sat virtual reality ii explanations phần 1 pot

The only thing to do is just to try the 5 answer choices: So, answer choice B must be correct, but just for the fun of it you can try the other 3: You could also have solved this by plug

Section 1 (Math)

1 A Here we have a basic algebra question The important thing to remember here is

that when solving an algebraic equation, you have to do the same thing to both

sides of the equation You’re given an equation with 4x + 2 on the left side of the

equal sign and the question stem asks about 4x + 8 What do you have to do to

4x +2 to make it look like 4x + 8? Right, just add 6 But since you’ve added 6 to

the left side of the equation, you have to add 6 to the right side of the equation also,

since you have to DO THE SAME THING TO BOTH SIDES! Adding 6 to the right

side of the equation gives you 26 + 6, or 32, answer choice (A)

A more mathematical way of expressing what we just did is to write it out like this:

4 x + 2 = 26

+ 6 + 6

4 x + 8 = 32

2 E You have to know a few facts about geometry here, and if you don’t, you might want

to do some review in your Math Reference Book First, you have to know that the

word “bisects” means “cuts in half.” So, if line EX bisects angle BXD, it cuts angle

is half of angle BXD, angle BXD must equal 60˚

Now that we’ve found the degree measure of angle BXD, we can find the degree

measure of angle CXB Angles CXB and BXD are supplementary angles because

they from line segment CXD So angle CXB plus angle BXD equals 180˚ Since

angle BXD is 60˚, angle CXB equals 180˚ – 60˚ or 120˚.

Even if you’ve forgotten most of what you ever learned about geometry, you can

eliminate two answer choices by just eyeballing the figure Figures are drawn to

scale on the SAT, so you can assume that the angle measurement is about what it

looks like You should know what a 90-degree angle looks like — the corner of this

piece of paper Angle CXB in the figure is clearly greater than 90˚, so choices (A)

and (B) can be eliminated So you could guess among answer choices (C), (D), and

(E)

3 C Here you have to know an important concept that appears fairly often on the SAT —

that when two quantities multiplied together equal 0, one or both of those quantities

is equal to 0 What that means in this question is that either x – 2 = 0 or x + 5 = 0

or they both equal 0 If x – 2 = 0, what does x equal? Just add 2 to both sides of

that equation to get x = 2 And do the same thing for x + 5 = 0 to get x = –5 So

You could also have solved this one by backsolving, or trying each answer choice in

the question You need to find all the possible values of x that make the given

equation true There are four numbers mentioned in the answer choices, in various

combinations They are –5, –2, 2, and 5 If x = –5, is the equation true? Yes, the

equation is then (–5 – 2)(–5 + 5) = (–7) × 0 = 0 So, choice (A) is a possibility, but

you can’t be sure if it’s right because you’re looking for ALL the values of x that

make the equation true However, since –5 does work, the answer choice must

contain –5, so it must be (A) or (C) What about 2? (2 – 2)(2 + 5) = 0 × 7 = 0, so if

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x is 2, the equation is also true Therefore, the correct answer choice must have both 2 and –5 so it’s choice (C) If you try plugging –2 or 5 into the equation, you’ll see that they don’t work

4 C Inequalities are very similar to equations The only difference between them is that

if you multiply or divide both sides of an inequality by a negative number, you must reverse the sign Here you have 2 inequalities, which you can solve the same way

as you would solve an equation In the first inequality you can subtract 2 from both sides:

a + 2 > 5

–2 –2

a > 3

The second inequality is similar It can be solved by adding 4 to both sides:

a – 4 < 1

+ 4 + 4

a < 5

Now you’ve figured out that a is less than 5 and a is greater than 3 There’s only one answer choice that is less than 5 and also greater than 3, answer choice (C)

