Statistics
There are 36 members in Mathematics Club. They are going to choose a student representative, who will be selected at random, but he/she CANNOT be one of the 7 committees of the club. What is the probability that Mary, who is a member, but NOT a committee, of the club will be chosen?
D
∵ Number of members but not committees = 36 – 7 = 29
∴ The required probability = 1/29
∴ The correct answer is D
There are 50 members in the Science Club. They are going to choose a student representative at random, but the representative cannot be one of the 12 committee members of the club. What is the probability that Alex, who is a member but not a committee member, will be chosen?
In the standard (x,y) coordinate plane, the graph of the line 6x + 5y – a = 0 passes through the line (2,–4). What is the value of a ?
*HINT* Plug-in and isolate a
6 × 2 + 5 × (–4) – a = 0
∴ a = 12 – 20 = –8
put this equation into point-slope form= 3y-6x=15
*HINT* point-slope form = y=mx+b
The taxi fee is $1.5 for the first 1 mile and $0.2 for each additional 0.25 mile. How much of a 3.75 miles ride?
∵ 3.75 = 1 × 1 + 0.25 × 11 ∴ 1.5 + 0.2 × 11 = 3.7 ∴ The cost is $3.7
A taxi charges $2 for the first 1 kilometer and $0.50 for each additional 0.5 kilometer. How much would a 3-kilometer ride cost?
Given ∠P >∠Q >∠R. The difference of the measures of∠P and∠Q is 90°. The difference of the measures of∠Q and∠R is 180°. The sum of the measures of∠R and∠S is 90°. What is the sum of the measures of∠P and∠S ?
*HINT* Draw a figure
(1) ∠P –∠Q = 90° (2) ∠Q –∠R = 180° (3) ∠R +∠S = 90° ∴ By adding (1), (2), and (3), we have ∠P +∠S = 90° (+ 180° + 90° = 360°)
Triangle ABC
∠A = 90° and ∠B= 45° What is the measure of ∠C
Which of the following data sets has the LEAST mean?
*HINT* mean=sum of all #/#s in data set
A. 1, 1, 1, 1, 5, 5, 5, 5
B. 1, 2, 3, 4, 5, 6, 7, 8
C. 1, 2, 2, 3, 3, 3, 4, 4
D. 3, 3, 3, 3, 3, 3, 3, 3
E. 0, 0, 2, 4, 6, 8, 10, 12
C. 1, 2, 2, 3, 3, 3, 4, 4 = 2.75
Which of the data sets has the least mode?
A. 1, 1, 1, 1, 5, 5, 5, 5
B. 1, 2, 3, 4, 5, 6, 7, 8
C. 1, 2, 2, 3, 3, 3, 4, 4
D. 3, 3, 3, 3, 3, 3, 3, 3
E. 0, 0, 2, 4, 6, 8, 10, 12
Kay will choose 1 novel at random from 80 novels on a bookshelf for her book report. There are four types of novels in the bookshelf and the number of each kind is given in the table below. What is the probability that the novel Kay chooses will be a romance or a mystery?
Science 25 | Romance 31 | Fantasy 17 | Mystery 7
The number of romance novels and mystery novels are 31 + 7 = 38
∴ The required probability is = 38/80 = 19/40
Liam will pick 1 book at random from 20 books on a shelf. The books are divided into four types:
What is the probability that Liam picks a mystery or a biography?
Given that A and B are (–4, 2) and (2, –1) respectively. What is the slope of AB , shown in the standard (x, y) coordinate plane?
*HINT* Draw!
By definition, the slope of AB is:
2 - (-1) / (-4) - 2 = 3/-6 = slope = -1/2
A and B are (2, -4) and (-1, -8)
What is the value of 4 (2 + 1) – 5 | 5 – 7 | ?
*HINT* Absolute value=absolute value from zero!
4 (2 + 1) – 5 | 5 – 7 | = 4 × 3 – 5 × | –2| = 12 – 5 × 2 = 12 – 10 = 2
2 (4 + 1) + |3 - 5|
There is a 7x13 foot garden. In that garden is a pond that measures 5x4 feet. How much of the rest of the garden is grassland?
*HINT* Draw
The area, in square feet, is 7 × 13 – (5 × 4) = 91 – 20 = 71ft
A stadium measures 20 x 15 feet. The playing field is 10 x 10 feet. How much of the space is seating?
In Harare on a certain Sunday, the highest temperature was 95°F and the lowest temperature was 68°F. What was the difference between the highest and the lowest temperatures, in degrees Celsius ?
*HINT* The relationship between the temperature C, in degrees Celsius, and the temperature F, in degrees Fahrenheit, is given by 5/9 (F-32)
Let Ch°C and Cl°C be the highest and the lowest temperatures in Harare on that Sunday respectively. ∵ 5/9 (95-32)=35hC ∵ 5/9 (77-32)+25lC ∵ Ch – Cl = 35 – 25 = 10
How much was a 67°F day in Celsius? *HINT* F to C = 5/9 (F-32)
In a department store, of 80 participants surveyed 18 answered that the service attitude was bad. Given that the only answers were yes/no, what % of customers found the service attitude to be good?
80 - 18 = 62 ∵ 62 / 80 = 77.5%
If out of 39 students, 12 are in band, what % of the students are in band?
In the standard (x,y) coordinate plane, the line represented by which of the following equations passes through (2,5) and is perpendicular to y = –2x + 4 ?
*HINT* perpendicular slope = opposite sign/reciprocal fraction
*HINT* point-slope form = y-y₁=m(x-x₁)
Let m be the slope of the required line. ∵ m × (–2) = –1 ∴ m = 1/2 ∴ The point-slope form gives y – 5 = 1/2 (x – 2) y – 5 = 1/2x – 1y = 1/2 x + 4
In the standard (x,y) coordinate plane, which of the following equations represents a line that passes through the point (1, 3) and is perpendicular to the line y=21x−1?
The prices of a can of soft drink and a bag of potato chips in a store are $1.2 and $3.5 respectively. If Tommy has $18 and he must buy one bag of potato chips, find the maximum number of cans of soft drink he can buy from the store.
Let n be the number of cans of soft drink he can buy.
∵ 1.2n + 3.5 ≤ 18 ∴ 1.2n ≤ 14.5 n ≤ 12.0833333333333 ∵ n is a positive integer. ∴ He can buy at most 12 cans of soft drink
A bottle of water costs $1, and a bag of chips costs $4. If Emma has $12 and she must buy one bag of chips, how many bottles of water can she buy with the remaining money?
Peter bought a chain of wooden fence of 72 m. He wanted to use all the fence to construct a fence for his rectangular garden. If the total width of the garden is 15 m (7.5 + 7.5), what is its length, in m?
*HINT* Set up an equation (2[l]+[w]=p)
Let x m be the length of the rectangle garden.
∵ 2(x + 15) = 72 ∵ x + 15 = 36 ∵ x = 21
Sarah has 24 meters of fencing to build a square garden. If she uses all the fencing, what is the length of each side of the square garden?
The fraction 3 13 is equivalent to the recurring decimal 0.230769 . What is the digit in the 8,002nd decimal place of 0.230769 ? (Note: The digit in the 5th decimal place of 0.230769 is 6.)
*HINT* Every 6th digit is recurring
The number 0.230769 recurrent every 6-digit. ∴ We consider the remainder when the number 8,002 is divided by 6. ∵ 8,002 = 1,333 × 6 with a remainder of 4 ∴ The digit in the 8,002nd decimal place of 0.230769 is equal to the digit in the 4th decimal place of 0.230769 . That is 7
The fraction 31 is equivalent to the repeating decimal 0.333…. What is the digit in the 100th decimal place of this decimal?
There are 13 Chinese songs and 9 English songs in a computer. Now 5 Chinese songs and 3 English songs are to be selected and added to a playlist. Which of the following expressions gives the number of different playlists that could be chosen from these 22 songs?
The number of ways to select 5 Chinese songs = 13C5. The number of ways to select 3 English songs = 9C3. ∴ The number of different playlists is (13C5)(9C3)
A music app has 10 pop songs and 8 rock songs. If you want to create a playlist with 4 pop songs and 2 rock songs, which expression represents the number of different playlists you can make?
The graphs of the functions 2x – y = – 6 and
x + 2y = 2 are shown in the standard (x, y)
coordinate plane below.
The domain of each function is all real numbers. What point do the graphs intersect?
*HINT* Draw a graph
*HINT* solve for one equation, substitute in to the next
2x-y=-6 AND x+2y=2
Solve the first equation for y: 2x−y=−6
Subtract 2x from both sides:−y=−2x−6
Multiply both sides by −1: y=2x+6
y=2x+6 [Equation A]
Substitute y=2x+6 into the second equation:
The second equation is: x+2y=2
Replace y with 2x+6: x+2(2x+6)=2
5x+12=2 x=-2
Substitute x=−2 back into y=2x+6:
y=2(−2)+6
y=−4+6 y=2
(-2,2)
x+y=5 and 2x−y=1
Let a, b and c be real numbers. Which of the following expressions is equivalent to | a × (–b) – c | ?
