maths 題 10點

a)
由於所有行數都係以等差數列排列...
而第三行與第十一行，中間相差八行...
所以得出：
(設每行相差的數目為a)

54 + 8(a) = 126
8a = 126 - 54
a = 72 / 8
a = 9

故每行相差9個座位...^^

b)
設第一行的座位數目為y：

y......................(第一行)
y + 9................(第二行)
y + 2(9)............(第三行)
...
y + (n-1)(9).......(第n行)

故此：
y + 18 = 54
y = 36

換句話說：
若果n = 15
第十五行的座位便是[y + 14(9)] = (36 + 126) = 162

總括而言：

所有座位加埋，便得出以下列式：

y + (y + 9) + [y + 2(9)] + [y + 3(9)] ... [y + 14(9)]
=15(y) + (9)(1 + 2 + 3 + ... + 14)
= 15(y) + (9)(105)
= 15(36) + (9)(105)
= 1485

故整間戲院共有1485個座位...^^

c)
從part(b)所知，
第n行的座位 = 36 + (n-1)(9)

所以：

36 + (n-1)(9) = 36 x 3
36 + (n-1)(9) = 108
(n-1)(9) = 108 - 36
n-1 = 72/9
n-1 = 8
n = 9

即第九行的座位係第一行坐位數目的三倍...^^

d)
若加建多三行，即加建第十六、十七、十八行的座位...
總共增加了：(設b為新增座位的數目)

b = [36 + (16-1)(9)] + [36 + (17-1)(9)] + [36 + (18-1)(9)]
b = [36 + (15)(9)] + [36 + (16)(9)] + [36 + (17)(9)]
b = [36 + 135] + [36 + 144] + [36 + 153]
b = 171 + 180 + 189
b = 540

即共增加了540個座位...^^

e)
設加建後的坐位比原本的設計增加了的百分比為z：

z = 540 / 1485 x 100%
z = 0.363636..... x 100%
z = 36.36%

This is a A.P. question :

(a) 9
(b) 1,485
(c) Row number 9
(d) 540
(e) 36.36...%

The details are as below :

(a) A = Row 1 and n = difference between rows
Row 3 : A + 2n = 54
Row 11 : A + 10n = 126

Row 11 - Row 3
8n = 72
n = 9
每行相差9個坐位

(b)
Row 3 : A + 2(9) = 54
A = 36
Row 1 = 36個坐位

Row 1 = A
...
Row 15 = A + 14(9) = 162個坐位

Summation of Row 1 to Row 15 = 15A + 105n = 15(36) + [(1+14)*14/2](9) = 1,485個坐位

i.e. :
Row 1 = 36個坐位
Row 2 = 36 + 9 = 45個坐位
Row 3 = 36 + 18 = 54個坐位
Row 4 = 36 + 27 = 63個坐位
Row 5 = 36 + 36 = 72個坐位
Row 6 = 36 + 45 = 81個坐位
Row 7 = 36 + 54 = 90個坐位
Row 8 = 36 + 63 = 99個坐位
Row 9 = 36 + 72 = 108個坐位
Row 10 = 36 + 81 = 117個坐位
Row 11 = 36 + 90 = 126個坐位
Row 12 = 36 + 99 = 135個坐位
Row 13 = 36 + 108 = 144個坐位
Row 14 = 36 + 117 = 153個坐位
Row 15 = 36 + 126 = 162個坐位

(c) Row 1 = 36個坐位
36個坐位 * 3 = 108個坐位 = Row 9

(d)
Row 16 = 36 + 15(9) = 171個坐位
Row 17 = 36 + 16(9) = 180個坐位
Row 18 = 36 + 17(9) = 189個坐位
or
3A + 48d = 3(36) + 48(9) = 540個坐位
Therefore, answer is 540個坐位

(e) Increase qty / Original qty :
540個坐位 / 1,485個坐位 = 36.36%

This is a A.P. question :

(a) 9
(b) 1,485
(c) Row number 9
(d) 540
(e) 36.36...%

The details are as below :

