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What is the distance/rate and work/rate formula?
- Distance = Rate * Time
- Work = Rate * Time
What are the 5 important steps to work/rate problems?
- 1) Always convert times to rates.
- 2) Rates are additive (follow simple fraction adding rules).
- 3) Once you have the rates, plug the info into the work/rate formula to solve for unknown.
- 4) When conditions change during the problem (or something added/taken away) always reset equations and solve w/ multiple equations
- 5) Time used is inversely proportional to the # of workers. More workers = less time needed (and vice versa).
What is the golden rule regarding work/rates if the question is regarding 1 complete job?
- You can convert the time taken to a rate, by realizing the rate is the inverse of time.
- ex: 6 hours for 1 person to complete a job.
- Rate = 1/6th of the job per hour.
What is the official guide's basic formula for solving work problems?
1/a + 1/b = 1/c
- 1 = the job, denominator = time in hours (for ex.)
- 1/a = time it takes person a to finish job alone
- 1/b = time it takes person b to finish job alone
- 1/c = time it takes both together to finish job
How do you solve mixture problems in which Product A is mixed with Product B?
Solve for the product of each separately and then combine.
What is the difference between simple annual interest and compounded interest? How do you solve for each?
Simple Annual Interest, interest is calculated on the principle only. (principle) * (interest rate) * (time).
- Compounded Interest is calculated on the principle AND any interest already earned.
- (principle) * (interest rate, for that period). Then add the new balance * interest rate for remaining period. To = total after interest compounded
If a price is discounted by n percent, then the formula is...
- discount price = (100 - n) % of the original price.
- Ex: 25% off of $30 -> (100-25) = (75)% * $30
Ex: if $24 is the discounted price after 25% off, what is the original price? 0.75p = $24, so p = $32
Profit $ =
selling price - cost ..... or (revenue - expenses)
- Ex: Cost = $30, Profit Goal = 50%, Retail = ?
- Retail - $30 = (0.5) ($30)
- Retail - $30 = $15 -> Retail = $45
Retail = ? Given Cost & Profit (Markup) Margin
Retail = (Cost * Markup) + Cost
Markup % (Margin) = ? Given Cost & Retail
Markup = (Retail - Cost) / Cost
Remember: Sets can be represented as Venn Diagrams or organized in a table.
Arithmetic Mean means what?
The Average = Sum of terms in set / # of terms in set
If all terms in a set are multiplied by a constant, the new average/mean can be derived by...
multiplying the original average/mean by the same constant.
- Ex Set: (3, 5, 6, 10) * 2 = 48 = mean 12
- Original mean = 6 (or 6 * 2) = 12 new mean
Median means what?
The middle # in a set.
If there are an even # of values in the set, it's the average of the two middle terms.
What is the first step in finding the median of a set?
Re-arrange #s in ascending or descending order.
If multiple sets are combined each having the same median, can you solve for the median of the combined set? Do you need to know the # of terms in each set?
Yes, you can solve. It's the same median as the others. No, in this case you don't need to know the # of terms in each set.
What is the relationship b/w the median & mean?
Generally no relationship.
Only exception is if the #s in a set are evenly distributed (have a constant # b/w each), then the mean will ALWAYS equal the median.
In evenly spaced sets in ascending order, how do you determine the mean & median?
Add the first & last # together & divide by 2. This = the median & mean.
If the word inclusive is including you need to what? How about if it's not inclusive?
If inclusive - add 1 to the total. If not inclusive subtract 1.
- ex: Range of 650-750 inclusive = ?
- 750-650 = 100 + 1 (inclusive) = 101
When having to sum multiple #s (perhaps in a range), what is a quick way to solve?
Take # of terms * avg = sum of terms
If you start a range with an odd term & end with an odd term then... (same applies to even terms)
you have 1 more of those.
- Ex: 2-10 (both are even). # of terms = 8 + 1
- (4 odd & 5 even)
Mode means ?
It is the most frequently occuring number(s). Note there can be multiple.
Ex Set: 2,2,3,3,4,4, - mode = 2,3,4
What is the range of a set?
- The difference b/w the largest & the smalles value in a set.
- Largest # - Smallest # = range.
What are the 3 properties of a range?
- 1) Range is always positive.
- 2) If the range = 0, then all #s must be identical
- 3) You can change any # in a set, except for the smallest & largest & it has no effect on the range.
What is Standard Deviation?
Tells you how closely the terms in a set are spread around it's mean (avg. #).
What are the 5 steps to finding a standard deviation of a set?
- 1) Find the mean (avg). Divide sum of results by # of terms.
- 2) Subtract ea. result from the mean (to find difference).
- 3) Square ea. difference & add together
- 4) Divide the sum of the squared differences by the # of terms.
- 5) Take the square root of the result.
The Standard Deviation or Avg. Squared Difference is also known as ...
Does the # of terms in a set matter in regards to standard deviation? Why/why not?
Yes, # of terms very important. If two sets have the same spacing, the # of terms will determine which Std. Dev is greater. If one has more terms, that has the greater deviation.
If same spacing & terms, deviation = same.
What effect does adding/subtracting a constant # for each element in a set have on the std. deviation?
None - std. deviation is same.
What are the only 2 things that can affect Std. Deviation?
Multiplication And Division
What are the 3 Std. Deviation Properties which may/may not have an effect on the std. deviation.
1) Multiplation by a # with an absolute value > 1 increases & < 1 decreases.
2) Division by a # with an absolute value >1 decreases & < 1 increases.
3) Changing the sign of the terms in the set or multiplying by -1 has no effect.
What are the 2 things that Std. Deviation can never be?
- 1) A Negative #
- 2) Greater than half the range of a set
When comparing standard deviation of sets, it's helpful if they have...
equal # of terms (in ea. set)
What is the equation for Venn Diagrams?
(x + y) +/- Both + Neither = Total
- Add the Both if word "only" is used
- Subtract the Both if (x+y) is > than the total