Sat - MC Grawhill part 86 potx

Math Smart Cards Math Smart CardsIf the area of square ABCD in the ﬁgure above is 100 ft2, then what is the cir-cumference of inscribed circle O?. Math MathMath Math Make a calendar: Th

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Math Math

Correct answer: 8

Cross-multiply: (4 + x) = (12 + x)(.6)

Distribute: 4 + x = 7.2 + 6x

Subtract 6x: 4 + 4x = 7.2

Subtract 4: .4x = 3.2

Divide by 4: x = 8

Common mistake: 4

This is the result if you forget to add the quantity

x to the denominator as well as the numerator.

4

+

x

x .

Formula/Concept:

The Triangle Inequality:

⏐B – A⏐ < C < ⏐B + A⏐

(B – A) < C < (B + A)

Substitute for A and B: (12 – 8) < C < (12 + 8)

Simplify: 4 < C < 20

This is the common error that can be made if

you mistakenly set 4 ≤ C ≤ 20.

Formula/Concept: Weighted Averages

If two numbers being averaged have different

“weights,” you must remember to account for

that when ﬁnding the average

This is the result if you mistakenly take the

10 82 6 90

16

820 540

( )+ ( )

A B C

To decrease a value by 10%, multiply by

0.90

Correct answer: 92 1999 = $100 (100)(1.25) = 125 2000 = $125 (125)(.8) = 100 2001 = $100 (100)(.8) = 80 2002 = $80 (80)(1.15) = 92 2003 = $92

Common mistake: 100 This is result if you simply add up the percent changes instead

of calculating the changes as above: 25 - 20 - 20 + 15 = 0

Average formula:

Multiply by 3: x + (x + 2) + (2x + 8) = 18

Combine like terms: 4x + 10 = 18

This is the result if you mistakenly think that there are 5 numbers instead of 3

x+ +(x 2)+(2x+8)=

Formula/Concept: Integer arithmetic

Remember that the sum of the numbers from -12 to 12 is 0, because the negative integers

“cancel” the positives 13 + 14 + 15 = 42, so n

must be 15 To ﬁnd the number of integers in the set, just subtract the ﬁrst from the last and add 1

This is the result if you forget that 0 is an

inte-sum average

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Math Smart Cards Math Smart Cards

If the area of square ABCD in the ﬁgure

above is 100 ft2, then what is the

cir-cumference of inscribed circle O?

In 1984, a share of stock in Black’s

Oil Trust cost $3 By 2000, it had

increased to $15 per share What is

the percent increase in the price of the

stock from 1984 to 2000?

If (x + 4)(x – 4) = 65, then what is the

value of x2?

O

If the third Friday in January occurs on the 15th, what is the date of the fourth Wednesday in January?

At a department store, all shirts are

priced at s dollars, but if you buy one

shirt at full price, you can buy any num-ber of additional shirts at a $2 discount

per shirt What is the cost of buying x

shirts at this sale?

l

A

E

C

D

x°

60 °

47 °

44 °

51 °

B

Note: Figure not drawn to scale

In the ﬁgure above, line segments AB –– , EB ––

CB –– , and DB –– intersect line l at point B What is the value of x?

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Math Math

Make a calendar:

This is the result if you mistakenly assume, without

drawing a calendar to conﬁrm, that the third

Wednes-day occurs before the third FriWednes-day In this particular

month, the ﬁrst Wednesday is after the ﬁrst Friday.

Formula/Concept: A linear angle

measures 180ⴗ

Correct answer: 85ⴗ

44ⴗ + xⴗ + 51ⴗ = 180ⴗ

Combine like terms: 95ⴗ + xⴗ = 180ⴗ

Subtract 95ⴗ: xⴗ = 85ⴗ

Common mistake: 47ⴗ

This is the result if you mistakenly think that

an-gles CBD and ABE are vertical anan-gles But vertical

angles must involve intersecting lines, not line

seg-ments There are no vertical angles in this ﬁgure

Formula/Concept: Translating words

into expressions

Correct answer: s + (x – 1)(s – 2)

The ﬁrst shirt costs s dollars, and each

addi-tional shirt costs (s – 2) dollars Don’t multiply

the (s – 2) by x, because this would account for

the ﬁrst shirt twice Only (x – 1) shirts are

priced at (s – 2) dollars.

Common mistake: s + (x)(s – 2)

M T W Th F Sa Su

11 12 13 14 15 16 17

18 19 20 21 22 23 24

25 26 27 28 29 30 31

Correct answer: 10␲

Area of square = (side)2 Substitute 100 for area: 100 = (side)2 Take square root: 10 = side of square

dia-meter d = side of square

Use circumference C = πd = π(10) = 10π

formula

Common mistake: 25π This is the result if you confuse the area formula with the circumference formula

Correct answer: 400%

Common mistake: 500%

This is the result if you mistakenly ﬁnd what per-cent 15 is of 3 instead of ﬁnding the perper-cent

change from 3 to 15.

