question #1: a right triangle with one length of 60 degrees has two leg lengths of x of and 2√3. if side 2√3 is across from the 60-degree angle, what is the measurement of x?

a) 2

b) 2√3

c) 4

d) 4√2

e) 4√3

here it is a good example of how drawing the triangle can be vital to getting the correct answer on a right triangle question. a 30-60-90 triangle has a ratio of x: x√3: 2x. if 2√3 is across from the 60-degree angle, then x = 2. the correct answer is (a).

question #2: if triangle abc is a 30-60-90 right triangle, which of the following sets could represent triangle abc’s side lengths?

a) 2, 2, 2

b) 2, 2, 2√2

c) 2, 2√2, 2√2

d) 2, 2√2, 2√3

e) 2, 2√3, 4

for each answer choice x = 2, so knowing that the ratio of a 30-60-90 is x: x√3 : 2x, we can plug x in to get: 2: 2√3 : 2(2) or 2: 2√3 : 4. the answer is (e).

now let’s look at an example with the 45-45-90 triangle:

question #3: which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?

recall that a fractional exponent is just another way of expressing a root. we know the ratio for a 45-45-90 is x: x: x√2, which means two of the sides must be equal. that eliminates d and e. out of the remaining choices, only (c) correctly expresses the ratio.

sometimes the act math test will make triangles more challenging by disguising them within larger figures. let’s look at an example for our final practice problem!

question #4: what is the area of the trapezoid in the figure provided?

(a) 210

(b) 360

(c) 440

(d) 570

(e) 620

the trapezoid consists of two smaller shapes: a triangle and a rectangle. draw a line from point a across this figure to form the two shapes.  find the area of each shape and then add them together to find the area of the trapezoid.  area of rectangle = length x width. therefore, the area of the rectangle is 18×20 = 360

therefore, area of the triangle = (¹/₂)(20)(21) = 210.  the total area of the trapezoid = 360+210 = 570, or (d).

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if you can’t figure out how to set up the algebra to solve, using an act math strategy like plugging in can definitely lead to better scores. use the answer choices to your advantage, trying each one out, by backsolving, then looking for the choice that confirms the information in the question-stem.

2) bubble in as soon as you find the answer

don’t wait until the end of the act test to mark your answers. if you accidentally run out of time and haven’t bubbled any in, you won’t get any points for answers circled in the test booklet, so get them down on that scantron as you go!

3) write a strong thesis for the act writing test

to ace the act writing test, make sure you only choose one side of the act writing prompt to support. don’t try to “hedge your bets” or have it both ways. choose the side for which you can think of the best examples, and make the best argument you can. remember, this is a test of your writing abilities, not your opinions!

4) don’t be tempted to cheat!

a surefire way to get a bad act score is to constantly compare yourself to the test-takers around you. the act is only one step in the college admissions process. don’t put so much pressure on yourself that you can’t focus. and definitely don’t copy your neighbor’s answers. your best guess is probably better than their’s!

5) don’t leave any questions blank

the act is a nice alternative to the sat for some test-takers because unlike the sat, the act has no wrong answer penalty. this is a very importance difference. this means you should answer every single act test question. don’t leave anything blank on your 2022世界杯入门名单s; set a standard and follow it throughout your test prep.

6) leave the pens at home!

use a soft lead no. 2 pencil with a good eraser. according to the official act website, a mechanical pencil or ink pen is not recommended. if you use one, you run the risk of your answer sheet not being scored accurately on test day.

7) skip the harder questions if you need to

if you find yourself spending more than 2 minutes on any one act test question, you might want to consider skipping it. better scores come from good pacing, and the most important act test tip is to manage your time wisely. in fact, time management is a very necessary skill you will need in college, so if you can demonstrate that ability in your college application with a good act score, you will already have a leg up!

8) underline, circle and jot down notes on the act reading test

these passages can be long. combat boredom, confusion, and tedium with “active reading.” look for the main idea of each paragraph, the author’s point of view, and the overall purpose of the passage as you read. don’t be afraid to circle, underline, or make marks in your test booklet!

9) don’t try to do complex calculations in your head

act the act math test by solving each problem methodically. it’s possible to make mistakes on even the most straightforward calculations, especially when you are under timing pressure.

10) remember some basic science-related vocabulary words

especially vocab like independent variables, dependent variables, direct variation, and indirect variation. look for how the data is presented and how the different variables interact with one another.