5 B You can solve this one very easily if you remember a convenient mathematical

shortcut A number is divisible by 3 if the sum of its digits is divisible by 3 Check your Math Reference Book for details about this if you don’t remember it This shortcut works with any multiple of 3, so the number 7,X 7X will be a multiple of 3 if the sum of its digits, or 7 + X + 7 + X, is a multiple of 3 7 + X + 7 + X = 14 + 2X,

but you can’t tell if that’s a multiple of 3 unless you know the value of X The only

thing to do is just to try the 5 answer choices:

So, answer choice (B) must be correct, but just for the fun of it you can try the other 3:

You could also have solved this by plugging in each of these five answer choices for

6 D Here you have to know the meaning of the word “equilateral.” This is an important

word in geometry It means “equal in length.” So an equilateral triangle is a triangle where all three sides are the same length, an equilateral quadrilateral is one where all four sides are the same length, etc So, the equilateral polygon in this question also has equal sides, but you’re not told how many sides the polygon has

However, you are told its perimeter, and perimeter is a word that you’re probably familiar with The perimeter of a polygon is the sum of the lengths of its sides

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In this case, all the sides of the polygon have the same length because it is

equilateral Does this tell you anything else about the perimeter? Yes, it does It

tells you that the perimeter is equal to the number of sides times the length of a

side Think about it like this — the perimeter of an equilateral triangle is just 3 times

the length of a side, the perimeter of a square is 4 times the length of a side, etc

So you just have to figure out which of the five numbers in the answer choices could

be multiplied by the number of the sides of the polygon to give you 21 The number

of sides of the polygon must be an integer, and the answer choices are all integers,

so you need 2 integers that when multiplied together give you 21 The only answer

choice that works is choice (D), 3, since 3 × 7 = 21

7 B Since this is a word problem, the first step is to translate it from words into math It’s

also a percent problem, so you can use the formula PERCENT × WHOLE = PART

The word “what” comes before the word percent (in the second line of the question

stem) so the percent is what you’re looking for The word “of” can be translated to

“times,” and since it comes before “Jorge’s monthly earnings” his earnings must be

the WHOLE The end of the question stem says “does he spend on rent” which we

can translate to “equals rent.” So, the question stem really says “What percent times

earnings equals rent?” In other words, earnings is the WHOLE, rent is the PART,

and we’re looking for the PERCENT

The first sentence gives us some numbers for these quantities It says that his

monthly earnings are $2000 and his monthly rent is $600 So, you can substitute

these numbers into the equation PERCENT × WHOLE = PART like this:

PERCENT =

Now you just have to make look like a percent You can get rid of the dollar

signs, since you have one in the numerator and one in the denominator (Units like

dollars, hours, etc cancel just like numbers when they’re in fractions.) If you divide

both the 600 and the 2000 by 100, you get (You can think of it as just chopping

off the same number of zeros from both numbers.) is the same as or 3 To

convert a decimal like 3 to a percent, just multiply by 100% So 3× 100% = 30%,

choice (B)

8 B Whenever you see an odd/even question (one in which you’re told a certain number

of numbers are odd or even and asked whether another number is odd or even) you

should always approach it the same way — by picking numbers Remember that

what is true for one odd or even number is generally true for all of them So pick the

most simple odd or even numbers for your variables and just see what happens

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$600



$2000

$600



$2000

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Here you’re told not only that a is an odd number but also that it’s negative, so you just pick a negative odd number for a –1 would be a good choice Since b is

positive as well as even, 2 would be a good choice for b Now all you have to do is plug those values into the answer choices (Just don’t forget the rules of multiplying and dividing by negative numbers.)

