This number always follows the word “of ” in a word problem.. ■ In the formula, the equal sign can be inter-changed with the word “is.” Examples: Finding a percentage of a given number:
Trang 1■ If given a percentage, write it in the numerator
position of the number column If you are not
given a percentage, then the variable should be
placed there
■ The denominator of the number column
repre-sents the number that is equal to the whole, or
100% This number always follows the word “of ”
in a word problem
■ The numerator of the number column represents
the number that is the percent
■ In the formula, the equal sign can be
inter-changed with the word “is.”
Examples:
Finding a percentage of a given number:
What number is equal to 40% of 50?
Solve by cross multiplying
100(x) = (40)(50) 100x = 2,000
1 1
0 0
0 0
x
= 2 1
,0 0
0 0 0
x = 20 Therefore, 20 is 40% of 50
Finding a number when a percentage is given:
40% of what number is 24?
Cross multiply:
(24)(100) = (40)(x) 2,400 = 40x
2, 4
4 0
00
= 4 4
0 0
x
60 = x Therefore, 40% of 60 is 24
Finding what percentage one number is of another:
What percentage of 75 is 15?
Cross multiply:
15(100) = (75)(x) 1,500 = 75x
1, 7
5 5 00
= 7 7
5 5
x
20 = x Therefore, 20% of 75 is 15
Ratio and Variation
A ratio is a comparison of two quantities measured in
the same units It is symbolized by the use of a colon—x:y.
Ratio problems are solved using the concept of multiples
Example:
A bag contains 60 red and green candies The ratio of the number of green to red candies is 7:8 How many of each color are there in the bag? From the problem, it is known that 7 and 8 share a multiple and that the sum of their prod-uct is 60 Therefore, you can write and solve the following equation:
7x + 8x = 60 15x = 60
1 1
5 5
x
= 6 1
0 5
x = 4
Therefore, there are (7)(4) = 28 green candies and (8)(4) = 32 red candies
Variation Variation is a term referring to a constant ratio in the
change of a quantity
■ A quantity is said to vary directly with another if
they both change in an equal direction In other words, two quantities vary directly if an increase
x
15
40 24
40
x
Trang 2in one causes an increase in the other This is also true if a decrease in one causes a decrease in the other The ratio, however, must be the same
Example:
Assuming each child eats the same amount, if
300 children eat a total of 58.5 pizzas, how many pizzas will it take to feed 800 children?
Since each child eats the same amount of pizza, you know that they vary directly Therefore, you can set the problem up the following way:
ChPiilzdzraen=
5
3 8
0
0 5
= 80
x
0
Cross multiply to solve:
(800)(58.5) = 300x 46,800 = 300x
46 3
, 0
8 0
00
= 3 3
0 0
0 0
x
156 = x
Therefore, it would take 156 pizzas to feed 800 children
■ If two quantities change in opposite directions,
they are said to vary inversely This means that as
one quantity increases, the other decreases, or as one decreases, the other increases
Example:
If two people plant a field in six days, how may days will it take six people to plant the same field?
(Assume each person is working at the same rate.)
As the number of people planting increases, the days needed to plant decreases Therefore, the relationship between the number of people and days varies inversely Because the field remains constant, the two expressions can be set equal
to each other
2 people × 6 days = 6 people × x days
2 × 6 = 6x
1 6
2
= 6
6x
2 = x
Rate Problems
You will encounter three different types of rate prob-lems on the SAT: cost, movement, and work-output
Rate is defined as a comparison of two quantities with
different unites of measure
Rate =
Examples:
mhoiluers,dhoolluarrs,pocousntd
Cost Per Unit
Some problems on the SAT will require you to calcu-late the cost of a quantity of items
Example:
If 60 pens cost $117.00, what will the cost of four pens be?
t#ootaflpceonsst= 1
6
1 0
7
=
To find the cost of 4 pens, simply multiply
$1.95 × 4 = $7.80
Movement
When working with movement problems, it is impor-tant to use the following formula:
(Rate)(Time) = Distance
Example:
A scooter traveling at 15 mph traveled the length of a road in 14of an hour less than it took when the scooter traveled 12 mph What was the length of the road?
