + 4 = 16 ]

9. Calculate Total Number of Students Now, add the number of males and females: [ \text{Total number of students} = m + (m + 4) = 12 + 16 = 28 ]