Algebra Practice Test

1. If the average of three numbers is V. If one of the numbers is Z and another is Y, what is the remaining number?
A. ZY - V
B. Z/V - 3 - Y
C. Z/3 - V - Y
D. 3V- Z - Y
E. V- Z - Y
2. Two cyclists start biking from a trail's start 3 hours apart. The second cyclist travels at 10 miles per hour and starts 3 hours after the first cyclist who is traveling at 6 miles per hour. How much time will pass before the second cyclist catches up with the first from the time the second cyclist started biking?
A. 2 hours
B. 4 ½ hours
C. 5 ¾ hours
D. 6 hours
E. 7 ½ hours

3. Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together?
A. 12 minutes
B. 15 minutes
C. 21 minutes
D. 23 minutes
E. 28 minutes
4. Mary is reviewing her algebra quiz. She has determined that one of her solutions is incorrect. Which one is it?
A. 2x + 5 (x-1) = 9, x = 2
B. p - 3(p-5) = 10, p = 2.5
C. 4 y + 3 y = 28, y = 4
D. 5 w + 6 w - 3w = 64, w = 8
E. t - 2t - 3t = 32, t = 8
5. What simple interest rate will Susan need to secure to make $2,500 in interest on a $10,000 principal over 5 years?
A. 4%
B. 5%
C. 6%
D. 7%
E. 8%
6. Which of the following is not a rational number?
A. -4
B. 1/5
C. 0.8333333...
D. 0.45
E.
7. A study reported that in a random sampling of 100 women over the age of 35 showed that 8 of the women were married 2 or more times. Based on the study results, how many women in a group of 5,000 women over the age of 35 would likely be married 2 or more times?
A. 55
B. 150
C. 200
D. 400
E. 600
8. John is traveling to a meeting that is 28 miles away. He needs to be there in 30 minutes. How fast does he need to go to make it to the meeting on time?
A. 25 mph
B. 37 mph
C. 41 mph
D. 49 mph
E. 56 mph
9. If Steven can mix 20 drinks in 5 minutes, Sue can mix 20 drinks in 10 minutes, and Jack can mix 20 drinks in 15 minutes, how much time will it take all 3 of them working together to mix the 20 drinks?
A. 2 minutes and 44 seconds
B. 2 minutes and 58 seconds
C. 3 minutes and 10 seconds
D. 3 minutes and 26 seconds
E. 4 minutes and 15 seconds
10. If Sam can do a job in 4 days that Lisa can do in 6 days and Tom can do in 2 days, how long would the job take if Sam, Lisa, and Tom worked together to complete it?
A. 0.8 days
B. 1.09 days
C. 1.23 days
D. 1.65 days
E. 1.97 days
11. Jim has 5 pieces of string. He needs to choose the piece that will be able to go around his 36-inch waist. His belt broke, and his pants are falling down. The piece needs to be at least 4 inches longer than his waist so he can tie a knot in it, but it cannot be more that 6 inches longer so that the ends will not show from under his shirt. Which of the following pieces of string will work the best?
A. 3 feet
B. 3 ¾ feet
C. 3 ½ feet
D. 3 ¼ feet
E. 2 ½ feet
12. The last week of a month a car dealership sold 12 cars. A new sales promotion came out the first week of the next month and the sold 19 cars that week. What was the percent increase in sales from the last week of the previous month compared to the first week of the next month?
A. 58%
B. 119%
C. 158%
D. 175%
E. 200%
13. If two planes leave the same airport at 1:00 PM, how many miles apart will they be at 3:00 PM if one travels directly north at 150 mph and the other travels directly west at 200 mph?
A. 50 miles
B. 100 miles
C. 500 miles
D. 700 miles
E. 1,000 miles
14. During a 4-day festival, the number of visitors tripled each day. If the festival opened on a Thursday with 345 visitors, what was the attendance on that Sunday?
A. 345
B. 1,035
C. 1,725
D. 3,105
E. 9,315