First, write what is known and unknown Unknown = time for scooter traveling
12 mph = x
Known = time for scooter traveling 15 mph =
x – 14
$1.95
pen
x units
y units
Trang 3the road does not change; therefore, you know
to make the two expressions equal to each other:
12x = 15(x – 14)
12x = 15x – 145
–15x –15x
– –
3 3
x
=
x = 54, or 114hours
Be careful, 114is not the distance; it is the time
Now you must plug the time into the formula:
(Rate)(Time) = Distance Either rate can be used
12x = distance
12(54) = distance
15 miles = distance
Work-Output Problems
Work-output problems are word problems that deal
with the rate of work The following formula can be
used of these problems:
(rate of work)(time worked) = job or part of job
completed
Example:
Danette can wash and wax two cars in six hours, and Judy can wash and wax the same two cars in four hours If Danette and Judy work together, how long will it take to wash and wax one car?
Since Danette can wash and wax two cars in six hours, her rate of work is , or one car every three hours Judy’s rate of work is there-fore , or one car every two hours In this problem, making a chart will help:
Rate Time = Part of Job Completed Danette 1
Since they are both working on only one car, you can set the equation equal to one:
1
3 x + 1
2 x = 1
Solve by using 6 as the LCD for 3 and 2:
6(13x) + 6(12x) = 6(1)
2x + 3x = 6
5
5x= 6
5
x = 115
Thus, it will take Judy and Danette 115hours to wash and wax one car
Special Symbols Problems
The SAT will sometimes invent a new arithmetic oper-ation symbol Don’t let this confuse you These prob-lems are generally very easy Just pay attention to the placement of the variables and operations being performed
Example:
Given a ∇ b = (a × b + 3)2, find the value of 1 ∇ 2
Fill in the formula with 1 being equal to a and 2 being equal to b.
(1 × 2 + 3)2= (2 + 3)2= (5)2= 25
So, 1 ∇ 2 = 25
Example:
b
c a
2
3 1
If = _ + _ + _ a − b a − c b − c
Then what is the value of
2 cars
4 hours
2 cars
6 hours
–415
–3
Trang 4Fill in variables according to the placement of
number in the triangular figure; a = 1, b = 2, and c = 3.
1 –32+ 1 –23 +2 –13= –13+ –1 + –1 = –213
Counting Principle
Some word problems may describe a possibilities for one thing and b possibilities for another To quickly solve, simply multiply a × b.
For example, if a student has to choose one of 8 different sports to join and one of five different com-munity service groups to join, we would find the total number of possibilities by multiplying 8 × 5, which gives us the answer: 40 possibilities
Permutations
Some word problems may describe n objects taken r at
a time In these questions, the order of the objects matters.
To solve, you will perform a special type of
calcu-lation known as a permutation The formula to use is:
n P r=
For example, if there are six students (A, B, C, D, E, and F), and three will be receiving a ribbon (First
Place, Second Place, and Third Place), we can calcu-late the number of possible ribbon winners with:
n P r=
Here, n = 6, and r = 3.
n P r= = 6P3 = = =
= 6 × 5 × 4 = 120
Combinations
Some word problems may describe the selection of r objects from a group of n In these questions, the order
of the objects does NOT matter.
To solve, you will perform a special type of
calcu-lation known as a combination The formula to use is:
n C r= n
r
P
!
r
For example, if there are six students (A, B, C, D,
E, and F), and three will be chosen to represent the
school in a nationwide competition, we calculate the number of possible combinations with:
n C r= n
r
P
!
r
Note that here order does NOT matter.
Here, n = 6 and r = 3.
n C r= n
r
P
!
r
= 6C3= 6
3
P
!
