Sat math Essentials

Greater than
means 
>
and 
less than
means 
<
.
Examples
The product of
x
and 9 is greater than 15.
x
9 > 15
When 1 is added to a number
x,
the sum is less than 29.
x
1 < 29
■
At least
means 
≥
and 
at most
means 
≤
.
Examples
The sum of a number 
x
and 5 is at least 11.
x
5 
≥
11
When 14 is subtracted from a number 
x
, the difference is at most 6.
x
14 
≤
6
■
To
square
means to 
use an exponent of 2
.
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P R O B L E M S O LV I N G
–
1 5 0

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Example
The square of the sum of
m
and 
n
is 25.
(
m
n
)
2
25
Practice Question 
If squaring the sum of
y
and 23 gives a result that is 4 less than 5 times 
y,
which of the following equations
could you use to find the possible values of
y?
a.
(
y
23)
2
5
y
4
b.
y
2
23 
5
y
4
c.
y
2
(23)
2
y
(4 
5)
d.
y
2
(23)
2
5
y
4
e.
(
y
23)
2
y
(4 
5)
Answer
a.
Break the problem into pieces while translating into mathematics:
squaring
translates to 
raise something to a power of 2
the sum of
y
and 23
translates to (
y
23)
So,
squaring the sum of
y
and 23
translates to (
y
23)
2
.
gives a result
translates to 
4 less than
translates to 
something 
4
5 times
y translates to 
5
y
So,
4 less than 5 times
y means 
5
y
4
.
Therefore,
squaring the sum of
y 
and 23 gives a result that is 4 less than 5 times
y translates to: (
y
23)
2
5
y
4.
A s s i g n i n g Va r i a b l e s i n Wo r d P r o b l e m s
Some word problems require you to create and assign one or more variables. To answer these word problems, first
identify the 
unknown
numbers and the 
known
numbers. Keep in mind that sometimes the “known” numbers won’t
be actual numbers, but will instead be expressions involving an unknown.
Examples
Renee is five years older than Ana.
Unknown
Ana’s age 
x
Known
Renee’s age is five years more than Ana’s age 
x
5
Paco made three times as many pancakes as Vince.
Unknown
number of pancakes Vince made 
x
Known
number of pancakes Paco made 
three times as many pancakes as Vince made 
3
x
Ahmed has four more than six times the number of CDs that Frances has.
Unknown
the number of CDs Frances has 
x
Known
the number of CDs Ahmed has 
four more than six times the number of CDs that Frances has 
6
x
4
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P R O B L E M S O LV I N G
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1 5 1

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Practice Question 
On Sunday, Vin’s Fruit Stand had a certain amount of apples to sell during the week. On each subsequent
day, Vin’s Fruit Stand had one-fifth the amount of apples than on the previous day. On Wednesday, 3 days
later, Vin’s Fruit Stand had 10 apples left. How many apples did Vin’s Fruit Stand have on Sunday?
a.
10
b.
50
c.
250
d.
1,250
e.
6,250
Answer
d.
To solve, make a list of the knowns and unknowns:
Unknown:
Number of apples on 
Sunday
x
Knowns:
Number of apples on 
Monday
one-fifth the number of apples on Sunday 
1
5
x
Number of apples on 
Tuesday
one-fifth the number of apples on Monday 
1
5
(
1
5
x
)
Number of apples on 
Wednesday
one-fifth the number of apples on Tuesday 
1
5
[
1
5
(
1
5
x
)]
Because you know that Vin’s Fruit Stand had 10 apples on Wednesday, you can set the expression for
the number of apples on Wednesday equal to 10 and solve for 
x
:
1
5
[
1
5
(
1
5
x
)] 
10
1
5
[
2
1
5
x
] 
10
1
1
25
x
10
125 
1
1
25
x
125 
10
x
1,250 
Because 
x
the number of apples on Sunday, you know that Vin’s Fruit Stand had 1,250 apples on
Sunday.
P e r c e n t a g e P r o b l e m s
There are three types of percentage questions you might see on the SAT:
1.
finding the percentage of a given number 
Example:
What number is 60% of 24?
2.
finding a number when a percentage is given 
Example:
30% of what number is 15?
3.
finding what percentage one number is of another number 
Example
: What percentage of 45 is 5?
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P R O B L E M S O LV I N G
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1 5 2

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To answer percent questions, write them as fraction problems. To do this, you must translate the questions
into math. Percent questions typically contain the following elements:
■
The 
percent
is a number divided by 100.
75% 
1
7
0
5
0
0.75
4% 
1
4
00
0.04
0.3% 
1
0
0
.3
0
0.003
■
The word 
of
means to multiply.
English: 10% 
of
30 equals 3.
Math:
1
1
0
0
0
30 
3
■
The word 
what
refers to a variable.
English: 20% of
what
equals 8?
Math:
1
2
0
0
0
a
8
■
The words 
is
,
are
, and
were
, mean equals.
English: 0.5% of 18 
is
0.09.
Math:
0
1
.
0
0
0
5
18 
0.09
When answering a percentage problem, rewrite the problem as math using the translations above and then
solve.
■
finding the percentage of a given number 
Example
What number is 80% of 40?
First translate the problem into math:
Now solve:
x
1
8
0
0
0
40
x
3
1
,2
0
0
0
0
x
32
Answer: 32 is 80% of 40
■
finding a number that is a percentage of another number 
Example 
25% of what number is 16?
First translate the problem into math: