Level 3 Difficult

Question 1

Determine combined percent increase/decrease

Answer 1

Start with 100 as a starting value.

(ex. A price rises by 10% one year and by 20% the next. What’s the combined percent increase?
Ans: Year one = 100 + (10% of 100) = 110
Year two = 110 + (20% of 110) = 132
From 100 to 132 is a 32% increase)

Question 2

Find the Original Whole before percent increase/decrease

Answer 2

Think of a 15% increase over x as 1.15x and set up an equation.

(ex. A 5% decrease results in 57,000. What’s the original whole?
0.95 x (original) = 57,000.
Original = 60,000)

Question 3

Solve a Simple Interest Problem

Answer 3

Interest is gained only based on the principal.

interest = (principal) x (interest rate as a decimal) x (time in years or fraction of a year)

Question 4

Solve a Compound Interest Problem

Answer 4

Interest is gained on the principal and any interest earned.

Final Balance) = (Principal) x (1 + (interest rate/C))^(time)(C
C = number of times compounded annually

Question 5

Solve a Remainders Problem

Answer 5

Pick a number that fits the given conditions and see what happens.

(ex. When n is divided by 7, the remainder is 5. What is the remainder when 2n is divided by 7?
To find a number that leaves a remainder of 7 when divided by 7, take itself and add 5 to it (in this case, 12). 2n = 24, when divided by 7, leave 3.)

Question 6

How to Solve a Digits Problem

Answer 6

Use logic and trial and error.

(ex. two two-digit numbers can’t add up to more than something in the 100s. Single digit numbers can’t add up to anything more than 18. etc.)

Question 7

Find a Weighted Average

Answer 7

Give each term the appropriate “weight.”
Don’t just average the averages.
(ex. The girls’ average score is 30. The boys’ is 24. There are twice as many boys as girls, what is the overall average?
Weighted avg = (1x30+2x24)/(3) = 26)

Question 8

Find the New Average when a number is Added or Deleted

Answer 8

Use the sum of the terms of the old average to help you find the new average.

(ex. Mike’s test score average is 80 after 4 tests. The fifth test is 100. What’s the new average?
Original sum = 4 x 80 = 320
320 + 100 = 420
420/5 = 84)

Question 9

Use the Original Average and New Average to figure out What was Added or Deleted

Answer 9

Use the sums.
Number Added = (new sum) - (original sum)
Number Deleted = (original sum) - (new sum)

Question 10

Find an Average Rate

Answer 10

Convert to totals.

Average A per B = Total A/Total B
Average speed = Total Distance/Time

Question 11

Solve a Combined Work Problem

Answer 11

Work Formula: The inverse of the time it would take everyone working together equals the sum of the inverses of the times it would take each working individually.

1/r + 1/s = 1/t
All variables must stand for units of time and must all refer to the amount of time it takes to do the same task.

Question 12

Determine a Combined Ratio

Answer 12

Multiply one or both ratios by whatever you need to in order to get the terms they have in common to match.

(If a:b is 7:3 and b:c is 2:5, what is a:c?
Multiply a:b by 2 and b:c by 3 to get a:b:c as 14:6:15. So a:c is 14:15.)

Question 13

Solve a Dilution or Mixture Problem

Answer 13

You can go straightforward or use the balancing method.

Straightforward: find totals and averages.
Balancing: Make the weaker and stronger substances balance.
(percent difference between the weaker solution and the desired solution) x (amount of weaker solution) = (percent difference between the stronger solution and the desired solution) x (amount of stronger solution)

Question 15

Solve a Group problem involving Both/Neither

Answer 15

Use this formula:

Group 1 + Group 2 + Neither - Both = Total

Question 16

How to Work with Factorials

Answer 16

If n is an integer greater than 1, then n factorial, denoted by n!, is defined as the product of all the integers from 1 to n.

Remember to factor or cancel whenever possible.

Question 17

Solve a Group problem involving Either/Or Categories

Answer 17

Organize the information in a grid. Then use simple arithmetic to fill in the remaining boxes until you get the number you are looking for.

Question 18

Solve a Permutation Problem

Answer 18

Factorials are useful. 
Permutation formula (for # of ways to arrange a smaller group within a larger group):
nPk = (n!)/(n-k)!
n = # in larger group
k = # you're arranging

Question 18

Solve a Combination Problem

Answer 18

If the order of arrangement of the smaller group does not matter, you are looking for combinations.
Use this formula:
nCk = n!/k!(n-k)!
n = (# in larger group)
k = (# you're choosing)

Question 19

Solve Probability Problems where probabilities must be multiplied

Answer 19

When all outcomes are all equally likely, the basic probability formula is:
Probability = (# of favorable outcomes)/(# of possible outcomes)

Question 20

How to Deal with Standard Deviation

Answer 20

SD is a measure of how spread out a set of number is (how much the numbers deviate from the mean).

1. Find the Average
2. Find the differences between the mean and each value in the set.
3. Square each of the differences.
4. Find the average of the squared differences.
5. Take the positive square root of the average.

SD questions can often be handled with estimation.