(A) (–1) + 2 = 1 (B) –(–1)(2) = 2 (C) (–1)(2) = –2

(E) 2 – (–1) = 2 + 1 = 3 The only one which is even and positive is answer choice (B)

9 D Here you have a word problem, and again the first step is to translate from words to

math You’re told that Patty uses 2 gallons of paint to cover 875 square feet That’s

a rate of 2 gallons per 875 square feet, or written a little more mathematically:

You have to find the number of gallons she’ll use for 4,375 sq ft One way to do that is to compare it with the rate you’re given by setting up a proportion:

=

=

Multiplying both sides by 4,375, we find that x = (4,375) = 2(5) = 10

10.A The question tells us that the volume of the rectangular box is completely taken up

by the smaller boxes We can find the volume of the large box because we know all its dimensions We also know the number of small boxes to be stored in the large box so we can divide the volume of the large box by this number to find the volume

of one of the smaller boxes The volume of the large box is length × width × height

which is 24 × 10 × 15 = 3600 Since there are 60 smaller identical rectangular

boxes, each one must have a volume of or 60 Only choice (A) has a volume

of 60, since 2 × 5 × 6 = 60, so choice

(A) must be correct

11.E Here we’re told that the sum of 4 numbers is between 53 and 57 Since the average

of a group of numbers is the sum of the numbers divided by how many numbers

there are, the average of this group of numbers is somewhere between and 57

4

53

 4

3600

 60

2

 875

x

 4,375

2

 875

x gallons



4,375 sq ft

2 gallons



875 sq ft

2 gallons



875 sq ft

2

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= 13 and = 14 , so the average must be greater than 13 and less

than 14 There’s only one answer choice that fits, 14, choice (E)

12.A Here you’ve got a rectangle with sides y and y + 4 and area 32 Let’s forget about

the circles for a minute This means that y(y + 4) = 32 You can make this into the

quadratic equation y2+ 4y – 32 = 0 and solve for y, but it’s much easier to just try to

guess the value of y What 2 numbers have 32 as their product where one is 4

greater than the other? The numbers 8 and 4 should come to mind So y = 4 and

the width of the rectangle is 4 and the length is 8 Now to the circles You have to

notice something important here — the diameter of each circle is the same length

as the width of the rectangle If you draw a vertical diameter in either circle you’ll

see that If you didn’t notice that don’t feel too bad — circles inscribed in squares or

rectangles are pretty common on the SAT so you’ll know what to look for next time

you see this type of figure So what’s the length of the diameter? That’s right, 4

So the radius is half of that, or 2, and the area is π× 22, or 4π, choice (A)

This is also a good eyeballing problem If the rectangle has an area of 32, then half

the rectangle has an area of 16 The area of either circle looks like it’s less than half

the area of the rectangle, or less than 16 Which answer choices are less than 16?

If you remember that π is a little more than 3 (which you should remember) then

you’ll know that only choices (A) and (B) are less than 16, so you can narrow it

down to 2 answer choices and guess

13.D Here you have to know something about coordinate geometry If your coordinate

geometry skills are rusty, do some review in your Math Reference Book

The easiest way to do this question is to estimate values for the x- and

y-coordinates of each point Look at point A We don’t have too much to go on, but it

looks like the x-coordinate of A is about 1 It also looks like the y-coordinate of

point A is about Is x – y < 0 in this case? No, because 1 – = 1, so choice

(A) is out Since point B is sitting right on the x-axis, its y-coordinate is 0 Its

x-coordinate looks like it’s about , and – 0 = , so (B) is out also Point C looks

like the point (1,–1 ), so x – y is 1 – (–1 ), which is positive Eliminate (C) (D) is

You might have wanted to approach this a little more theoretically Under what

conditions would the expression x – y be negative? If x is a positive number, or 0,

then x – y would be negative if y were greater than x Are there any points which

meet those requirements? No, because although points A, B, and C all have

positive x-coordinates, they also all have y-coordinates which are smaller than their

respective x-coordinates What about when x is negative? If x is negative and y is

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positive, then x – y must be negative, so all you have to do is find a point where x is

negative (to the left of the y-axis) and y is positive (above the x-axis) and you’ve found the answer, again choice (D)

