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- Quant - Miscellaneos

Kunal K.

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Interest Formulas

Compound interest = P * (1+r/n)^nt

P = principal

r = rate (in decimal form)

n = number of times per year

t = number of years

BUT you don't need to know this formula!

Treat these problems like successive percent problems.

e.g. after 2 years of 8% annual compound interest, the new amt is Principal*1.08*1.08

** Be careful of when the principal compounds versus the rate. Everytime the principal compounds, the amount increases. e.g. if you have $100 that earns 8% annual interest and compounds quarterly, then by Q3 it has compounded 6%

Percent Change = Change in value / Original Value

Smart numbers

Sometimes I will have to plug in numbers to solve a problem, and I'll need to choose a number

The best number (aka smart number) is going to be one which is a multiple of the denominators of all fractions in the problem.

Problem: If tank #1 is 4/5 full and tank #2 is 1/2 full, and if all liquid in tank #2 is moved to tank #1, then how full is tank#1?

The smart number needed to solve this problem is 10!!

**You can ONLY pick ONE smart number per problem - NEVER PICK MORE THAN ONE SMART NUMBER!!!! You can't do this because the ratio of the two smart numbers can vary

Comparing fractions using cross-multiplication

Is 3/13 bigger or smaller than 2/9?

Sometimes I can quickly determine this (because I'm the shit...). Other times, I waste time figuring stupid stuff like this out.

So, to quickly solve, use cross-multiplication!!

3/13 ??? 2/9

9x3 ??? 13x2

27 ??? 26

3/13 > 2/9!!!!!!

Notice how important it is to avoid mixing up sides during the calculations!

Mixture charts

When I am adding/subtracting ingredients from two different items, a mixture chart is a great way to figure out in an organized fashion what the new %'s are for each ingredient in the overall mixture.

Think of mixture charts like a simple, straightforward table. See p. 114-115 of the FDP book for examples

+ The columns show the original, change, and new (original + change = new)

+ The rows show the ingredients - e.g. alcohol, water, total solution

+ The individual cells are the volumes for each ingredient

*** Fill out what you know, then solve for unknowns!

Decimals to memorize

This will help me (a) identify solutions faster, (b) easily convert decimals to fractions so I can do math easily, (c) help me avoid careless errors. The decimals I need to know are:

+ Multiples of 1/6, Multiples of 1/7, Multiples of 1/8, Multiples of 1/9, and Multiples of 1/11

+ 1/6 --> 0.16, 0.33, 0.50, 0.66, 0.83

+ 1/7 --> 0.14, 0.29, 0.43, 0.57, 0.71, 0.86

+ 1/8 --> 0.125, 0.250, 0.375, 0.50, 0.625, 0.75, 0.875

+ 1/9 --> 0.11, 0.22, 0.33, 0.44, 0.55, 0.66, 0.77, 0.88

+1/11 --> 0.09, 0.18, 0.27, 0.36, 0.45, 0.54, 0.63, 0.72, 0.81, 0.90

This is a step-by-step bulletproof way to convert any decimal to a fraction, which is VERY useful if I can't figure out the fraction

Step 1: Count the #'s to the right of the decimal

Step 2: Get rid of the decimal - you now have the numerator

Step 3: The denominator starts with a 1, and then has as many 0's as you counted numbers in step 1

Terminating decimals

Terminating decimals can all be written as:

Some integer ÷ some power of 10

Rates and Work

(2) Rate × Time = Work

If you are dealing with 2 separate and distinct bodies that are moving towards each other, away from each other, or in the same direction at diff rates, then the calculations could take a lot of time to work through.

***For all 3 of the above scenarios, one strategy to save time is to create a third RT = D equation for the rate at which the distance between the 2 bodies is changing

e.g. Two people are both going in the same direction but one is moving at 8 mph and the other is moving at 5 mph. Thus, they are decreasing the distance between them at a rate of 3 mph.

If you are dealing with two workers who are both doing the same work, then you can add the rates at which they're doing the work.

e.g. Lucas finishes something in 1/3 of an hour, and Martha finishes something in 1/2 an hour, so combined they can complete some task in 5/6 of an hour.

When working distance and work problems, ALWAYS write rates so that the time appears on the bottom.

It may sound obvious, but if I take a bad shortcut and forget to do this, I could easily miss the problem (as practice problems have shown).