Question 21

How to Multiply/Divide Values with Exponents Powers

Answer 21

Add or Subtract the Exponents
x^a x x^b = x^(a+b)
x^c/x^d = x^(c-d)

Question 22

How to handle a value with an Exponent Raised to an Exponent

Answer 22

Multiply the Exponent

(x^a)^b = x^ab

Question 23

How to handle Powers with a base of Zero and Powers with an Exponent of Zero

Answer 23

1. Zero raised to any nonzero exponent equals zero.
2. Any nonzero number raised to the exponent zero equals 1.
3. Zero raised to the zero power is undefined.

Question 24

How to handle Negative Powers

Answer 24

A number raised to the exponent -x is the reciprocal of that number raised to the exponent x.
n^-1 = 1/n, (n^-2) = 1/n^2, and so on.

Question 25

How to handle Fractional Powers

Answer 25

Fractional exponents relate to roots. For instance, x^1/2 = root of x.
x^1/3 = cubed root of x.
x^2/3 = cubed root of x^2, and so on.

Question 26

How to handle Cube Roots.

Answer 26

The cube root of x is just the number that when used as a factor 3 times gives you x.

1. Both positive and negative numbers have only one cubed root.
2. The cube root of a number is always the same sign as the number itself.

Question 27

How to Add, Subtract, Multiply, and Divide Roots

Answer 27

You can add/subtract roots only when the parts inside the radical are identical.
root of 2 + 3(root of 2) = 4(root of 2)
root of 2 - 3(root of 2) = -2(root of 2)
root of 2 + root of 3 cannot be combined.

To multiply/divide roots, deal with what’s inside the radical and outside the radical separately.
(2 times root of 3)(7 times root of 5) = (2 x 7)(root of 3 x 5) = 14(root of 15)

Question 28

How to Simplify a Radical

Answer 28

1. Look for factors of the number under the radical sign that are perfect squares; then find the square root of those perfect squares.
2. Keep simplifying until the term with the square root sign is as simplified as possible: there are no other perfect square factors inside the radical.
3. Write the perfect squares as separate factors and “unsquare” them

Question 29

How to solve certain Quadratic Equations

Answer 29

1. Manipulate the equation so that it equals 0.
2. Factor the left side using reverse FOIL (Find two numbers whose product is the constant and whose sum is the coefficient of the term without the exponent).
3. Break it into two simple expressions.
4. Find the values for the variable that make either expression = 0.

Question 30

How to solve Multiple Equations

Answer 30

Combine, solve, and rearrange in such a way to get something closer to what you are looking for.

Question 31

How to solve a Sequence Problem

Answer 31

The nth term in the sequence is generated by performing an operation, which will be defined on either n or the previous term in the sequence.

Question 32

How to solve a Function Problem

Answer 32

1. An algebraic expression of only one variable may be defined as a function, usually symbolized by f or g, of that variable.
2. Input numbers into the function to get an output.

Question 33

How to handle Graphs of Functions

Answer 33

On a graph, f(x) becomes the y-coordinate.
If an equation needs to be chosen based on a given graph, plug in obvious/easy points (such as (0,1) into the functions and eliminate answer choices.

Question 34

How to handle Linear Equations

Answer 34

y = mx + b
m = the slope of the line = rise/run
b = the y-intercept 

If written in the form x + y = C, solve for y to get it into the proper form.
Or pick obvious/easy points and plug into the equation to eliminate answer choices.

Question 35

How to find the x and y-intercepts of a line

Answer 35

Remember, the y-intercept (when x = 0) is b in the equation y = mx + b.
The x-intercept is when y = 0.

Question 36

How to find the Maximum and Minimum lengths for a Side of a Triangle

Answer 36

If you know n = the lengths of two sides of a triangle, you know that the third side if somewhere in between the positive difference and the sum.
(ex. if two sides are 7 and 3, the third side is in between 4 and 10.)

Question 37

How to find one angle or the sum of all the Angles of a Regular Polygon

Answer 37

“Regular” means all angles of the polygon are equal.

Sum of the interior angles in a polygon with n sides = (n-2) x 180.

Degree measure of one angle in a regular polygon with n sides = ((n-2) x 180)/n

Question 38

How to find the Length of an Arc

Answer 38

An Arc is a fraction of a circle’s circumference.
Use the equation:
Length of an arc = (n/360) x 2pi(r)
n = degree of angle from center point of circle the arc is on
r = radius of circle

Question 39

How to find the Area of a Sector

Answer 39

A Sector is a fraction of a circle's area.
Use the equation:
Area of sector = (n/360) x pi(r^2)
n = degree of angle in center of circle.
r = radius of circle

Question 40

How to find the dimensions or area of an Inscribed or Circumscribed Figure

Answer 40

Look for the connection: is the diameter the same as a side or a diagonal?
(ex. if the corners of a square touch the circle, the circumference of the circle is the same as the diagonal of the square)

Question 41

How to find the Volume of a Rectangular Solid

Answer 41

Volume = length x width x height

Question 42

How to find the Surface Area of a Rectangular Solid

Answer 42

Add the area of each face together.
Surface Area = 2(lw) + 2(wh) + 2(lh)
l = length, w = width, h = height

Question 43

How to find the Diagonal of a Rectangular Solid

Answer 43

Use the Pythagorean Theorem twice, or find special triangles.

Question 44

How to find the Volume of a Cylinder

Answer 44

Volume = area of the base x height = pi(r^2)h

Question 45

How to find the Surface Area of a Cylinder

Answer 45

Surface Area = 2pi(r^2) + 2pi(rh)