Answers & Explanations

1. D: The average of the three numbers may be written as (^Z+Y+x)/3=V, where x represents the value of the third number. Solving for x will give the value of the remaining number. Multiplying both sides of the equation by 3 gives Z + Y + x = 3V. Subtraction of Z and Y, from both sides of the equation gives x = 3V - Z - Y. The value of the remaining number is 3V - Z - Y.
2. B: The intersection of the graphs of the equations, y = 6x and y = 10x - 30, represents the time (x) and distance (y), where the second cyclist catches up with the first cyclist. The point of intersection is (7½, 45). Thus, after 7½ hours from the time the first cyclist starts and 4½ hours from the time the second cyclist starts, the second cyclist catches up with the first cyclist.
3. B: The amount of time it takes the three of them to fill the pool may be represented by the equation, ¹/₃₀+¹/₄₅+¹/₉₀=¹/_t, where t represents the number of minutes. Solving for t gives t = 15. Thus, after 15 minutes, the three of them will fill the pool, when working together.
4. E: The correct solution is t = -8. When adding t to -5t, it looks like she forgot to include the negative sign on 4t, which gave an incorrect solution of positive 8.
5. B: Simple interest is represented by the formula, I = Prt, where I represents the interest amount, P represents the principal, r represents the interest rate, and t represents the time. Substituting 2,500 for I, 10,000 for P, and 5 for t, gives the equation, 2,500 = 10,000(r)(5). Thus, r = 0.05, or 5%.
6. E: √2 has a decimal expansion that does not terminate or repeat (1.414213562…). Thus, it is an irrational number.
7. D: The following proportion may be used to solve the problem: ⁸/₁₀₀=^x/₅₀₀₀. Solving for x gives x = 400. Thus, 400 women, out of the random sample of 5,000, will likely have been married 2 or more times.
8. E: The following equation may be used to find the speed at which he needs to travel: 28/x=1/2. Thus, x = 56. He needs to travel 56 mph, in order to make it to the meeting on time.
9. A: The amount of time it takes the three of them to mix the 20 drinks may be represented by the equation, 1/5+1/10+1/15=1/t, where t represents the number of minutes. Solving for t gives t=30/11, which equals 2.73 minutes. There are 60 seconds in a minute, so multiply 60 by 2.73 minutes to get 163.8 seconds. Divide that by 60, and it comes to approximately 2 minutes and 44 seconds.
10. B: The amount of time it will take the three of them to finish the job, when working together, may be modeled by the equation, 1/4+1/6+1/2=1/t, where t represents the number of days. Solving for t gives t=12/11, or 1.(09). Thus, it will take the three of them 1.09 days to finish the job.
11. C: The inequality, 40 ≤ x ≤ 42, represents his situation. A length of 3 1/2 feet equals 42 inches, which satisfies the inequality.
12. A: The percent increase may be represented as (19-12)/12, which equals 0.583 ?. Thus, the percent of increase was approximately 58%.
13. C: The Pythagorean theorem may be used to solve the problem. The vertical distance of the plane traveling north, after 2 hours, is 300 miles. The horizontal distance of the plane traveling west, after 2 hours, is 400 miles. Thus, the following equation represents the distance between the planes, at 3 P.M.: 300²+400²=c². Solving for c gives √250,000=c, or c = 500. After 2 hours, the planes are 500 miles apart.
14. E: The problem represents a geometric sequence, with a common ratio of 3. Thus, the problem may be modeled with the equation, a₄=345^.3^4-1, where a₄=9,315. The problem may also be solved by writing the sequence, 345, 1035, 3105, 9315, and identifying the value of the fourth term as the number in attendance for Sunday, or the fourth day.

U.S. High School Students

Report: Chinese Third-Graders Falling Behind U.S. High School Students in Math, Science

Experts say the average Chinese third-grader is now, alarmingly, barely able to compete with a U.S. high school senior.

CHESTNUT HILL, MA—According to an alarming new report published Wednesday by the International Association for the Evaluation of Educational Achievement, third-graders in China are beginning to lag behind U.S. high school students in math and science.
The study, based on exam scores from thousands of students in 63 participating countries, confirmed that in mathematical and scientific literacy, American students from the ages of 14 to 18 have now actually pulled slightly ahead of their 8-year-old Chinese counterparts.
“This is certainly a wake-up call for China,” said Dr. Michael Fornasier, an IEA senior fellow and coauthor of the report. “The test results unfortunately indicate that education standards in China have slipped to the extent that pre-teens are struggling to rank among even the average American high school student.”
“Simply put, how can these third-graders be expected to eventually compete in the global marketplace if they’re only receiving the equivalent of a U.S. high school education?” Fornasier added.
Fornasier stressed that while the gap is not yet dramatically sizable, it has widened over the past two years after American high schoolers tested marginally higher in algebra, biology, and chemistry than, shockingly, most of China’s 8- and 9-year-olds.
“For decades, young children in China have scored at the expected level of their peers in American high schools, so this is a very worrying drop in performance,” said Fornasier, adding that the majority of Chinese third-graders are now a full year behind the average U.S. 12th-grader in their knowledge of calculus. “In the chemistry portion of the exam, for example, Chinese children proved to be slightly deficient compared to American teenagers in their understanding of the periodic table, molecular structure, and the essential principles of atomic theory.”
“And even when they did test at the same level in mathematics, it often took Chinese elementary school students 10 to 15 minutes longer to do simple things like factor a polynomial equation or compute the derivative of a continuous function,” Fornasier added. “That just isn’t normal.”
In addition to disappointing marks from grade school children in China, 10-year-olds in Germany, South Korea, Japan, Switzerland, and New Guinea also reportedly tested an average of three percentage points lower than U.S. high school seniors in physics, with education officials from each country expressing deep concerns about the increasingly mediocre quality of their primary schools.
In light of the alarming study, many in China have called for considerable reforms of the country’s education system, including implementing far stricter standards for teachers, investing in better learning materials, and increasing the length of school days.

