Question: The average height of 4 boys is $150$ cm. When a new boy joins, the average height increases by $2$ cm. What is the height of the new boy?
1. **State the problem:** We know the average height of 4 boys is $150$ cm. When a new boy joins, the average height becomes $150 + 2 = 152$ cm for 5 boys.
2. **Formula for average:** Average height $= \frac{\text{Sum of heights}}{\text{Number of boys}}$
3. **Calculate the sum of heights of the original 4 boys:**
$$\text{Sum of 4 boys} = 4 \times 150 = 600 \text{ cm}$$
4. **Calculate the sum of heights of all 5 boys after the new boy joins:**
$$\text{Sum of 5 boys} = 5 \times 152 = 760 \text{ cm}$$
5. **Find the height of the new boy:**
$$\text{Height of new boy} = \text{Sum of 5 boys} - \text{Sum of 4 boys} = 760 - 600 = 160 \text{ cm}$$
**Final answer:** The height of the new boy is $160$ cm.