Sometimes you have a problem which involves an old average, a new average, and an additional entry in the data set that has causes the old average to shift (to the new average).

How do you solve these problems?

There's a quick and easy way to solve problems that involve (a) an overall average, and (b) >= 2 groups which each have their own averages

Take this problem: A mixture of "lean" ground beef (10% fat) and "super-lean" ground beef (3% fat) has a total fat content of 8%. What is the ratio of "lean" to "super-lean"?

Another way to solve the problem is to work it out algebraically

10x + 3y = 8 (x+y)

Eventually you realize that 2x=5y, and x=5 when y=2

ALWAYS ALWAYS solve these weighted average problems through using differentials, and picking #'s!!

Let's say you have a charity that sold an average of 66 tickets to each member. The female members averaged 70 tickets per member. The male:female ratio is 1:2.

What is the average number of tickets sold by the male members?

SD's tell you how far from the mean the data points typically fall.

Variance is the square of SD (Variance = SD^2)

+ Small SD means data is clustered near the mean

+ Large SD means data is spread out widely

Set 1 = {5, 5, 5, 5} --> avg spread = 0, SD = 0

Set 2 = {2, 4, 6, 8} --> avg spread = 2, SD = sqrt(5)

Set 3 = {0, 0, 10, 10} --> avg spread = 5, SD = 5

"Evenly spaced sets" are numbers that increase/decrease by a fixed amt (aka the increment)

The following properties apply to all evenly spaced sets:

Strategies for dealing with these sets:

If you have to count the number of items in a set, use this formula:

(Last - First) ÷ Increment + 1

**You add 1 because of the "off-by-1 bug"

You divide by the increment so that you don't overcount if the set has an increment of 3

Let's say you have to sum the integers from 20 to 100 inclusive. What's a quick way to do this?

Doing this sum takes way too much time on the GMAT. YOU HAVE TO ELIMINATE THE TIME!

Therefore, use the rules for evenly spaced sets in order to create shortcuts!

+ The average of an odd number of consecutive integers is always an integer

{1, 2, 3, 4, 5} --> 3

+ The average of an even number of consecutive integers is never an integer

{1, 2, 3, 4} --> 2.5

Remember, overlapping sets are all about describing two pieces of independent information for a single group of people.

Tip #1: Be careful when a problem says something like "20% of all students love dogs" vs. "20% of those students who have money love dogs". These two statements describe two different numbers! One relates to the "grand total" i.e. total number of students; while the other relates only to a subtotal, i.e. the students who have money.

--> Don't write "20" in my table, write (1/5 of subtotal)

Overlapping sets - Table tips and tricks (2/2)

Tip #3: Be prepared to use variables to relate 2 distinct squares to one another

Tip #4: Be careful when the problem says "25% of students are males who took math" vs. "25% of the males had taken math." These 2 statements are different enough that they lead to 2 different tables!

+ Tables are used for problems that have two sets. We've seen this described briefly in the previous flash card.

+ Venn diagrams are used for problems that have three sets. These problems usually involve 3 teams/clubs, and there are people that are either ON or NOT ON these teams/clubs.

Take this problem as an example: There's students taking 3 electives - gym, music, and art. 20 in music, 23 in gym, 24 in art.

How many kids are in 2 electives when 5 take all 3 and the total # kids is 52?

+ To find out how many kids are in 2 courses, you must realize that your only goal is to bring the total # kids down to 52.

+ When accounting for the kids in the middle, you went from 20 + 23 + 24 to (20-5) + (23-5) + (24-5) + 5 = 15 + 18 + 19 + 5. Thus, to find the # of kids in both groups, run a similar calculation.

Sometimes you'll have to plug in answer choices to find the right answer (rather than doing math that is tedious). There's a right way and wrong way to do this.

Grouping problems involve drawing items out of a large pool and grouping them together somehow. They look like this:

e.g. There's a conference where groups are being formed. You need 1 person from Division A, 2 from Division B, and 3 from C. There are 20 people from A, 30 from B, and 40 from C. What is the smallest # of people who will not be able to be assigned to a group?

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

(2) For any set of consecutive integers with an ODD # of items, the sum of all the integers is ALWAYS a multiple of the # of items.

--> This works b/c the sum equals the average times the # of items. For odd-type sets, the average is an integer, so the sum is a multiple of the # of items.

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