1.一場聚會中有飲料3款，可樂平均2人飲一支，七喜平均3人飲一支，橙汁平均4人飲一支，共飲去117支飲料。求參加該聚會的人數。
2.有100粒糖分別放於5個容器A.B.C.D.E中
A+B=52粒
B+C=43粒
C+D=34粒
D+E=30粒
求5個容器中，各放了多少粒糖。
3.小強年齡的平方加上小明的年齡為62;小明年齡的平方加上小強的年齡為176,求兩人分別為幾歲。
麻頪列出步驟,THX
1.
Let no. of person = x
No. of Coke = x/2
No. of 7 - up = x/3
No. of Orange Juice = x/4
x/2 + x/3 + x/4 = 117
6x + 4x + 3x = 117 x 12 = 1404
x = 1404/13 = 108 = no. of persons.
2.
A + B + C + D + E = 100
Since D + E = 30
so A + B + C + 30 = 100
A + B + C = 70
Since A + B = 52
so 52 + C = 70
C = 70 - 52 = 18.
B = 43 - C = 43 - 18 = 25
18 + D = 34
D = 34 - 18 = 16
16 + E = 30
E = 30 - 16 = 14.
So A = 27, B = 25, C = 18, D = 16 and E = 14. Total = 100.
3.
Let Keung's age = x. Ming's age = y
x^2 + y = 62 ........(1)
y^2 + x = 176 ......(2)
From (1)
y = 62 - x^2
Notice that both x and y must be positive integers. So only possibilities are :
x y
1 61
2 58
3 53
4 46
5 37
6 26
7 13.
From these 7 sets, only the set x = 7, y = 13 meets equation (2).
So Keung is 7 and Ming is 13.
1. 平均每 12 人飲可樂 6支，七喜 4支，橙汁 3支，共 13支飲料，
117＝13 x 9，12 x 9＝108
所以，參加該聚會的有 108人。
2014-07-17 08:41:36 補充：
2. A＋B＋C＋D＋E＝100 ⋯⋯ (a)
A＋B＝52 ⋯⋯⋯⋯⋯⋯⋯⋯ (b)
B＋C＝43 ⋯⋯⋯⋯⋯⋯⋯⋯ (c)
C＋D＝34 ⋯⋯⋯⋯⋯⋯⋯⋯ (d)
D＋E＝30 ⋯⋯⋯⋯⋯⋯⋯⋯ (e)
(a)－(b)－(d) 得 E＝14
(a)－(c)－(e) 得 A＝27
從 (b), (e) 再得 B＝25, D＝16
再從 (c) 或 (d) 可得 C＝18
所以容器 A 放了 27粒糖、B 放了 25粒糖、C 放了 18粒糖、D 放了 16粒糖、E 放了 14粒糖。
2014-07-17 08:42:09 補充：
3. 設小強 x歳，小明 y歳，則
x^2＋y＝62 ⋯⋯⋯ (f)
y^2＋x＝176 ⋯⋯ (g)
(g)－(f) 得
y^2－x^2＋x－y＝176－62
==> (y－x)(y＋x)－(y－x)＝114
==> (y－x)(y＋x－1)＝1*114＝2*57＝3*38＝6*19
==> (x, y)＝(57, 58) 或 (28, 30) 或 (18, 21) 或 (7, 13)
代入 (f)，只有 (x, y)＝(7, 13) 符合。
所以，小強 7歳，小明 13歳。
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Answers & Comments
Verified answer
1.
Let no. of person = x
No. of Coke = x/2
No. of 7 - up = x/3
No. of Orange Juice = x/4
x/2 + x/3 + x/4 = 117
6x + 4x + 3x = 117 x 12 = 1404
x = 1404/13 = 108 = no. of persons.
2.
A + B + C + D + E = 100
Since D + E = 30
so A + B + C + 30 = 100
A + B + C = 70
Since A + B = 52
so 52 + C = 70
C = 70 - 52 = 18.
B = 43 - C = 43 - 18 = 25
18 + D = 34
D = 34 - 18 = 16
16 + E = 30
E = 30 - 16 = 14.
So A = 27, B = 25, C = 18, D = 16 and E = 14. Total = 100.
3.
Let Keung's age = x. Ming's age = y
x^2 + y = 62 ........(1)
y^2 + x = 176 ......(2)
From (1)
y = 62 - x^2
Notice that both x and y must be positive integers. So only possibilities are :
x y
1 61
2 58
3 53
4 46
5 37
6 26
7 13.
From these 7 sets, only the set x = 7, y = 13 meets equation (2).
So Keung is 7 and Ming is 13.
1. 平均每 12 人飲可樂 6支，七喜 4支，橙汁 3支，共 13支飲料，
117＝13 x 9，12 x 9＝108
所以，參加該聚會的有 108人。
2014-07-17 08:41:36 補充：
2. A＋B＋C＋D＋E＝100 ⋯⋯ (a)
A＋B＝52 ⋯⋯⋯⋯⋯⋯⋯⋯ (b)
B＋C＝43 ⋯⋯⋯⋯⋯⋯⋯⋯ (c)
C＋D＝34 ⋯⋯⋯⋯⋯⋯⋯⋯ (d)
D＋E＝30 ⋯⋯⋯⋯⋯⋯⋯⋯ (e)
(a)－(b)－(d) 得 E＝14
(a)－(c)－(e) 得 A＝27
從 (b), (e) 再得 B＝25, D＝16
再從 (c) 或 (d) 可得 C＝18
所以容器 A 放了 27粒糖、B 放了 25粒糖、C 放了 18粒糖、D 放了 16粒糖、E 放了 14粒糖。
2014-07-17 08:42:09 補充：
3. 設小強 x歳，小明 y歳，則
x^2＋y＝62 ⋯⋯⋯ (f)
y^2＋x＝176 ⋯⋯ (g)
(g)－(f) 得
y^2－x^2＋x－y＝176－62
==> (y－x)(y＋x)－(y－x)＝114
==> (y－x)(y＋x－1)＝1*114＝2*57＝3*38＝6*19
==> (x, y)＝(57, 58) 或 (28, 30) 或 (18, 21) 或 (7, 13)
代入 (f)，只有 (x, y)＝(7, 13) 符合。
所以，小強 7歳，小明 13歳。