Intro to Word Problems
Assigning Variables
1) do not use generic variables. Choose variables that will help you keep track of the numbers
2) sometimes it helps to choose the smallest value as the variable
3) sometimes it helps to choose the target value as the variable
4) if all the values are related to one quantity, choose that quantity as the variable
Writing Equations
1) translate from words into math
2) remember that you may have to do a certain amount of algebraic set-up before you can write the equation you will solve
Number of Variables
1) if there are two quantities in a word problem, it may save you time to use only one variable
2) if there are 3+ quantities in a word problem, you will always want to relate all of them to single variable and construct a single equation
Age Questions
1) age questions can be tricky when different mathematical relationships among the ages are specified at different times
2) choose variables to represent the age now, and use addition & subtraction to create expressions for ages at other times
Motion Questions
1) D = RT is the fundamental equation for moving things; know how to solve for R and for T
2) keep track of the units of numbers when you plug in
3) unit conversion can be written as fractions equal to 1; we can multiply & divide by them
4) know the common unit conversions, and change units so that all units in the problem are consistent
E.g. 1ft = 12in; 1h = 3600s
Average Speed
1) when finding an average velocity, do not fall into the trap of thinking it is a simple numerical average of two things
2) you have to find the D and T of each leg of the trip and add across legs to find the total D and the total time
3) the average velocity = (total distance)/divided by (total time)
Multiple Traveler Questions
1) when a word problem involves multiple travelers, multiple trips, or a trip with multiple legs, remember that each traveler, each trip, and/or each leg deserves its own D = RT equation
2) sometimes, you will be able to solve for all the quantities in one equation and use those numbers to help solve the other equations
3) more often, you will have to use the techniques for solving 2+ equations with 2+ unknowns (substitution & elimination)
Shrinking and Expanding Gaps
1) going in opposite directions: ADD
2)going in the same direction: SUBTRACT
3) think about the directions to determine shrinking vs. expanding
4) solving a D = RT for the gap itself can enormously simplify such problems
Work Questions
1) For work-related word problems, use A = RT
2) For proportion-related work problems, you need matching units on each side and you need to understand operations with proportions
3) For problems with multiple workers/machines, create rates for each and add the rates
Growth and Decay
Mixture Questions
1) concentration is the ratio of solute to total solution, most often expressed as a percent
2) we can change concentration by adding water or adding pure solute to a solution, or by mixing two solutions of two different concentrations
3) if we mix two solutions in unknown amounts to get a known total of known concentration, we set up simultaneous equations: one of the total amount, and one for the amount of solute
Intro to Sets and Venn Diagrams
1) Venn diagrams can be helpful in problems with two overlapping sets
2) the problem may give a Venn diagram, but if it does not, one is often helpful for solving this kind of question
3) remember to be careful in interpreting wording: ‘all in X’ includes those who are also in Y, but ‘all those in X only’ excludes those also in Y
Double Matrix Method 画表格
1) the double matrix method can enormously simplify problems in which each member is placed into two different kinds of categories
2) the entries in rows sum to the row totals in the rightmost column
3) the entries in the columns sum to the column totals in the bottom row
4) the grand total, in the lower right-hand box, is everyone; this equals the sum of the row totals and equals the sum of the column totals
Three Criteria Venn Diagrams
1) in populations in which each individual can be a member of any of three categories, use a 3-way Venn diagram
2) remember that many of the common categories will involve more than one section of the diagram
3) work from the central region outward
Intro to Sequences
1) a sequence is an ordered list of numbers
2) in the notation a_n, n is the ‘index’, that is , the place on the list
3) an entire infinite sequence can be specified simply by giving an algebraic formula for a_n in terms of n
Arithmetic Sequences
1) an arithmetic sequence is one in which the terms have a common difference
2) any evenly spaced list is an arithmetic sequence. Other examples include consecutive multiples of a number, consecutive odds or evens, and numbers which, when divided by the same divisor, have the same remainder
3) the nth term of any arithmetic sequence is
a_n = a_1 + d*(n-1)
Recursive Sequences
1) in recursive sequences, each term a_n is defined in terms of one or two previous terms (a_(n-1) and maybe a_(n-2))
2) the numerical values of one or two terms will always be specified
3) with a recursive sequence, there is no way to jump immediately to the value of term such as a_6. Instead, we have to find each and every term from the start up to the desired term
Inclusive Counting
1) we use inclusive counting whenever the situation demands that both endpoints, the lowest value and the highest value, are part of what we are counting
2) we perform the ordinary subtraction of high - low, and then +1 for the included lower endpoint
Sums of Sequences
Any evenly-spaced list that has N items, with a lowest term a_1 and a final term a_N
Sum of list = N(a_1 + a_N)/2 = (N/2)(a_1 + a_N)
Consecutive Integers
some basic facts:
1) a set of n consecutive integers will always contain one number divisible by n
2) if n is odd, then the sum of a set of n consecutive integers will always be divisible by n
3) in a set of 3 consecutive integers you could have two evens and one odd, or two odds and one even. In a set of 4 consecutive integers you must have two evens and two odds
Backsolving
1) when all five answer choices are numbers, one alternative strategy is to solve by backsolving
2) start with answer choice (c). Try this as the answer to the prompt question, and see if it works in the scenario
3) if (c) does not work, us the information about 'too big' or 'too small' to eliminate other answers
Picking Numbers
when to apply
variables in the question
variables in the answer options
what are 'smart number'
NOT 0 or 1
NOT numbers that already appear in the answers
Ideally, realistic
And if you still end up with duplicates... just tweak your numbers
