※ 引述《liamasso (字神帝國)》之銘言：
感謝康老師，以上非數學內容43
: ▲ 數學
: ◆ 選擇題：給一個圓，中心點(0,0)，半徑5，問有多少個整數點在圓上。答案(C) 12
x平方+y平方=5平方的整數解
依x的絕對值大小排列 解的數目
x=正負5 y=0 2
x=正負4 y=正負3 2*2= 4
x=正負3 y=正負4 2*2= 4
x=0 y=正負5 2
共12
: ◆ 比大小：0<x<y<1，問 xy 跟 x/y 誰大
x大於零，所以參賽者雙方同除以x不影響雙方孰大孰小
除掉x之後就是y和1/y比大小
答案是 1/y比較大 (x/y 比 xy 大)
: ◆ 比大小：Given x is positive integer，g > 0: g^(x)+g^(-x) 跟 1 誰大
算幾不等式
a=g^x, b=g^(-x)
(a+b)/2 >= 根號(ab)=根號1=1
所以a+b最少是2 (當a=b的時候，也就是g=1時)
所以那個式子比1大
: ◆ 比大小：(3^(5))^(7) 跟 3^(5*7) 誰大
指數律 左式 = 3的35次方
所以一樣大
: ◆ 比大小：0<x<y<1，A: xy；B x/y (B)
重複題，答案確認B無誤
: ◆ a square table cloth is to cover a circular table with radius x, what is
: the required side of the table cloth (2x)
正方形內接半徑x的圓形，正方形邊長2x
: ◆ committee A has 5 people, B has 7 and there are 5 in both A & B. (a) there
: are total 12 people in either A or B or both (b) there are 2 people in B but
: not in A (c) all the members in A are in B. (b) (c)
人 1 2 3 4 5 6 7
委 A A A A A
委 B B B B B B B
(a)錯誤 應為7人
(b)7-5=2 正確
(c)正確
: ◆ line k x-intercept is a, y-intercept is b. which of the following
: individually determine the slope of line k is positive (a) a+b<0 (b) a-b<0
: (c)ab<0 (c)
x-int = a
y-int = b
斜率-b/a
所以兩intercept異號 <=> 斜率為正
(a)(b)兩選項無關乎a b的正負，兩者可同號可異號
只有(c)正確
: ◆ K: 3 5 7 9 11 x, 7<=x<=9；L: 3 5 7 9 11 y, 8<=y<=10
: (A) median of K (B) median of L
(A) 為(x+7)/2
(B) 為(y+7)/2 當 8<=y<=9
或8 當 9< y<=10
兩者可大可小
x 7 9
y 8 8
(A) 7 8
(B) 7.5 7.5
大 (B) (A)
: ◆ 5 books each has the pages: 480 490 x 510 520. What is the greatest
: possible value of x that the median and the average are the same____ (550)
這題畫median函數圖形就OK了，另外也可以先把資料平移縮小(減500再除以10)
新資料變成 -2 -1 x 1 2 (但沒有規定x一定只能在-1到1之間，這樣寫出題者的刻意)
median的函數圖形y=md(x) 在x在-1到1之間就會是x md(x)=1 if x>1; md(x)=-1 if x<-1
算術平均數的函數y=mu(x) = (-2 -1 +x +1 +2)/5 = x/5
md(x)=mu(x)有三個解(剛好對應md(x)的三段)
x/5=-1 ==> x=-5 最小解
x/5= x ==> x= 0 中間解
x/5= 1 ==> x= 5 最大解 然後再把x的資料復原回去，乘以10再加500 答案550
: ◆ A certain chemical reaction that will turn 1/4 of the thing into another
: and the remaining 3/4 stay the same. Now B is 4 times of A, what is the ratio
: of A and B after this reaction happen? 7:13
原先B:A = 4:1 = 16:4 (乘四倍為了數字好算)
發生化學反應後
16B=4A+12B
4A=3A+ 1B
----------
7A+13B
A:B (英文應該是ratio of A to B) = 7:13
: ◆ a candy store has two candy: one cost 70 cents per pound another cost 90
: cent per pound. The owner of the store now mix a 85 cent per pound candy with
: 30 pound of the 90 cent, what is the total pound of the mixture candy? (a)40
: (b) 45 (c) 50 (d) 55 (e) 60
這題題目看起來有點怪怪的，容我改寫
The owner mix 30 pound of the 90 cent per pound candy with some 70 cent per
pound candy to make a mixture worth 85 cent per pound. What is the total pound
of the mixture candy?
單價 磅數
0.90 30
0.70 x
-------------
0.85 x+30
0.90*30 + 0.70*x = 0.85*(x+30) ==>x=10
或可用加權平均數概念看出 貴的糖:便宜的 = 3:1 ==>x=10
題目所問為x+30=40
: ◆ there are 12 tables, each can sit maximum people of 4. For 50 people to be
: seated, what is the maximum number of table that sit only 1 person? 4
題目數字有誤，12張桌子都坐滿也只能坐48人，再容我改數字 桌數13 人數40
有S1張桌子只坐1人
有S4張桌子坐滿4人
S1+ S4 <= 13
S1+4S4 = 40
S4=9，S1=4 (我是用答案去反向工程的，或有誤，歡迎指正)
: ◆ A: the number of the odd numbers between square root(12) and 12^2 B:70
: ans: C
root(12) = 3.XX
12^2 = 144
4~144之間奇數數目為70 兩式相等
: ◆ (pie chart) Agriculture:2% Finance:5% Manufacture:8% Managing:15%
: others:60%(還有兩個忘了)
: (1) what is the proportion of agriculture to the shaded area (the items
: mentioned above other than the "other" session) ans:5% choice: 2% 2.5% 5%
: .......
應該有漏掉一個10%的因為有數字的那幾個加起來才30%
2%/(2%+5%+8%+15%+10%) = 2%/40%=1/20=5%
: (2) if there are 187500 people in managing, how many people are there in
: finance are more than people in the agriculture? ans:37500(填空)
187500=15%
finance - agriculture = 3% ==> 15%的1/5
答案就是187500/5 = 37500
: (3) if people in agriculture earn 35000, and that in manufacture is 45000,
: what is the average that these two earn? ans: 43000 (選擇)
加權平均數
agriculture:manufacture = 2%:8% = 1:4
加權平均 = 45000 - (1/5)*(45000-35000)=43000
: 一個圖表題組，給4500受試者填8個領域的滿意度。題目根據圖表問若20%的人都填高滿意
: 度給其中兩個領域那有幾趴的人都選低滿意度在這兩個領域。
(沒圖)
: ◆ 51!-50!/50!-49!=? 50*50/49
上下同除49!即可，很基本
: ◆ 1、2、3、4排列組合的四位數大於3000的機率? 1/2
開頭可以是1 2 3 4 沒有其他特別限制或要求此四者機率一樣
所以開頭3或4的機率就是1/2
: ◆ W、Q兩個立方體給其高(30&25)與底面積比例，問Q體積是W的幾分之幾? 5/8
應該是長方體(rectangular solid)而非立方體(cube)，下面一樣反向工程把題目補全
高(的比例) 底面積比 體積比
W 6 w 6w
Q 5 q 5q
6w:5q = 8:5 ==> w:q=4:3
: 圖表有490/480/510/520問最大另一個數值可讓這5個平均值與median相同? (550)
重複了，前面已答過
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天天都要算數學！
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