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the pythagorean theorem can be represented as in which a and b are the two shorter sides and c is the hypotenuse (the name given to the side across from the 90-degree angle).  to save time on the act math test, it helps to memorize and recognize the common pythagorean triplets. these are common ratios between the sides that come up often in right triangles. the most common are 3:4:5 and 5:12:13. these ratios will also be true for any multiples of 3:4:5 and 5:12:13 such as 6:8:10 or 10:24:26.

for example, if you are told a right triangle has a hypotenuse of 10 and one side with a length of 6, you can tell that the third side is 8. that’s because it must fit a 6:x:10 ratio, which if we divide by two  becomes 3:(x/2):5. this looks a lot like 3:4:5, so x/2 = 4, and x = 8. definitely a good time-saver!

there are also two right triangles that are very important to know called the special right triangles. these are so called because the ratio of their sides never changes. the first is a 30-60-90 triangle. its sides will always be in a ratio of x: x√3 : 2x. the other special triangle is the 45-45-90 triangle. its sides will always be in a ratio of x: x: x√2.

it’s important to remember that for the 30-60-90 triangle, the hypotenuse is the side that has the ratio of 2x. don’t confuse it with the 45-45-90 ratio, and think that the x√3 should be there!

here’s  great question that illustrates how knowing these special right triangle ratios can be invaluable:

which of the following sets of three numbers could be the side lengths, in yards, of a right triangle containing a 45° angle?

here, even if we forgot the 45-45-90 ratios, we can eliminate any choice that doesn’t meet the pythagorean theorem, since the question tells us we’re dealing with a right triangle.

to choose between (c), (d), and (e), notice that a right triangle with one 45 degree angle must have another 45 degree angle. 180 – 90 – 45 = 45. this means that the triangle is isosceles, and the correct answer choice must contain two equal values. the correct answer is (c).

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how is each experiment set up? make sure you understand the method for each experiment. what tools/processes/chemicals are used?

what does the data show? pay close attention to the results of each experiment. usually this is presented in tables or charts. how do the different variables relate to one another? draw arrows on the tables to show the trends.

how do the experiments differ? there will be certain elements common to both experiments, and one or more elements will change from experiment 1 to experiment 2. circle the new information. then focus on the results – do the variables interact similarly or differently in the 2^nd experiment? is the range of data greater or smaller?

try a sample passage and practice question!

passage i

a student wishes to discover for herself what effects, if any, pressure and temperature have on the solubility of various solutes in water.  the solute of a solution is the substance that is dissolved by the solvent.

experiment 1

at a constant pressure of 770 torr, a student tested the solubility of five compounds at different temperatures.  in order to do this, the student started with 100g of water and gradually added the solutes, in five separate trials, until no more of that solute could dissolve.  all of the solutes tested in this experiment were in the solid state unless otherwise indicated.  the results of this experiment are summarized in table 1.

experiment 2

a second experiment was conducted at a constant temperature of 25˚c under different pressures.  in order to do this, the student gradually increased the pressure of the solutions while determining the effect on solubility.  the solvents tested were in either solid or liquid state.  the results of this experiment are summarized in table 2.

question

the solubility of co2 is tested at 25° and 75 torr. according to the data, its solubility in g solute/100 g of water is most likely closest to what value?

(a) 5

(b) 10

(c) 45

(d) 50

according to table 2, the pressure and solubility of a gas are directly proportional. since the solubility of co2 is 20 g/100 g water at 150 torr, its solubility at 75 torr should be half of 20, or 10. the correct answer is (b).

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large numbers and very small decimals are often expressed with exponents using scientific notation. scientific notation involves writing the number as a product of a decimal and the number 10 raised to a certain power. the reason scientific notation is used is that is saves space. who would want to write .000000000000000000547, when  saves us a lot more room?

the value of the exponent indicates the number of places the decimal moves. in our example above, we moved the decimal 19 places to the right, so the exponent was a positive 19.

golden rules of scientific notation: positive exponents move to the right. negative exponents move to the decimal to the left. try a practice question on your own!

scientists testing a certain atomic reaction expected it to take place in seconds. in fact, the reaction actually lasted 100 times longer than they expected. how long did the reaction actually last, in seconds?

let’s start by writing out  since we have a negative exponent, we know the decimal will move to the left. .00000034 = 

now we would multiply the decimal by 100. since there are two zeroes in 100, the decimal will move two places to the right. the answer would be .000034. to rewrite that in scientific notation, we can move the decimal 5 places to the right again, which would be a negative exponent of 5. the answer is d.

a faster way to think about this question is to know that .  we can express the solution as:  if you remember your exponent rules, when we multiply exponents with the same base, we can add the exponents. -7 + 2 = -5. again, this matches choice (d).

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research summaries involve one or more experiments conducted by a group of students or scientists. often some type of phenomena will have been observed and the experiments will be set up to investigate how certain factors affect the phenomena. often a graph, table or figure will accompany the description of the experiments to show the results. here are some strategies to help you with these!

make sure to underline the purpose & method for each experiment as you read (don’t wait until you finish reading everything or you won’t remember!). the purpose tells you why the scientists are conducting the experiment. what are they trying to find out? look for verbs like “to study…” or “to examine…” in the first explanatory paragraph. that is often where the description of the purpose can be found. the method for each experiment will be described in the paragraphs that follow. make sure to make note of what is similar and what is different between the two experiments if there is more than one. sometimes the scientists will change one or more factors between the experiments to see if the results change.

factors, also known as variables, are important elements in these experiments. these are often things like temperature, ph, pressure, time, distance, etc. depending on the way the variable is being used in each experiment, it can be called either dependent or independent. independent variables are those factors that are controlled by the scientists. did the scientists increase the heat in the experiment? did they add or remove pressure? if the scientists were the ones controlling the variable, it is independent.  dependent variables are what the scientists observed changing. let’s say that when the scientists increased the heat in our hypothetical experiment, the time also increased.

to analyze the results, make sure to read all the text provided, including the title of each table. what is in each column? make sure you understand how the labels in the data correspond to the descriptions in the passage. if there is a graph, read what is on the x-axis and y-axis. you should be doing this anyway to understand the factors, but it’s very important to read every single word on the page before diving into those questions! don’t skip over anything just because it looks confusing or has unknown verbiage. a huge part of success on the act science test is simply knowing where to look for the right answer to the question!

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but first, remember that the formula for average speed = total distance / total time. it might help to remember it as as = d / t, or come up with your own fun mnemonic device!

suzanne drove 40 miles to see her aunt and was going 20 mph. it took her 2 hours to get to her aunt’s house. then, she left and drove another 30 miles to the pet store, but this time only drove at 10 mph. if it took suzanne 3 hours to arrive at the pet store, what was her average speed in miles per hour for the entire car ride from her home to the pet store? 

(a) 10

(b) 11

(c) 12

(d) 14

(e) 15

average speed = total distance / total time.

40 miles + 30 miles so the total distance was 70 miles. suzanne drove for 2 hours + 3 hours so the total time was 5 hours. 70/5 = 14.

the average speed for the whole trip was 14 mph.  the correct answer is (d).

notice how the test-maker has made this problem tricky! the average speed in this problem is 14 mph, which is different from  simply taking the mean of the two speeds. if we had just averaged the two speeds (10mph and 20mph) we would have gotten 15mph. average speed is a weighted average. since suzanne spent more time in the problem going 10mph than 20mph, it makes sense that the average speed would be closer to 10mph.

let’s try another word problem!

students at thomas jefferson high school boarded the bus for a field trip that went 15mph through a 30 mile section of the city. the bus then stopped for lunch in a suburb before continuing on a 3 hour tour of countryside at a constant speed of 10mph. finally, the bus drove 40 miles straight back to the high school. if the trip back to thomas jefferson high school took two hours, approximately what was the average speed for the entire field trip?

(a) 11

(b) 12

(c) 13

(d) 14

(e) 15

to find the average speed of the bus, we know we will need to find the total distance and the total time, so we can start by using another formula (distance = rate x time) to help us find the pieces we’re missing for each part of the trip.

for the first part of the field trip: 30 miles = 15mph x t, so we know that t = 2 hours.  for the middle part of the trip, we know that d = 10mph x 3 hours, so we know that d = 30 miles. for the last part of the trip, we know that 40 miles = r x 2 hours, so we know that r = 20mph.  now we can find the total distance and the total time:

total distance = 30 miles + 30 miles + 40miles = 100 miles.

total time = 2 hours + 3 hours + 2 hours = 7 hours.

the average speed = 100 miles/ 7 hours = 14.28mph. since the question used the word “approximately,” we can round to the nearest integer: 14. the correct answer is (d).

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as you read each act science passage, look closely for keywords that help you identify each scientist’s opinions. what does the author like? what does he dislike? what adjectives and adverbs does he use to describe the various scientific topics? he may feel positively about one thing, and negatively about another. feel free to use smiley faces or positive and negative symbols to mark up the passages and make sense of them, and ask yourself questions to help you summarize the author’s point of view. don’t feel like you have to take extensive, detailed notes. a few short notes or underlined sections will be enough to help you refer back! the important thing is to read actively, not passively.

try a sample conflicting viewpoints question asking us to comprehend point of view!

practice

scientist 1
the evolution of the “modern” humans, homo sapiens was a result of parallel evolution from populations of homo erectus and an intermediary of some sort. this process occurred in africa, europe and asia with some genetic intermixing among some members of these populations. there is clear anatomical evidence for this theory when comparing certain minor anatomical structures of homo erectus populations with modern humans from these areas. these anatomical differences are so minor, this is clear evidence that modern humans must have evolved separately in africa, europe and asia. this is the “multi-generational hypothesis.”

which of the following best states the basis for the belief of scientist 1?