14.C Remember – when you’re given an odd-shaped figure, think of it as a combination of

ordinary figures So, think of quadrilateral ABCD as 2 right triangles pushed together and work with the triangles one at a time You’re given the lengths of the legs of triangle BCD but not the length of the hypotenuse Notice that the 2 triangles have the same hypotenuse – this is very important The other triangle,

the length of Going back to triangle BCD, if the legs have lengths 6 and 8, how long is the hypotenuse? You could plug 6 and 8 in for a and b in the Pythagorean

equation a2 + b2 = c2 which would tell you that the hypotenuse has length 10, but you should have recognized the 6-8-10 right triangle without having to do that (It will save you a lot of time on the SAT if you memorize the ratios of the common Pythagorean triplets.) Now you know the hypotenuse of both triangles In triangle

and c) to get a value of 53, but again you should have recognized this as a multiple of the Pythagorean ratio 1:3:2 However you do it, side AD has length

53, answer choice (C)

15.E There’s a couple of different ways to solve this one You could sketch a coordinate

plane and plot points C and D Then simply draw a line that connects them Which

of the answer choices could be on this line? The only one which is anywhere near the line is (1,4) choice (E)

You could do it a little more formally and use the two points given to find the equation of the line (Your Math Reference Book goes into more detail about this topic.) The slope of a line is the change in y divided by the change in x The

change in y is

2 – 5, or –3, and the change in x is –1 – 2, or –3, so the slope of the line is , or

1 You can find the equation of a line with the formula y = mx + b where m is the

slope and b is the y-intercept (b is the value of y when x = 0.) Plugging in our value

of 1 for the slope results in y = x + b, and you can find b by just plugging in the coordinates of any point that you know is on the line If we plug in the point (2, 5)

we get 5 = 2 + b, so b = 3 and the equation of the line is y = x + 3 Only the point

(1, 4) fits into this equation so again the correct answer is (E)

16.E Here you’re given the repeating decimal 0476190476190 You have to notice

what exactly it is that repeats It’s the six digits 047619 Since these six digits will repeat forever, the first six digits after the decimal point are 047619, the next six digits, or the 7th through 12th digits after the decimal point, are also 047619, then the 13th through 18th digits, etc Therefore the digits in place numbers 6, 12, 18, and every other multiple of 6 will be a 9 That means that the 48th digit will be a 9, since 48 is a multiple of 6 If the 48th digit is a 9, then the 49th is 0, the 50th is 4, and the 51st is 7, choice (E)

–3

 –3

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17 B The only thing you really need to remember for probability questions is that the

probability of an event occurring is the number of desirable outcomes divided by the

number of possible outcomes (See your Math Reference Book for more details

about probability.) In this question, Sharon is choosing 1 coin at random out of

many How many? Well, there are 6 + 5 + 10 or 21 coins at the beginning, but

you’re told that Sharon has already discarded one of them so there must be 20

coins left So the number of possible outcomes is 20, since she will be randomly

choosing only 1 out of the 20 Since she wants to choose a quarter, the number of

desired outcomes is simply the number of quarters There were 6 quarters to start

with, and since you’re told that the coin she discarded is not a quarter, there must

still be 6 quarters So the number of quarters is 6, and the probability of choosing

one is 6 out of 20, or , or , answer choice (B)

18.C There are a number of different ways to do this one One way is to think about what

you know about number properties You have two numbers, a and b One is an

integer and the other isn’t You have to combine them in a way that results in an

integer If you take a look at the answer choices you’ll see that (A) adds them

together, (B) subtracts the noninteger from the integer, (C) multiplies them together,

(D) subtracts the noninteger from twice the integer, and (E) divides the noninteger

by the integer We don’t know what number a is, but it doesn’t really matter since

we know that it could be any integer except 0 What happens if you add or subtract

a noninteger (such as a fraction like ) from a? The resulting number could not be

an integer It might help to draw a number line:

You can see that whatever the value of b, adding or subtracting it from a will leave

you on the number line somewhere between 2 integers You can therefore eliminate

choices (A) and (B) Choice (D) is similar, except that you are subtracting b from 2a

instead of a But if a is an integer, then so is 2a, and you’ve already determined that

subtracting b from an integer will not result in an integer, so choice (D) is out That

leaves (C) and (E) Can you multiply an integer and a non-integer together and

getan integer? Try to come up with some numbers that do that What about 2 and

? Those two numbers multiplied together give you a result of 1, which is an

integer So do 3 and , or 8 and , and many other combinations So you’ve

found an answer choice that could give an integer result, which is all you need, so

choice (C) must be correct

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a – 1

a

a + 1 a + 2

You could have tried picking numbers for a and b from the start, but that’s chancy in

this particular question since most combinations would give you noninteger results for all 5 answer choices For example if you had picked 1 for a and for b, the

result would be a fraction in every answer choice Then you’d just have to keep trying different combinations of numbers until you found one that gave an integer result in one of the five answer choices, but again you’d find that only choice (C) does

19.E Equilateral isa word you have to know for the SAT It means “equal in length.” So

an equilateral triangle is a triangle whose 3 sides are all equal in length So if the expressions 3x + 1 and x + 7 represent sides of an equilateral triangle, they must be equal We can use this to set up an algebraic equation: 3x + 1 = x + 7, which can

be solved for x The result is x = 3, and all you have to do is plug that into either expression That will tell you that the length of a side of this equilateral triangle is

10, so the perimeter (which you should know is the sum of the length of the sides) is equal to 10 + 10 + 10, or 30, choice (E)

20.D Since this word problem has a lot of details, it’s important to look at them one at a

time But first, you need the overall picture It helps to think about what’s going on

in a general sense — you’re trying to figure out how much money it will cost to run a certain air conditioner for 90 days What information are you given? The number of hours per day the air conditioner is on, which is called h hours, and how much it costs to run the air conditioner per hour What is that cost? It’s 12 cents per hour if you run it for up to 8 hours, then after 8 hours it goes down to 10 cents an hour You know that you’re running it h hours per day, but why does it give you that 8 < h <

24? Because they want h to be greater than 8 so that you have to deal with the change from 12 to 10 cents per hour, and because they want h to be less than 24

since there are only 24 hours in a day So h is a number like 10 or 15 One good way to start is to figure out how much it costs to run this air conditioner for just one day The first 8 hours cost a different amount than the rest of the day, right? In the first 8 hours it costs 12 cents per hour, for a total of 8 × 12 cents for the whole 8 hours Since it looks like nothing’s been multiplied out in the answer choices, leave the numbers the way they are instead of multiplying them out You’ve probably noticed that all the answer choices have the number 12 in them That’s because the question stem asks for the cost in dollars So, in dollars those first 8 hours will cost 8 times 12, since 12 dollars is the same as 12 cents What about the remaining hours in the day? Suppose h is 10 Then the air conditioner is on for 10 hours per day, and it costs 8 × 12 dollars for the first 8 hours and how much for the remaining 2? Right, 2 × 10 cents, or 2 × 10 dollars What if h were 15? Then the daily cost would be 8 × 12 dollars for the first 8 hours and 7 × 10 dollars for the next 7 hours In both cases, you had to subtract 8 from the total number of hours per day in order to find the number of hours that cost 10 cents per hour So, we can generalize from that to say that no matter what the value of h, the first 8 hours will

cost 8 × 12 dollars and the rest of the hours will cost (h – 8) × 10 dollars Writing this whole thing a little more formally, you can write that the cost in dollars of running the air conditioner for 1 day is:

1

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(8)(.12) + (h – 8)(.10)

What’s left in order to finish this one? Just to determine how much it will cost for 90

days That’s pretty easy; all you have to do is multiply the above expression by 90

to get

90[(8)(.12) + (h – 8)(.10)] = 90(8)(.12) + (90)(h – 8)(.10)

Do any of the answer choices look like this? Well no, not exactly, but the second

term in the above expression, (90)(h – 8)(.10), can be rewritten as 9(h – 8), since

