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發問:
一間戲院進行改建,原本的圖側顯示整個戲院共有15行,第三行和第十一行分別有坐位54 和 126個,若戲院坐位的設計以等差數列的概念為基礎。請計算: a)每行相差多少個坐位? b)整間戲院共有多少個坐位? c)哪一行坐位是第一行坐位數目的三倍? d)若戲院準備加建三行,共會增加多少個坐位? e)加建後的坐位比原本的設計增加了多少個百分比?
最佳解答:
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%
其他解答:
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%
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maths 題 10點發問:
一間戲院進行改建,原本的圖側顯示整個戲院共有15行,第三行和第十一行分別有坐位54 和 126個,若戲院坐位的設計以等差數列的概念為基礎。請計算: a)每行相差多少個坐位? b)整間戲院共有多少個坐位? c)哪一行坐位是第一行坐位數目的三倍? d)若戲院準備加建三行,共會增加多少個坐位? e)加建後的坐位比原本的設計增加了多少個百分比?
最佳解答:
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%
其他解答:
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%
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