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Sum/Product of Consecutive Integers

The average of an odd number of consecutive integers will always be an integer.

The average of an even number of consecutive integers will never be an integer.

Example: What is the sum of all the integers from 20 to 100, inclusive?

(100+20)/2 = 60. The average of the set is 60.
Number of terms = 100 – 20+1 = 81.
Therefore sum = 60*81 = 4860

The product of k consecutive integers is always divisible by k factorial (k!).

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Divisibility & Primes

To find the LCM: use prime factorization and take the numbers with the highest power of repeated factors (i.e. 2³ instead of 2²)

To find the HCF: use prime factorization and take the numbers with the lowest power of repeated factors (i.e. 3² instead of 3³)

BEWARE of factors with power 0!

To find the total number of factors:

Factor pair method (too cumbersome for large numbers)
Add 1 to the exponent of each unique prime factor and multiply them together.

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Odds & Evens/Positives & Negatives

Use algebra to represent odds and evens

An arbitrary even number can be written as 2n, and an arbitrary odd number can be written as 2n+1 or 2n-1.

When inequalities containing 0 exist, test positive/negative cases to help solve the problem.

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Absolute value

|x-y| can be interpreted as the distance between x and y.

Solve algebraically OR test using appropriate numbers.

Example: Girls must be between 48 inches and 54 inches exclusive. Write an absolute equation for the heights of the girls, if x represents a girls height.

Find the average of 48 and 54 = 51 inches
Difference between 51 and a limit is 3.
Inequality: |x-51|<3

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Basic Inequalities

You can divide by a variable only if x > 0.

Question: Is 0.4x > 0.3x?
If you know: x > 0
Then you know: Question becomes “Is 0.4>0.3”. Answer is yes.

Multiplying 2 inequalities (but not dividing them!)

If x < y, z < w
And you know: x,y,z,w > 0
Then you know: xz < yw

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Rates

Rate x Time = Distance [RT = D]
Rate x Time = Work [RT = W]

In multiple RTD/RTW problems:

Rates can be added.
Times cannot be added.
Rates cannot be averaged directly - Use total distance/total time

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Relative rates:

Two people walking towards each other at different rates (speeds), distance between them shrinks at the rate of R₁+R₂.
Two people walk away from each other at different rates, distance between them grows at the rate of R₁+R₂.

Always express rate/work as job per unit time.

For combined work problems that involve time relations, find the combined rate, then use algebra to solve for t.

Use at RTW chart like the one above for complicated problems.

R	x	T	=	W
7/4	x	t	=	1

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For population problems, create a population chart:

Time elapsed	Population (in million)
10 mins ago	20
NOW	40
In 10 mins	80
etc	etc

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Advanced Rates & Work

Can be solved using the multiple RTD or RTW chart.
Take a step-by-step approach, thereby ensuring you don’t miss out on important information.
Make the task algebraic and hence computable.

Exponential growth/decay

Population doubling= S.2^t/I
Population tripling = S.3^t/I

Where S is the starting value, t is the time from now, I is the interval of doubling/tripling.

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Ratios

They establish a relationship between two or more items; they do not provide enough information on their own to determine the exact quantity of each item.

Often provide enough information to answer Data Sufficiency questions.

If two quantities have a constant ratio they are in direct proportion to each other, and vice versa.

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Averages

The effect of a new term in a set of terms can be computed using a shortcut

Change in mean = new term – old mean / new total number of terms.

A residual is the difference between a data point and the mean.

Positive residual for a number larger than the mean, negative residual for a number smaller than the mean.

For any set of numbers, the residuals sum to 0.

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Combinatorics

Fundamental Counting Principle

If you must make a number of separate decisions, then multiply the ways to make each individual decision to find the number of ways to make all the decisions.

2 types of bread (brown, white)
3 fillings (tuna, chicken, turkey)
Therefore, 2 x 3 = 6 different sandwiches possible.

Method also works for more than 2 decisions.

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For problems in which certain choices are restricted and/or affect other choices, choose the most restricted option first.

Example: 5 digit lock code to be chosen. First and last digits must be odd, and no repetitions allowed.
5 ways of picking first digit (1, 3, 5, 7, 9).
No repetition, hence 4 ways of picking last digit.
No repetition, hence 8 ways of picking second digit, 7 ways of picking third digit, and 6 ways of picking fourth digit.
Therefore total number of codes possible: 5 x 4 x 8 x 7 x 6 = 6720.

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Slot method

Draw empty slots corresponding to the choices that need to be made.
Fill in each slot with the number of options for that slot. Pick the most restricted slots first.
Use the Fundamental Counting Principle to find the total number of combinations.

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Use factorials to find the number of ways of putting n distinct objects in order, if there are no restrictions.

Example: Anagrams.

The number of arrangements of a set of items is the factorial of the total number of items, divided by the factorial(s) corresponding to sets of repeated items.

Use the Anagram Grid in cases of multiple repetition and where we care about who is in and out of a group, not the internal order of the chosen group.

Use the Glue Method for problems in which items or people must be next to each other by pretending that the items “stuck together” are actually one larger item.

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Insert slides from work here

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Advanced Comb/Prob/Stats

Permutations: a selection of items from a pool, where the order of items matter.

A-B-C is different to B-A-C.
Given by n!/(n – r)!

Combinations: a selection of items from a pool, where the order of items do not matter.

A-B-C is the same as B-C-A.
Given by n!/(n – r)!r!

If switching the elements in a chosen set creates a different set, it is a permutation. If not, it is a combination.

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Set Theory & Venn Diagrams

Create a double set matrix

Usually easier for problems with two sets.
Use nice round numbers for the totals (i.e. 100 if the problem uses percentages, LCM for fractions).

Use Venn diagrams for problems with more than two sets.

Work from the inside out.

	Men	Women	TOTAL
Employed
Unemployed
TOTAL			100

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Minor Problem Types

Optimization:

maximizing or minimising a quantity by choosing extreme values of related quantities

Grouping:

putting people or items into different groups to maximize or minimize some characteristics.

Scheduling:

planning a timeline to coordinate events according to a set of restrictions.

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Be aware of both explicit constrains (restrictions actually stated in the text) and hidden constraints (restrictions implied by real-world limitations).

If 40 people need to be put in groups of 6, hidden constraints mean the number of people in each group must be a positive integer.

Choose the highest/lowest values of the variables that you're allowed to select in order to maximize/minimize quantities.

Be cautious when it comes to rounding. Some problems require a rounding up, while others require a rounding down.