Percent change=15 3− × =

Percent change=final−original×

original 100%

Formula/Concept: FOILing

(a + b)(a – b) = a2– ab + ab + b2

(x + 4)(x – 4) = 65

FOIL: x2– 4x + 4x – 16 = 65

Combine like terms: x2– 16 = 65

This is result if you mistakenly solve for x instead

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If w divided by 1⁄4is equal to 32, then

what is the value of w?

At Holston Hospital, a “team” consists

of 1 resident and 2 medical students If

there are 4 residents and 5 medical

students in the emergency department,

how many different teams could be

formed?

If the average of 5, 6, 8, x, and 8 is 6, then

what is the median of the set?

Point W is on line segment XY

such that If WY = 12

then what is XY?

XW

WY = 3 4

For all values of x, let x = Which of the following is equal

to – ? (A)

(B) (C) (D) (E)

x − 2 4

When 34 is divided by 6, the remainder

is n What is the remainder when

n is divided by 2?

6

8 14 24 26

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Math Math

Plug in 12 for WY:

Cross-multiply: 4XW = 36

Divide by 4: XW = 9

Solve for XY: XY = 9 + 12 = 21

This is the result if you solve for XW instead of

XY.

XW

12

3 4

=

XW

WY = 3 4

Formula/Concept: New Symbols/Functions

Correct answer: E

Common mistake: A

This is the result if you do not notice that the

answer choices are also in boxes

Formula/Concept: Remainders

6 34 30 5

4

4 2

0

4 0

34 34 2

4

32

10 2 4

8

4 2

34 10 8 2 6 26 26 2

4

24

4 6

fraction, multiply

by its reciprocal

Write an equation: wⴜ1⁄4= 32 Multiply by the reciprocal: wⴛ 4 = 32

This is the result if you divide 32 by 1⁄4instead

of multiplying it by 1⁄4

Formula/Concept: The Fundamental

Counting Principle

Correct answer: 4ⴛ 5 ⴛ 4 ⴜ 2 = 40

The number of options for choosing a resident

is 4, since there are 4 residents The number of different pairs of interns is 10, because there

are 5 options for intern A and then 4 options for intern B, but since choosing AB is the same

as choosing BA, we must divide this set by 2.

Common mistake: 100 or 80

This is the result of 4 ⴛ 5 ⴛ 5 or 4 ⴛ 5 ⴛ 4

Formula/Concept: Median = middle number

Mean = average

Multiply by 5: 5 + 6 + 8 + x + 8 = 30

Combine like terms: 27 + x = 30

Subtract 27: x = 3

Find the median: 3, 5, 6, 8, 8

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What is the area of the triangle in

the ﬁgure above?

The slope of line l is –1

⁄2 If two points

on line l are (2, 4) and (x, –2), what is

the value of x?

There are 25 students in Ms Jamison’s

4th-grade homeroom class If 12 of her

students have a cat, 19 have a dog, and

every student has at least 1 pet, how

many students have both a cat and

a dog?

(5,6)

(7,0) (14,0) x

y

A printer can produce 50 pages in

3 minutes At this rate, how many pages can it print in 300 minutes?

If f(x) = (x – 2)2, what is the range of this function over the domain –2 ≤ x ≤ 3?

If b varies inversely as the square of c and directly as a, and b = 4 when c = 4 and a = 8, then what is the value of b when a = 18 and c = 6?

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Math Math

Correct answer: 5,000

Set up a ratio:

Cross multiply: 15,000 = 3x

Divide by 3: 5,000 = x

This is the result if you set up the ratio improperly

50 3

300 pages minutes

minutes pages

=

x

50

pages minutes

The range of a function is the set of all of the

possible outputs, or “y-values.”

Correct answer: 0 ≤ y ≤ 16

Plug in the integer values of the

domain and ﬁnd the range:

f (-2) = 16; f (–1) = 9; f (0) = 4;

f (1) = 1; f (2) = 0; f (3) = 1 This

yields a range of 0 ≤ y ≤ 16 Or

you could graph it:

Common mistake: 1 ≤ y ≤ 16

This results if you try to ﬁnd the range by

plug-ging in only the two domain endpoints, ignoring

the points in between

Formula/Concept: Direct and Inverse

Variation

Cross-multiply: 64 = 8k

Divide by 8: k = 8

Set up new

equation:

Common mistake: 1⁄2

x

y

5

5 -5

-5

b ka c

k

( )

2 4 82

4

6

144

2 2

The base of the triangle is the distance from (7, 0) to (14, 0), which is 14 – 7 = 7 The height

is the distance from the x-axis to (5, 6), which

is 6 – 0 = 6

Area = 1⁄2(7)(6) = 21

Common mistake: 17.5

This is the common error that can be made if you use 5 as the height instead of 6

