1) The document contains several math problems involving algebra, geometry, percentages, and calculations with means, medians, and modes. 2) Geometry problems include calculating side lengths of triangles using trigonometric functions like sine and solving for missing angles. 3) Algebra problems involve solving equations for unknown variables and calculating percentages of changes. 4) Statistics problems calculate the mean, median, and mode of data sets.

1. a =
3+𝑏
5+2𝑏
a (5+2b) = 3+b
5a+2ab = 3+b
3+b-2ab = 5a
b(1-2a) = 5a-3
b =
5𝑎−3
1−2𝑎

2. log10 3 = 0.4771 → 100.4771 = 3
log10 𝑥 = 3.4771 → 103.4771
= 𝑥
𝑥 = 103
× 100.4771
= 1000 × 3
= 3000

3. 𝑎2 + 𝑏2 = 𝑐2
∆𝐶𝐷𝐹: 𝐷𝐹2
= 52
+ (5 3)2
𝐷𝐹 = 10
DE = 10
tan 𝜃 =
𝐶𝐹
𝐶𝐷
=
5
5 3
𝑎𝑛𝑔𝑙𝑒 𝐶𝐷𝐹 = 𝑎𝑛𝑔𝑙𝑒 𝐶𝐷𝐹 = 𝜃
𝜃 = 30 °
𝜃 = 30°
∆𝐷𝐸𝐹
𝑎𝑛𝑔𝑙𝑒 𝐶𝐷𝐹 =
180 ° − 30 °
2
= 75 °

4. mean =
3 4 +6 10 +9 4 +12(2)
20

5. 4,5,6

6. V = 144 𝜋
V hemisphere =
2
3
𝜋𝑟3
144 𝜋 =
2
3
𝜋𝑟3
r = 6

7. 𝑥 = original price
𝑦 = selling price
𝑦 − 𝑥
𝑥
× 100 % = 20 %
𝑦 - 𝑥 = 0.2 𝑥
𝑦 = 1.2 𝑥−−−−①
0.9 𝑦 - 𝑥 = 24 −−②

8. ?
?
CD = 𝐵𝐷 = 15 cm
angle ABC = 75°
angle ABE = 30°
angle EBD = 30°
angle CBD = 15°
∆𝐴𝐵𝐷
sin 30° =
𝐴𝐵
𝐵𝐷
1
2
=
𝐴𝐵
15
𝐴𝐵 = 7.5 cm
∆𝐴𝐵𝐸
sin 60° =
𝐴𝐵
𝐵𝐸
3
2
=
7.5
𝐵𝐸
𝐵𝐸 =
15
3
=
15 3
3
= 5 3 cm

9. = 9 − 1 2 + 2 − 8 2
= 8 2 + −6 2
AB = 10 AC = 𝐵𝐶 = 50
= 5 2
𝑎2 + 𝑏2 = 𝑐2
1
4
=
𝐴𝐷𝐵
2𝜋𝑟