A. | –ab + c | B. | ab – c | C. | ab + c | D. –| ab + c | E. –| ab – c |
By the definition of absolute value, we have | a × (–b) – c | = | –(ab + c) | = | ab + c |
Let x and y be real numbers. Which of the following expressions is equivalent to ∣x×(–y)+z∣?
A. ∣–xy+z∣ B. ∣xy+z∣ C. ∣xy–z∣
D. -∣xy+z∣ E. –∣xy–z∣
15,000 cm3 of water is poured into a rectangular tank of length 60 cm and width 25 cm. Find the depth of water in the tank, in centimeters.
Let d cm be the depth of the water in the tank. ∵ 60 × 25 × d = 15,000 ∴ d = 10
Find the volume of liquid that a tank where length = 10cm width = 55cm and depth = 3cm
It is known that the assessed value is 4/5 of the property value and the yearly tax is computed as $2.5 per $100 of assessed value. If the yearly tax of a property is $1,200, what is the property value?
Let $x be the property value. ∵ x * 4/5 * 2.5/100 = 1,200 ∴ x = 60,000 ∵ The property value is $60,000
A house is assessed at 43 of its actual value. The yearly tax is $3 per $100 of the assessed value. If the yearly tax is $900, what is the actual value of the house?
A factory produces light bulbs with a mean lifespan of 1,200 hours and a standard deviation of 100 hours. The lifespans of the light bulbs are normally distributed.
*HINT* empirical rule (68-95-99.7 rule) and z-scores
The z-score tells us how many standard deviations a value is from the mean. The formula for the z-score is: z=σX−μ
For 1,100 hours: z1=100/1,100−1,200=100−100=−1
For 1,300 hours: z2=100/1,300−1,200=100/100=1
The z-scores of -1 and 1 correspond to the following cumulative probabilities:
To find the percentage of light bulbs that last between 1,100 and 1,300 hours, subtract the smaller cumulative probability from the larger one:
Percentage=0.8413−0.1587=0.6826 OR 68.2%
A class of 30 students took a math test. The mean score was 75, and the standard deviation was 10. The teacher decides to curve the scores by adding 5 points to each student’s score.
*HINT* New Mean=Original Mean+Standard Deviation
In the standard (x,y) coordinate plane, a point P(–3,2) is translated 4 units to the right to Q, find the coordinates of Q.
*HINT* left-right = x up-down = y
Coordinates of Q is (–3 + 4,2 + 0) = (1,2)
In the standard (x,y) coordinate plane, a point A(2, 5) is moved 3 units to the right to point B. What are the coordinates of point B?
The product of (3x + 4) and the binomial f(x) is the polynomial 18x2 + 45x + 28. What is f(x) ?
*HINT* divide by original
18x2 + 45x + 28 = (3x + 4)(6x + 7) c f(x) = 6x + 7
The product of (2x+3) and the binomial g(x) is the polynomial 6x2+13x+6. What is g(x)?
International Commerce Centre (ICC) is a 118-storey, 1,587 feet commercial building in Hong Kong. It is the world’s 11th tallest building by height now. The total floor area is around 2.8 million square feet and the area of each office floor is roughly 35,000 square feet. In the figure below, an observer found the angle of elevation at V to the top of the ICC to be 35°.
Which of the following expressions is equivalent to the length, in feet, of UV ?
*HINT* Draw
In △UVW, we have ∵ tan35° = WU/UV ∴ UV = 1,587tan35 OR 2,266.5 feet
A skyscraper is 1,800 feet tall. If an observer stands x feet away from the building and measures an angle of elevation of 40° to the top, which expression represents the distance x from the observer to the building?
Starting next year, Miss Chan plans to increase the number of flowers in the garden each year so that the numbers form an arithmetic sequence. In 4 years, there will be 63 flowers in the garden. What is the common difference of the arithmetic sequence?
*HINT* an = a1+(n-1)d
Let d be the common difference and Tn be the general term of the arithmetic sequence in n years. ∵ The first term is 36 and T4 = 63 ∴ 36 + (4 – 1)d = 63 3d = 27 d = 9
A plant grows 2 cm taller every year. If it is 10 cm tall this year, how tall will it be in 3 years? What is the common difference in its height each year?