Cross-multiply: 2(–6) = –1(x – 2)

This is the result if you mistakenly calculate “run over rise” for the slope: x x

2 1

−

= −

− =

− −

− =

−

2 4 2

1 2

slope rise run

−

Formula/Concept: Venn Diagrams

Let x represent the number of

students with both cats and dogs:

(12 – x) + (19 – x) + x = 25

Combine like terms: 31 – x = 25

Subtract 31: –x = –6

Divide by –1: x = 6

Cats Dogs

12 – x x 19 – x

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If 5 – 2 ⏐x⏐> –9, ﬁnd the range of

possible values for x.

–2, 0, 2, –2, 0, 2, –2, 0, 2,

The sequence above continues

accord-ing to the pattern shown What is the

sum of the ﬁrst 31 terms of this

sequence?

The sum of a set of 7 integers is 67 If

each of these numbers must be less

than 14, what is the smallest possible

value of any one number in the set?

If -1 < w < 0, then which of the

follow-ing statements is true?

(A) w < w2< w3

(B) w2 < w < w3

(C) w < w3< w2

(D) w3 < w2< w (E) w3 < w < w2

A jar contains only red and white mar-bles If the probability of randomly selecting a white marble from the jar is 1

⁄4and there are 15 red marbles in the jar, how many white marbles are there?

Jim takes the same train to and from work each day One winter day, during the morning commute, the train aver-aged 80 miles per hour During the evening commute, due to ice on the track, the train averaged only 48 miles per hour If Jim spent 2 hours in the train that day, how many miles is the train ride to work?

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Math Math

Correct answer: C

Plug in a number for w:

Plug in –1⁄2for w: w = –1⁄2

w2= (–1⁄2)2=1⁄4

w3= (–1⁄2)3= –1⁄8

Rank the values of w, w2, w3: –1⁄2< –1⁄8<1⁄4

Common mistake: A or E

You might choose E if you forget that –1⁄8> –1⁄2

You might choose A if you forget that squaring

any negative number will result in a positive

number

Formula/Concept: Ratios and Probability

If the probability of selecting a white marble is 1⁄4,then

the probability of selecting a red marble is 1–1⁄4=3⁄4

Set up a ratio:

Cross-multiply: 60 = 3(total)

Divide by 3: 20 = total = white + red

Solve for white: 20 = white + 15

Subtract 15: white = 5 marbles

This is the result if you answer for the total number

of marbles instead of the number of white marbles.

Formula/Concept: Distance = Rate × Time

Correct answer: 60 (because he takes the same

train, the distance is the same going in both directions.)

Set them equal: 80t = 48(2 – t)

Plug in 75 for t: d = 80(.75) = 60 miles

red total =3 total = 4

4

Solving absolute value inequalities

Correct answer: –7< x < 7

5 – 2⏐ x ⏐ > –9

Subtract 5: –2⏐ x ⏐ > –14

Divide by -2: (swap the inequality) ⏐ x ⏐ < 7

Take away absolute value: –7 < x < 7

Common mistake: x > 7 or x < –7

This is the common error that can be made if you forget to swap the inequality when dividing

or multiplying by a negative number

Sequence problems: ﬁnding the sum of a random number of terms

Correct answer: –2

The pattern repeats every 3 digits, and the sum of each repetition is –2 + 0 + 2 = 0 The pattern occurs 31 ÷ 3 = 101⁄3times, or 10 with remainder 1

The 10 full repetitions have a sum of 10(0) = 0

Since the 31st term is –2, the sum is 0 + –2 = –2

This is the common error that can be made if you forget to add the 31st term after ﬁnding out that the pattern occurs 101⁄3times

Formula/Concept: Numerical Reasoning

Correct answer: –11

a + b + c + d + e + f + g = 67

If you want g to be as small as possible, then

make the sum of the other numbers as large as

possible: Substitute 13 for a through f:

13 + 13 + 13 + 13 + 13 + 13 + g = 67

Combine like terms: 78 + g = 67

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Math Smart Cards Roots Smart Cards

If the rectangular solid above has a

volume of 162 cubic inches, what is the

surface area of the solid?

Which point corresponds to the result

when the numbers corresponding to

points E and B are multiplied?

9

6 h

b° c°

Z m

l

66°

106°

magnanimous

Given the graph of y = f (x) shown

above and the transformed graph on the right, what is the equation of the new function?

5

y

x

5

y

Instructions for studying with Root Smart Cards

1 st point: Deﬁne the word on the front of

the card

2 nd point: Give the meaning of the root in

bold

3 rd point: List at least three words that

contain the root

Note: Figure not drawn to scale

In the ﬁgure above, if l m and n p,

what is the value of b + c?

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