Pythagoras Philosophy

 Pythagoras Philosophy


Like Thales, Pythagoras is rather known for mathematics than for philosophy. Anyone who can recall math classes will remember the first lessons of plane geometry that usually start with the Pythagorean theorem about right-angled triangles: a²+b²=c². In spite of its name, the Pythagorean theorem was not discovered by Pythagoras. The earliest known formulation of the theorem was written down by the Indian mathematician Baudhayana in 800BC. The principle was also known to the earlier Egyptian and the Babylonian master builders. However, Pythagoras may have proved the theorem and popularised it in the Greek world. With it, his name and his philosophy have survived the turbulences of history.

Pythagoras founded a society of disciples which has been very influential for some time. Men and women in the society were treated equally -an unusual thing at the time- and all property was held in common. Members of the society practised the master’s teachings, a religion the tenets of which included the transmigration of souls

It could be said that Pythagoras saw the study of mathematics as a purifier of the soul, just like he considered music as purifying. Pythagoras and his disciples connected music with mathematics and found that intervals between notes can be expressed in numerical terms. They discovered that the length of strings of a musical instrument correspond to these intervals and that they can be expressed in numbers. The ratio of the length of two strings with which two tones of an octave step are produced is 2:1.

Music was not the only field that Pythagoras considered worthy of study, in fact he saw numbers in everything. He was convinced that the divine principles of the universe, though imperceptible to the senses, can be expressed in terms of relationships of numbers. He therefore reasoned that the secrets of the cosmos are revealed by pure thought, through deduction and analytic reflection on the perceptible world.

This eventually led to the famous saying that “all things are numbers.” Pythagoras himself spoke of square numbers and cubic numbers, and we still use these terms, but he also spoke of oblong, triangular, and spherical numbers. He associated numbers with form, relating arithmetic to geometry. His greatest contribution, the proposition about right-angled triangles, sprang from this line of thought: Pythagoras Proposition“The Egyptians had known that a triangle whose sides are 3, 4, 5 has a right angle, but apparently the Greeks were the first to observe that 3²+4²=5², and, acting on this suggestion, to discover a proof of the general proposition. Unfortunately for Pythagoras this theorem led at once to the discovery of incommensurables, which appeared to disprove his whole philosophy. In a right-angled isosceles triangle, the square on the hypotenuse is double of the square on either side.

No texts by Pythagoras are known to have survived, although forgeries under his name — a few of which remain extant — did circulate in antiquity. Critical ancient sources like Aristotle and Aristoxenus cast doubt on these writings. Ancient Pythagoreans usually quoted their master's doctrines with the phrase autos ephe ("he himself said") — emphasizing the essentially oral nature of his teaching.

Pythagoras believed that all relations could be reduced to number relations. As Aristotle wrote:-

The Pythagorean ... having been brought up in the study of mathematics, thought that things are numbers ... and that the whole cosmos is a scale and a number.

Pythagoras studied properties of numbers which would be familiar to mathematicians today, such as even and odd numbers, triangular numbers, perfect numbers etc. However to Pythagoras numbers had personalities which we hardly recognise as mathematics today

Each number had its own personality - masculine or feminine, perfect or incomplete, beautiful or ugly. This feeling modern mathematics has deliberately eliminated, but we still find overtones of it in fiction and poetry. Ten was the very best number: it contained in itself the first four integers - one, two, three, and four [1 + 2 + 3 + 4 = 10] - and these written in dot notation formed a perfect triangle.

Pythagoras transformed the study of geometry into a liberal education, examining the principles of the science from the beginning and probing the theorems in an immaterial and intellectual manner: he it was who discovered the theory of irrational and the construction of the cosmic figures.