90 × 10 = 9, which gives us an answer of 90(8)(.12) + 9(h – 8), which is choice (D)

21.B This is a hard algebra question where you’re given 2 equations and an inequality

and asked to find the value of another expression Let’s start with the

easiest-looking equation, = 6 If = 6 then xy = That means that the inequality,

neither x nor y is equal to 0and so it’s OK to divide by x and y Now we know the

value of xy but we need the value of x + y What can we do with the other

equation? This is the sum of two fractions You know that you can’t add 2 fractions

unless they have the same denominator, so we need to find a common denominator

You can always find a common denominator of two fractions by multiplying the two

denominators together x times y is xy, so our common denominator is xy

Multiplying by gives us , and multiplying by gives us , so we get:

This means that x + y = 5xy Since xy = , x + y = 5 × = What we’re looking

for is , or divided by 5, or , choice (B)

22.B The important thing here is not to be intimidated by the weird symbol and just solve

this one step at a time You’re told that x with a box around it, or stands for the

smallest integer greater than x What exactly does that mean? Let’s pick a number

and try to figure that out Let’s say that x is What’s the smallest integer that is

greater than ? Some integers that are greater than are 1, 2, 3, etc Which one

is smallest? The number 1, of course If you think of a number line, 1 is the next

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integer to the right of So it looks like you just have to find the next integer to the right of x on the number line in order to find Let’s do the 2 numbers with boxes

around them separately What’s ? The next integer to the right of 5.1 on the number line is 6, so = 6 What about ? The next integer to the right of –5.1 on the number line is –5 Be careful here, you might have thought it was –6, but –6 is less than –5.1 (It is to the left of –5.1 on the number line) So

– = 6 – (–5) = 6 + 5 = 11, choice (B)

23.C This is a complicated triangles problem, but if you do it one step at a time and use

everything you know about triangles, you should be able to figure it out

The figure shows two small triangles joined together to form a larger triangle Since the only side you know the measure of is AB, it’s probably a good idea to start by just looking at the triangle ABD Angle BDC is a right angle, and since angles BDC

and BD Alie on a straight line, BD Amust also be a right angle You’re given that angle D ABis 60˚ Therefore, angle ABD must be 30˚ and you have a 30-60-90 right triangle You probably remember (and if you don’t, you should!) that the lengths of the sides of a 30-60-90 right triangle are always in the ratio 1:3:2 AB is the

hypotenuse, or longest side of triangle ABD, so it must correspond to the “2” in the 1:3:2 ratio Since 8 = 4 × 2, the other two sides must be 4 × 1, or 4, and 4 × 3,

which can just be written 43 Which side is which? The smallest side must be opposite the smallest angle, so the side with length 4 must be AD, which is opposite

the 30˚ angle, and the remaining side, BD must measure 43

Now it’s time to work with the other small triangle You were given that angle BDC is

90˚ and that angle BCD is 45˚, so angle DBC must also be 45˚, which makes triangle BCD a 45-45-90 right triangle We just figured out that BD measures 43

Another Pythagorean ratio that you should remember is the one for 45-45-90 triangles, 1:1:2 Even if you didn’t remember that one, you could probably figure out that sides BD and DC must be equal in length since they are both opposite 45˚ angles So if BD measures 4 3, then DC also measures 43 The area of a right triangle is just (leg1)(leg2), or × 43 × 43, or × 16 × 3, or 24, choice (C)

If you were really stuck on this one, you could have eyeballed it Since you’re given that AB is 8, you could use the edge of your answer grid as a ruler to compare the lengths of AB and DC You’d find that DC is a little bit shorter than AB, probably somewhere around 7 You could compare AB with BD and you’d notice that BD is

also a little bit shorter than AB and again guess 7 for the length of BD Since DC is

around 7 and BD is around 7, the area of BCD is around × 7 × 7, or × 49, or

241, which is closest to 24, choice (C)

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