Question 9: Heights of six students: $160cm, 175cm, 142cm, 136cm, 148cm, 182cm$. Find the mean.

Sum of heights $= 160 + 175 + 142 + 136 + 148 + 182 = 943$.
Mean $= 943 / 6 \approx 157.17cm$.

Question 10: Cricketer scores 98 runs in 19th innings, average increases by 4 runs. What will be his average score after 19th innings?

Let average before 19th innings be $A$. Total runs in 18 innings $= 18A$.
After 19th innings: $(18A + 98) / 19 = A + 4$.
$18A + 98 = 19A + 76$.
$A = 98 - 76 = 22$.
Average after 19th innings $= A + 4 = 22 + 4 = 26$.

Question 34: Average age of 9 students and teacher is 16 years. First 4 students' average is 19. Last 5 students' average is 10. What is the age of the teacher?

Total age of 9 students and teacher $= 10 \times 16 = 160$ years.
Total age of first 4 students $= 4 \times 19 = 76$ years.
Total age of last 5 students $= 5 \times 10 = 50$ years.
Total age of 9 students $= 76 + 50 = 126$ years.
Age of teacher $= 160 - 126 = 34$ years.

Question 35: Average weight of A, B, C is 45 kg. Average weight of A, B is 40kg. Average weight of B, C is 43kg. Find weight of B.

$A+B+C = 3 \times 45 = 135$ kg.
$A+B = 2 \times 40 = 80$ kg.
$B+C = 2 \times 43 = 86$ kg.
$(A+B) + (B+C) = 80 + 86 = 166$ kg.
$(A+B+C) + B = 166$ kg.
$135 + B = 166$.
$B = 166 - 135 = 31$ kg.

Question 36: Average of 45 numbers is 150. A number 46 was wrongly written as 91. Find the correct average.

Total sum (incorrect) $= 45 \times 150 = 6750$.
Correct sum $= 6750 - 91 + 46 = 6705$.
Correct average $= 6705 / 45 = 149$.

Question 37: A class of 95 students has average marks 75%. Five new students came with marks 70%, 68%, 42%, 83%, 57%. Find new average marks.

Total marks of 95 students $= 95 \times 75 = 7125$.
Marks of new students $= 70 + 68 + 42 + 83 + 57 = 320$.
New total marks $= 7125 + 320 = 7445$.
New total students $= 95 + 5 = 100$.
New average marks $= 7445 / 100 = 74.45\%$.

LCM & HCF

Question 11: LCM of two numbers is 4 times their HCF. Sum of LCM and HCF is 125. One number is 100. Find the other.
- Let HCF be $H$. Then LCM is $4H$.
- $H + 4H = 125 \Rightarrow 5H = 125 \Rightarrow H = 25$.
- LCM $= 4 \times 25 = 100$.
- Product of two numbers = LCM $\times$ HCF.
- $100 \times \text{other number} = 100 \times 25$.
- Other number $= 25$.
Question 12: LCM of two numbers is 936. HCF is 4. One number is 72. Find the other.
- Product of two numbers = LCM $\times$ HCF.
- $72 \times \text{other number} = 936 \times 4$.
- Other number $= (936 \times 4) / 72 = 936 / 18 = 52$.

Ratios & Proportions

Question 13: Number of boys in a class is four times the number of girls. Which number can represent the total number of children?
- Let number of girls be $G$. Number of boys $= 4G$.
- Total children $= G + 4G = 5G$.
- The total number of children must be a multiple of $5$.