Theory: They taught that in Number only is absolute certainty to be found; that Number is the Essence of all things; that things are only a copy of Numbers; nay, that in some mysterious way, Numbers are things themselves. This Number theory was probably worked out from the fundamental conception, that, after destroying or disarranging every other attribute of matter, there still remains the attribute Number; we still can predicate that the thing is one. With this doctrine of Number was intimately connected that of the Finite and Infinite, corresponding respectively with the Odd and Even in Number; and from a combination of this Finite and Infinite it was taught that all things in the Universe result. The abstract principle of all perfection was One and the Finite; of imperfection, the Many and the Infinite

The Ethical teaching of the Pythagoreans was of the purest and most spiritual kind; Virtue was regarded as a harmony of the soul, a conformity with or approximation of the Deity; Self-restraint, Sincerity, and Purity of Heart were especially commended; and Conscientiousness and Uprightness in the affairs of life would seem to have been their distinguishing characteristics.

The Pythagorean system was carried on by a succession of disciples down to about 300 BC, when it seems to have gradually died out, being superseded by other systems of philosophy

The TEN PRINCIPLES of PYTHAGORAS Also known as the table of Opposites

    limit unlimited
    odd even
    one plurality
    right left
    male female
    at rest moving
    straight crooked
    light darkness
    good bad
    square oblong

OTHER QUOTES…

    Pythagoras calls drunkenness an expression identical with ruin.
    The fundamental element of the world is the number 1.
    Among what he calls his precepts were such as these:
    Do not stir the fire with a sword.
    Do not sit down on a bushel.
    Do not devour the heart.
    In the time of Pythagoras that proverbial phrase « Ipse dixit » was introduced into ordinary life.
    Animals share with us the privilege of having a soul.

Pythagoras had also spoken about the five solid figures, called the mathematical solids. He said that…

    The earth is made from a cube,
    Fire from the pyramid,
    Air from the octahedron,
    Water from the eicosahedron,
    And the sphere of the whole (the Aither) from the dodecahedron.

Pythagoreans held that the cosmos ‘Breathed in ‘from the Infinite Breath outside it: « aither ».

Pythagoras derived the world from the fire and the fifth element.

Philolaus the Pythagorean says that

    FIRE is at the centre, calling it the heart of the universe;
    second comes the COUNTER-EARTH, and
    third the inhabited earth which in its revolution remains opposite the counter-earth,

wherefore the inhabitants of this earth do not see those of the other.

« They call the earth a star as being itself an instrument of time, for it is the cause of day and night.

Day it creates by being lit up on the side which turned toward the sun, and night through the cone of his shadow.

    COUNTER EARTH was the name given by the Pythagoreans to the MOON and also HEAVENLY EARTH.»
    Pythagoras maintained that the fire was at the centre of the cosmos. Many Pythagoreans followed his example and approved this ancient insight of HELIOCENTRICITY.

Pythagoras had also a certain opinion about the Nature of the Soul. He said one day: « In the case of the soul, the three parts that have to be brought into accord are of course reason,

Calculus - Center of Mass

Center of Mass

In this section we are going to find the center of mass or centroid of a thin plate with uniform density ρ.  The center of mass or centroid of a region is the point in which the region will be perfectly balanced horizontally if suspended from that point.

So, let’s suppose that the plate is the region bounded by the two curves  and  on the interval [a,b].  So, we want to find the center of mass of the region below.

We’ll first need the mass of this plate.  The mass is,

Next we’ll need the moments of the region.  There are two moments, denoted by M_x and M_y.  The moments measure the tendency of the region to rotate about the xand y-axis respectively.  The moments are given by,

Equations of Moments

The coordinates of the center of mass, , are then,

Center of Mass Coordinates

where,

Note that the density, ρ, of the plate cancels out and so isn’t really needed.

Let’s work a couple of examples.

Example 1  Determine the center of mass for the region bounded by ,  on the interval . Solution Here is a sketch of the region with the center of mass denoted with a dot. Let’s first get the area of the region.                                                           Now, the moments (without density since it will just drop out) are,    The coordinates of the center of mass are then,                                                                Again, note that we didn’t put in the density since it will cancel out. So, the center of mass for this region is .

Example 2  Determine the center of mass for the region bounded by  and . Solution The two curves intersect at  and  and here is a sketch of the region with the center of mass marked with a box. We’ll first get the area of the region.                                                           Now the moments, again without density, are                       The coordinates of the center of mass is then,                                                               The coordinates of the center of mass are then, .