3
= 3×1220× 1 = 1260= 20
Probability
Probability is expressed as a fraction and measures the likelihood that a specific event will occur To find the probability of a specific outcome, use this formula: Probability of an event =
Example:
If a bag contains 5 blue marbles, 3 red marbles, and 6 green marbles, find the probability of selecting a red marble
Probability of an event =
= 5 +33 + 6
Therefore, the probability of selecting a red marble is
1
3 4
.
Multiple Probabilities
To find the probability that two or more events will occur, add the probabilities of each For example, in the problem above, if we wanted to find the probability of drawing either a red or blue marble, we would add the probabilities together
Number of specific outcomes
Total number of possible outcomes
Number of specific outcomes
Total number of possible outcomes
6 × 5 × 4 × 3 × 2 × 1
3 × 2 × 1
6!
(3)!
6!
(6 – 3)!
n!
(n – r)!
n!
(n – r)!
n!
(n – r)!
Trang 5The probability of drawing a red marble = 134 and the probability of drawing a blue marble = 154 So,
the probability for selecting either a blue or a red = 134
+ 154= 184
Helpful Hints about Probability
■ If an event is certain to occur, the probability is 1
■ If an event is certain not to occur, the probability
is 0
■ If you know the probability of all other events occurring, you can find the probability of the remaining event by adding the known probabili-ties together and subtracting from 1
Trang 6P a r t 1 : F i v e - C h o i c e Q u e s t i o n s
The five-choice questions in the Math section of the SAT will comprise about 80% of your total math score
Five-choice questions test your mathematical reason-ing skills This means that you will be required to apply several basic math techniques for each problem In the math sections, the problems will be easy at the begin-ning and will become increasingly difficult as you progress Here are some helpful strategies to help you improve your math score on the five-choice questions:
■ Read the questions carefully and know the answer being sought In many problems, you will
be asked to solve an equation and then perform
an operation with that variable to get an answer
In this situation, it is easy to solve the equation and feel like you have the answer Paying special attention to what each question is asking, and then double-checking that your solution answers the question, is an important technique for per-forming well on the SAT
■ If you do not find a solution after 30 seconds, move on You will be given 25 minutes to answer
questions for two of the Math sections, and 20 minutes to answer questions in the other section
In all, you will be answering 54 questions in 70 minutes! That means you have slightly more than one minute per problem Your time allotted per question decreases once you realize that you will want some time for checking your answers and extra time for working on the more difficult prob-lems The SAT is designed to be too complex to fin-ish Therefore, do not waste time on a difficult problem until you have completed the problems you know how to do The SAT Math problems can
be rated from 1–5 in levels of difficulty, with 1 being the easiest and 5 being the most difficult The following is an example of how questions of
vary-math section on a past SAT The distribution of questions on your test will vary
1 1 8 2 15 3 22 3
2 1 9 3 16 5 23 5
3 1 10 2 17 4 24 5
4 1 11 3 18 4 25 5
5 2 12 3 19 4
6 2 13 3 20 4
7 1 14 3 21 4
From this list, you can see how important it is
to complete the first fifteen questions before get-ting bogged down in the complex problems that follow After you are satisfied with the first fifteen questions, skip around the last ten, spending the most time on the problems you find to be easier
■ Don’t be afraid to write in your test booklet That is what it is for Mark each question that
you don’t answer so that you can easily go back to
it later This is a simple strategy that can make a lot of difference It is also helpful to cross out the answer choices that you have eliminated
■ Sometimes, it may be best to substitute in an answer Many times it is quicker to pick an
answer and check to see if it is a solution When
you do this, use the c response It will be the
mid-dle number and you can adjust the outcome to
the problem as needed by choosing b or d next,
depending on whether you need a larger or smaller answer This is also a good strategy when you are unfamiliar with the information the problem is asking
■ When solving word problems, look at each phrase individually and write it in math lan-guage This is very similar to creating and
assign-ing variables, as addressed earlier in the word problem section In addition to identifying what
is known and unknown, also take time to trans-late operation words into the actual symbols It is best when working with a word problem to