(a) A = Row 1 and n = difference between rows
Row 3 : A + 2n = 54
Row 11 : A + 10n = 126

Row 11 - Row 3
8n = 72
n = 9
每行相差9個坐位

(b)
Row 3 : A + 2(9) = 54
A = 36
Row 1 = 36個坐位

Row 1 = A
...
Row 15 = A + 14(9) = 162個坐位

Summation of Row 1 to Row 15 = 15A + 105n = 15(36) + [(1+14)*14/2](9) = 1,485個坐位

i.e. :
Row 1 = 36個坐位
Row 2 = 36 + 9 = 45個坐位
Row 3 = 36 + 18 = 54個坐位
Row 4 = 36 + 27 = 63個坐位
Row 5 = 36 + 36 = 72個坐位
Row 6 = 36 + 45 = 81個坐位
Row 7 = 36 + 54 = 90個坐位
Row 8 = 36 + 63 = 99個坐位
Row 9 = 36 + 72 = 108個坐位
Row 10 = 36 + 81 = 117個坐位
Row 11 = 36 + 90 = 126個坐位
Row 12 = 36 + 99 = 135個坐位
Row 13 = 36 + 108 = 144個坐位
Row 14 = 36 + 117 = 153個坐位
Row 15 = 36 + 126 = 162個坐位

(c) Row 1 = 36個坐位
36個坐位 * 3 = 108個坐位 = Row 9

(d)
Row 16 = 36 + 15(9) = 171個坐位
Row 17 = 36 + 16(9) = 180個坐位
Row 18 = 36 + 17(9) = 189個坐位
or
3A + 48d = 3(36) + 48(9) = 540個坐位
Therefore, answer is 540個坐位

(e) Increase qty / Original qty :
540個坐位 / 1,485個坐位 = 36.36%

(a) if a = row 1
and n = difference per row
then a + 2n = 54 (row 3), and a + 10n = 126 (row 11)

(step 1)
a + 2n = 54
a + 10n = 126

(step 2)
-a - 2n = -54
a + 10n = 126

(step 3)
8n = 72

(step 4)
n = 9

so answer is 9

(b)
since n = 9
then a + 2(9) = 54
then a = 36 (row 1)

row 1=36
row 2=36+9=45
row 3=45+9=54
row 4=54+9=63
row 5=63+9=72
row 6=72+9=81
row 7=81+9=90
row 8=90+9=99
row 9=99+9=108
row 10=108+9=117
row 11=117+9=126
row 12=126+9=135
row 13=135+9=144
row 14=144+9=153
row 15=153+9=162

45+54+63+72+81+90+99+108+117+126+135+144+153+162 = 1,485
so answer is 1,485

(c) row 9

(d)
row 16=162+9=171
row 17=171+9=180
row 18=180+9=189

171 + 180 + 189 = 540
so answer is 540

(e) 540 / 1,485 = 36.36% .......................................................................................

(a) if a = row 1
and n = difference per row
then a + 2n = 54 (row 3), and a + 10n = 126 (row 11)

(step 1)
a + 2n = 54
a + 10n = 126

(step 2)
-a - 2n = -54
a + 10n = 126

(step 3)
8n = 72

(step 4)
n = 9

so answer is 9

(b)
since n = 9
then a + 2(9) = 54
then a = 36 (row 1)

row 1=36
row 2=36+9=45
row 3=45+9=54
row 4=54+9=63
row 5=63+9=72
row 6=72+9=81
row 7=81+9=90
row 8=90+9=99
row 9=99+9=108
row 10=108+9=117
row 11=117+9=126
row 12=126+9=135
row 13=135+9=144
row 14=144+9=153
row 15=153+9=162

45+54+63+72+81+90+99+108+117+126+135+144+153+162 = 1,485
so answer is 1,485

(c) row 9

(d)
row 16=162+9=171
row 17=171+9=180
row 18=180+9=189

171 + 180 + 189 = 540
so answer is 540

(e) 540 / 1,485 = 36.36%