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11 Cards in this Set

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What is the best way to solve problems with unspecified amounts?
Pick numbers.

Example:

If the length of the side of a cube decreases by two-thirds, by what percentage will the volume of the cube decrease?

Solution:

Pick numbers!
Volume of cube = s^3 = 27
Therefore s = 3.

If side decreases by 2/3 therefore 1/3*3 = 1

New volume = 1^3 = 1
Percent change = (27-1)/27 = 96.3%
What is the general formula for a sequence who's terms are seperated by a constant?
kn+x

k is the difference between successive terms.
n is the nth term.
x is a real number.

If the difference between successive terms is always the same, the rule wil take the form of kn+x, where k and x are real numbers and k is equal to the difference between successive terms.

Example:

16, 20, 24, 28

The difference between the terms is 4 (i.e k=4).

kn+x
4+x=16
x=12

Thus the formula is 4n+12
What is the general formula for a sequence who's terms are seperated by a constant of a constant?
If the difference between thedifference between successive terms is always the same, the rule will take the form of an^2+bn+c, where a, b, and c are real numbers.

Example:

18, 27, 38, 51
9, 11, 13
2, 2

Thus since the difference of the difference is a constant we know that the sequence will take the form an^2+bn+c

Thus the first term in the sequence is a(1)^2+b(1)+c=18

Thus the second term in the sequence is a(2)^2+b(2)+c=27

Thus the third term in the sequence is a(3)^2+b(3)+c=38

Thus we have 3 equation and 3 unknown we can solve for a, b and c.

a(1)^2+b(1)+c=18
a(2)^2+b(2)+c=27
a(3)^2+b(3)+c=38

Therefore a = 1, b = 6 and c = 11.

Thus the formula for this sequence is n^2+6n+11.
When would you use another method to determine a sequence?
If the problem seems to require too much computation consider an alternative method.

Example:

If each number in a sequence is three more than the previous number, and the sixth number is 32, what is the 100th number?

Solution:

We know that S6=32 and we know that between 100th term and the 6th term there are 94 jumps of 3.

Thus 94*3=282
Therefore the 100th term must be 282 units greater than the 6th term. (i.e S100=32+282 = 314)
What can an absolute value be thought as?
The distance from zero.

Example:

abs(-5) = 5 or five units from 0.
What is the best way to visualize inequalities problems?
Draw a number line.
What is the best way to solve combined inequality problems when no real numbers are given?
Line them up and use LICHLUC.

List
Change
Line up
Combine

Example:

Give the u<t, b>r, f<t, and r>t is b>u?

List and Change:
u<t, r<b, f<t and t<r

Line them up:
u<t
r<b
f<t
t<r
Combine
u<t<r<b
Therefore t<b.
What is the best way to solve inequalities problems when real numbers are given?
Plug in extreme values.

Example:

Given that 0<=x<=3 and y<8, which of the following could NOT be the value of xy?

A. 0
B. 8
C. 12
D. 16
E. 24

Lower limit:

We see that xy has no lower limit because y has no lower limit.

Upper limit:

We see the upper limit of x is <=3 and y<8.


Thus the product xy<24 because y cannot be 8. It is useful to write 3*(less than 8) is less than 24.

Therefore the answer is E.
What does PTSTEACH stand for?
It's an acronym for solving VICs.
Pick Number
Tracking chart.
Solve
Teach: Test each answer choice.

Example:

Jack bought x pounds of candy at d dollars per pound. If he ate w pounds of his candy and sold he rest to Jill for m dollars per pound, how much money did Jack spend, in dollars, on the candy that he ate himself?

A. xd-wm
B. xm-wd
C. wd-xm+wm
D. xd+xm-xw
E. 2xd-xm

Pick numbers and use a tracking chart.
--------------------
| Variable | Number |
--------------------
| x | 10 |
| d | 3 |
| w | 7 |
| m | 2 |
--------------------

Solve:

Jack bought 10 pounds of candy at $3 per pound. Therefore, he spent $30. Jack ate 7 pounds of his candy and sold the rest to Jill. Therefore, he sold 3 pounds to Jill, Jack sold the 3 pounds to Jill at $2 per pound. Therefore, Jilly paid $6 for the candy.

How much money did Jack spend on the candy that he ate himself?

Jack spent $30 on candy and sold $6 worth of that candy to Jill.

Therefore, Jack spent $24 on the candy that he ate himself.

Target is $24.

TEACH: Test each answer choice.

A. xd-wm=10*3-7*2=16 incorrect.
B. xm-wd=10*2-7*3=-1 incorrect.
C. xd-xm+wm=10*3-10*2+7*2=24 correct.
D. xd+xm-xw=10*3+10*2-10*7=-20 incorrect.
E. 2xd-xm=2*10*3-10*2=40 incorrect

Thus the answer is C.
What are the 3 rules for picking numbers?
1. Never pick 1 or 0.
2. Pick numbers that are different.
3. Pick small numbers.
True or false picking numbers is the best way to solve VICs.
False.

It depends on the problem and what you're comfortable with. If the problem cannot be solved in 15 seconds using algebra use the picking numbers technique.

The problem below shows both strategies.

If (abc)/72=2/d, which of the following expression is equivalent to ab-2?

A. 72
B. 72/(cd)
C. 144/(cd)
D. a(144-2cd)/(cda)
E. (144a-2cd)/cd

--------------------
| Variable | Number |
--------------------
| a | 2 |
| b | 3 |
| c | 4 |
| d | 6 |
--------------------

Therefore ab-2=2*3-2=4
Solving

A. 72 incorrect
B. 3 incorrect
C. 6 incorrect
D. 4 correct
E. 10 incorrect

Thus the answer is D.

This can also be solved algebraically.

(abc)/72=2/d --> ab=144/cd

Thus ab-2= 144/cd-2

Simplfying results in

(144-cd)/cd and multiplying both numerator and denominator by "a" yields:

a(144-cd)/(acd) which is D.