1. molecular changes are more important than anatomical differences
2. molecular changes are less important than anatomical differences
3. molecular changes are more important than anatomical similarities
4. molecular changes are less important than anatomical similarities

solution

scientist 1 states: “these anatomical differences are so minor, this is clear evidence that modern humans must have evolved separately in africa, europe and asia.” clearly, scientist 1’s focus is on the “differences.” the answer is (b).

helpful hint

if you are having trouble finding what the overall point of view is for the passage or paragraph, go back to the very first sentence and the very last sentence – often they contain the most obvious clues.
 

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learning the basics of number properties is crucial for success on the act math test. here, you’ll learn some basic terminology to help you on the test.

a factor, or divisor, of a number is a positive integer that evenly divides into that number. for example, 4 is a factor of 12 because 12/4 is an integer, 3. all integers have 1 and themselves as factors. for example there are six factors for the number 12: 1, 2, 3, 4, 6, 12.

the gcf, or greatest common factor, is the biggest factor that two numbers have in common. think of gcf questions as: what is the biggest number that divides evenly into both?

a prime number is a special kind of integer that has exactly two factors: itself and 1.

remember that 1 is not prime, because it only has one factor, itself. 2 is a special number because it is the smallest prime number and the only even prime number. the first few prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.

a multiple is a number that is evenly divisible by another integer. for example, 6 is a multiple of 3. 10 is a multiple of 5. the multiple is always larger than the other integer. the other integer would be considered a factor of the multiple. every number is a multiple of itself.

for example, for the number 12, possible multiples are 12, 24, 36, 48, 60, etc. the lcm, or lowest common multiple, is the smallest number for which both given numbers are factors.
 

practice question

what is the lcm of 12 and 9?

9 = 3 x 3, and 12 = 3 x 4, so the lcm must be 3 x 3 x 4 = 36. by saying that the lcm of 12 and 9 is 36, you are saying that 36 is the smallest number that both 12 and 9 will divide into evenly. notice how the lcm is not necessarily just the product of the two integers.
 

the act math rules of divisibility

in addition to factors and multiples, it helps to know some basic divisibility rules – a great shortcut on certain long division problems!

• a number is divisible by 2 if it’s an even number.
• a number is divisible by 3 if the sum of the digits is divisible by 3.
• a number is divisible by 4 if the last two digits are divisible by 4.
• a number is divisible by 5 if the last digit is either 0 or 5.
• a number is divisible by 6 if it’s divisible by both 2 and 3.
• a number is divisible by 9 if the sum of the digits is divisible by 9.

that’s all for this lesson! brush up on your math basics, then tackle more challenging concepts, like matrices on the act.

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all fractions are rational numbers. a rational number is a number that can be expressed as a ratio of two integers. therefore all integers are also rational numbers. rational numbers either have no decimal (4/1), or have a terminating or repeating decimal (1/4 = .25, 1/3 = .3333).

equivalent fractions are fractions that simplify to the same form. for example, 6/8, and ¾. in a fraction, when you multiply the numerator and the denominator by the same non-zero integer, it’s like multiplying the entire fraction by 1. the value of the fraction doesn’t change.

to simplify fractions, you need to understand the lcm, lowest common multiple, and the lcd, least common denominator. the lcd is the lcm (the smallest number that is also a multiple of both integers). for example, if we were to solve: 2/3 + ¼  – 1/2. the lcd is going to be the smallest number that 3, 4, and 2 divide into evenly. since 2 is already a factor of 4, we can choose 12. since 3, 4, and 2 will all evenly divide into 12.

when you multiply fractions, you can multiply the numerators straight across, then the denominators straight across. before you multiply, you can also cancel out numbers across the numerators and denominators. for example: 1/5 x 5/6 x 12/2 x 1/3 = ?. one way to start if by crossing out the 5 in the denominator of the first fraction, and the 5 in the numerator of the second fraction. that gives us:

1/1 x 1/6 x 12/2 x 1/3. it’s a much easier question now, and we can cross out the 6 in the denominator of the 2^nd fraction if we divide it out of 12, the numerator in the third fraction to give us:

1/1 x 1/1 x 2/2 x 1/3 = 1/3.

a complex fraction is one that contains fractions in the numerator and the denominator. for these, find the lcm of all the denominators to simplify.

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