The image presents several math problems related to the concept of the mean (average). Problem 3 asks to find the mass of the fifth man, given that the mean mass of five men is 76 kg and the masses of four of the men are 72 kg, 74 kg, 75 kg, and 81 kg. Problem 4 asks to find the length of the sixth rod, given that the mean length of 6 rods is 44.2 cm and the mean length of 5 of them is 46 cm. Problem 5 consists of two parts: a) Find x, given that the mean of 3, 7, 8, 10, and x is 6. b) Find x, given that the mean of 3, 3, 7, 8, 10, x, and x is 7. Problem 6 asks to find: a) the total height of 12 men, given that the mean height of 12 men is 1.70 m. b) the total height of 8 women, given that the mean height of 8 women is 1.60 m. c) the mean height of the 20 men and women.

1. Problem Description

The image presents several math problems related to the concept of the mean (average).

Problem 3 asks to find the mass of the fifth man, given that the mean mass of five men is 76 kg and the masses of four of the men are 72 kg, 74 kg, 75 kg, and 81 kg.

Problem 4 asks to find the length of the sixth rod, given that the mean length of 6 rods is 44.2 cm and the mean length of 5 of them is 46 cm.

Problem 5 consists of two parts:

a) Find x, given that the mean of 3, 7, 8, 10, and x is

6. b) Find x, given that the mean of 3, 3, 7, 8, 10, x, and x is

7. Problem 6 asks to find:

a) the total height of 12 men, given that the mean height of 12 men is 1.70 m.

b) the total height of 8 women, given that the mean height of 8 women is 1.60 m.

c) the mean height of the 20 men and women.

2. Solution Steps

Problem 3:

Let the mass of the fifth man be $m$.

The mean mass of the five men is given by:

$\frac{72 + 74 + 75 + 81 + m}{5} = 76$

$302 + m = 5 \times 76$

$302 + m = 380$

$m = 380 - 302$

$m = 78$

Problem 4:

Let the length of the sixth rod be $l$.

The mean length of the 6 rods is given by:

$\frac{\text{Total length of 6 rods}}{6} = 44.2$

Total length of 6 rods = $6 \times 44.2 = 265.2$

The mean length of 5 rods is given by:

$\frac{\text{Total length of 5 rods}}{5} = 46$

Total length of 5 rods = $5 \times 46 = 230$

Length of the sixth rod = Total length of 6 rods - Total length of 5 rods

$l = 265.2 - 230$

$l = 35.2$

Problem 5 a:

The mean of 3, 7, 8, 10, and x is

6. So,

$\frac{3 + 7 + 8 + 10 + x}{5} = 6$

$28 + x = 5 \times 6$

$28 + x = 30$

$x = 30 - 28$

$x = 2$

Problem 5 b:

The mean of 3, 3, 7, 8, 10, x, and x is

7. So,

$\frac{3 + 3 + 7 + 8 + 10 + x + x}{7} = 7$

$\frac{31 + 2x}{7} = 7$

$31 + 2x = 7 \times 7$

$31 + 2x = 49$

$2x = 49 - 31$

$2x = 18$

$x = \frac{18}{2}$

$x = 9$

Problem 6 a:

Mean height of 12 men = 1.70 m

Total height of 12 men = Mean height × Number of men

Total height of 12 men = $1.70 \times 12 = 20.4$ m

Problem 6 b:

Mean height of 8 women = 1.60 m

Total height of 8 women = Mean height × Number of women

Total height of 8 women = $1.60 \times 8 = 12.8$ m

Problem 6 c:

Total height of 20 people (12 men and 8 women) = Total height of 12 men + Total height of 8 women

Total height of 20 people = $20.4 + 12.8 = 33.2$ m

Mean height of 20 people = $\frac{\text{Total height of 20 people}}{20}$

Mean height of 20 people = $\frac{33.2}{20} = 1.66$ m

3. Final Answer

Problem 3: 78 kg

Problem 4: 35.2 cm

Problem 5 a: x = 2

Problem 5 b: x = 9

Problem 6 a: 20.4 m

Problem 6 b: 12.8 m

Problem 6 c: 1.66 m