標題:
一條好難的數學題(20pt)
有一個人,佢入了去7-11買了4樣野,佢唔小心張4個物品的價錢乘咗,就要比7.11美金,知後佢清醒番,就將4個物品的價錢加咗佢,都是要比7.11美金,咁果4件物品分別是幾多錢 ?+?+?+?=7.11 ?x?x?x?=7.11 我需要的是個答案and解釋
最佳解答:
w + x + y + z = 7.11 wxyz = 7.11 w, x, y, z are rational numbers let w = a/b, x = c/d, y = e/f, z = g/h consider prime factorization of 711 and 100 aceg/bdfh = 711/100 = 3^2 * 79 / 2^2 * 5^2 note that 79/1, 79/2, 79/4, 79/5, 79/10, 3*79/20, 3*79/25 > 7.11 so one of the item's price can be 79/20 = 3.95 79/25 = 3.16 79/50, 79/100, 3*79/50, 3*79/100 I tried 3.95 for a long time, and failed. For z = 3.16, w + x + y = 3.95, wxy = 2.25 Since the factor 79 is taken a, c, e can take 1, 3, 3 or 1, 1, 9 consider the 1st case (9 looks too large) Since the factor 25 is taken b, d, f can take 1, 2, 2 or 1, 1, 4 consider again the 1st case (so many 1 as divisors, looks cannot give us 3.95) Guess y = 3/2 = 1.5, w + x = 2.45, wx = 0.375 solve this system of equations w = 1.2, x = 1.25 Done! i.e. the prices are 1.2, 1.25, 1.5 and 3.16 2007-03-26 02:18:00 補充： 本來係4個未知數, 2條方程, 應該係有好多答案 但係冇理由seven會賣樣標價係3.141592653589 o既o野... 所以應該係有理數(rational number) 而每一個整數可以寫成一連串質數(prime number)o既積(product) 呢兩點就係解題o既要點 又由於4個未知數+2條方程 o既關係, 難免要估o下 其實可能可以找到o的理由排除某些數, 唔駛亂估亂試
其他解答:
w + x + y + z = 7.11 wxyz = 7.11 w, x, y, z are rational numbers let w = a/b, x = c/d, y = e/f, z = g/h consider prime factorization of 711 and 100 aceg/bdfh = 711/100 = 3^2 * 79 / 2^2 * 5^2 note that 79/1, 79/2, 79/4, 79/5, 79/10, 3*79/20, 3*79/25 > 7.11 so one of the item's price can be 79/20 = 3.95 79/25 = 3.16 79/50, 79/100, 3*79/50, 3*79/100 I tried 3.95 for a long time, and failed. For z = 3.16, w + x + y = 3.95, wxy = 2.25 Since the factor 79 is taken a, c, e can take 1, 3, 3 or 1, 1, 9 consider the 1st case (9 looks too large) Since the factor 25 is taken b, d, f can take 1, 2, 2 or 1, 1, 4 consider again the 1st case (so many 1 as divisors, looks cannot give us 3.95) Guess y = 3/2 = 1.5, w + x = 2.45, wx = 0.375 solve this system of equations w = 1.2, x = 1.25 Done! i.e. the prices are 1.2, 1.25, 1.5 and 3.16
一條好難的數學題(20pt)
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發問:有一個人,佢入了去7-11買了4樣野,佢唔小心張4個物品的價錢乘咗,就要比7.11美金,知後佢清醒番,就將4個物品的價錢加咗佢,都是要比7.11美金,咁果4件物品分別是幾多錢 ?+?+?+?=7.11 ?x?x?x?=7.11 我需要的是個答案and解釋
最佳解答:
w + x + y + z = 7.11 wxyz = 7.11 w, x, y, z are rational numbers let w = a/b, x = c/d, y = e/f, z = g/h consider prime factorization of 711 and 100 aceg/bdfh = 711/100 = 3^2 * 79 / 2^2 * 5^2 note that 79/1, 79/2, 79/4, 79/5, 79/10, 3*79/20, 3*79/25 > 7.11 so one of the item's price can be 79/20 = 3.95 79/25 = 3.16 79/50, 79/100, 3*79/50, 3*79/100 I tried 3.95 for a long time, and failed. For z = 3.16, w + x + y = 3.95, wxy = 2.25 Since the factor 79 is taken a, c, e can take 1, 3, 3 or 1, 1, 9 consider the 1st case (9 looks too large) Since the factor 25 is taken b, d, f can take 1, 2, 2 or 1, 1, 4 consider again the 1st case (so many 1 as divisors, looks cannot give us 3.95) Guess y = 3/2 = 1.5, w + x = 2.45, wx = 0.375 solve this system of equations w = 1.2, x = 1.25 Done! i.e. the prices are 1.2, 1.25, 1.5 and 3.16 2007-03-26 02:18:00 補充： 本來係4個未知數, 2條方程, 應該係有好多答案 但係冇理由seven會賣樣標價係3.141592653589 o既o野... 所以應該係有理數(rational number) 而每一個整數可以寫成一連串質數(prime number)o既積(product) 呢兩點就係解題o既要點 又由於4個未知數+2條方程 o既關係, 難免要估o下 其實可能可以找到o的理由排除某些數, 唔駛亂估亂試
其他解答:
w + x + y + z = 7.11 wxyz = 7.11 w, x, y, z are rational numbers let w = a/b, x = c/d, y = e/f, z = g/h consider prime factorization of 711 and 100 aceg/bdfh = 711/100 = 3^2 * 79 / 2^2 * 5^2 note that 79/1, 79/2, 79/4, 79/5, 79/10, 3*79/20, 3*79/25 > 7.11 so one of the item's price can be 79/20 = 3.95 79/25 = 3.16 79/50, 79/100, 3*79/50, 3*79/100 I tried 3.95 for a long time, and failed. For z = 3.16, w + x + y = 3.95, wxy = 2.25 Since the factor 79 is taken a, c, e can take 1, 3, 3 or 1, 1, 9 consider the 1st case (9 looks too large) Since the factor 25 is taken b, d, f can take 1, 2, 2 or 1, 1, 4 consider again the 1st case (so many 1 as divisors, looks cannot give us 3.95) Guess y = 3/2 = 1.5, w + x = 2.45, wx = 0.375 solve this system of equations w = 1.2, x = 1.25 Done! i.e. the prices are 1.2, 1.25, 1.5 and 3.16
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