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Understanding Percentages and Financial Concepts, Exams of Mathematics

A.T. Still University of Health Sciences (ATSU)Mathematics

A comprehensive overview of various concepts related to percentages, including their basic uses, appropriate and common abuses, as well as formulas for calculating absolute and relative changes, comparisons, and compound interest. It delves into the nuances of percentage calculations, highlighting the importance of maintaining a consistent reference value and the potential pitfalls of averaging percentages. Additionally, the document covers key financial concepts such as installment loans, credit cards, mortgages, and prepayment strategies. The detailed explanations and examples make this document a valuable resource for understanding the practical applications of percentages and their role in personal finance.

Typology: Exams

2024/2025

Available from 09/12/2024

DrShirleyAurora
DrShirleyAurora 🇺🇸

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Quantitative Reasoning Test 1
Definition of percentage -
Per 100, how many parts per 100
Hundredths -
5% 5/100 or .05 are all the same in hundredths
Tenths -
3/10 or .3 or 30% notice the decimal in in the tenth place
5% of a class of 40 students get an A equals how many students? -
5/100 * 40/1= 200/100 or 2 (students)
What if you have a % in your decimal like .003% ? -
Take the number & write it over 100 to make a fraction- .003/100
The Three basic uses of percentages -
1. A portion of something; to express a fraction of a total or a ratio. 2. How something, a
value, has changed or is different. 3. A compared value or comparison of two different numbers.
(Percentages are always a comparison; either of the total, a previous value or 1 out of 2 values to
some standard value which signifies 100% which is the standard value always goes in the
denominator)
Appropriate uses of percentages -
1. To describe change ie. population growth, rising price, pay comparisons. 2. To describe a
fraction of the total (work force). 3. To compare the performance or cost of products.
Common abuses of percentages -
1. Shifting reference values (percentages fail if you don't have a standard & that standard
must be constant or already known) 2. less than nothing, 2. Don't average percentages unless both
tests happen to have the exact same number of questions.
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Quantitative Reasoning Test 1

Definition of percentage - Per 100, how many parts per 100 Hundredths - 5% 5/100 or .05 are all the same in hundredths Tenths - 3/10 or .3 or 30% notice the decimal in in the tenth place 5% of a class of 40 students get an A equals how many students? - 5/100 * 40/1= 200/100 or 2 (students) What if you have a % in your decimal like .003%? - Take the number & write it over 100 to make a fraction- .003/ The Three basic uses of percentages -

  1. A portion of something; to express a fraction of a total or a ratio. 2. How something, a value, has changed or is different. 3. A compared value or comparison of two different numbers. (Percentages are always a comparison; either of the total, a previous value or 1 out of 2 values to some standard value which signifies 100% which is the standard value always goes in the denominator) Appropriate uses of percentages -
  2. To describe change ie. population growth, rising price, pay comparisons. 2. To describe a fraction of the total (work force). 3. To compare the performance or cost of products. Common abuses of percentages -
  3. Shifting reference values (percentages fail if you don't have a standard & that standard must be constant or already known) 2. less than nothing, 2. Don't average percentages unless both tests happen to have the exact same number of questions.

Absolute Change Describing a Change in a value - describes the actual increase or decrease from a reference value to a (new) smaller or larger value. Absolute change=NV - RV new value minus the reference value (reference value is always the previous value, or, the New value minus the old, or new value-reference value Absolute Change for Comparisons of two different vaues - the actual difference between the compared value and the reference value. Absolute difference = compared value - reference value. AD=CV-RV Relative Change that Describes a Change - is a fraction that describes the size of the absolute change in comparison to the reference value: Relative change = the absolute change divided by the reference value which = the new value minus the reference value divided by the reference value. Relative Difference for Comparisons - describes the size of the absolute difference as a fraction of the reference value. Relative difference = Absolute difference divided by the reference value which is the compared value minus the reference value divided by the reference value. (pg 132) When a quantity doubles in value it's relative change is - 1 (RV) or 100% extra; the 100% is the difference between them so it's (RV or 1)+100% more When a quantity triples in value, its relative change is - 2 or 200% extra or loss if it's negative When a quantity quadruples in value, its relative change is - 3 or 300% extra or loss if it's negative. If $250 is increased by 500% you would multiply $250 by 500% + the original $250 because it 'Increased' The Relative difference formula gives a fraction which you can convert to a percentage by - Multiplying it by 100%

Translating a more than statement into an of statement - Retail prices are 100% + 25% = 125% of wholesale prices. You replace the of with multiplication. The retail price =125% times the wholesale price. If the compared value is P% more than the reference value then - the compared value = 100% + P% times the reference value and the reference value = the compared value divided by 100% +P% If the compared value is less than the reference value then - Use 100-P instead of + (plus) Shifting Reference Values - The reference value shifts during the problem; it can be higher or lower in the first calculation versus the second. Consider this statement, Your investment lost 60% the first year but gained 75% the second, so you're 15% ahead. An investment of $1,000 that lost 60% of its value would leave you with $400, in the second year your investment gained 75%, of $400, which would be $300, for a total of $700. This is less than your original investment not a gain of 15% overall. Don't average percentages; why? - Unless both tests or averages have the exact same numbers they wouldn't be an accurate average. You should NEVER average percentages. If you got 70% on one test and 90% on the second would your average be 80%? - Not unless they both had the exact same number of questions. If one has 10 questions & you got 7 correct, and the other has 100 & you got 90 correct, the combined total would be 97 correct out of 110 which is 88.2% because 97 divided by 110 is 0.882. pg 139 New or Changed Value equals - Reference Value times the percentage of Compared Value equals - Reference Value times the percentage of Simpson's Paradox -

It is possible for a set of data to give different results in each of several groups than it does when the groups are taken together. Overall results are divided into unequally sized groups. Check to see if groupings are unequal, look at total number of information. Numbers can deceive unless they are examined with great care. Numbers may not lie, but they may be deceiving when interpreted improperly. Examples of Simpson's Paradox -

  1. Groups of mild & severe acne with no clear rationale of how they were defined. 2. Who Played better, Shaq or Vince in the 1st half & 2nd half? Even if Shaq has a higher percentage in both halves at 40% & 70%, Vince who had 25% & 70% had a better overall shooting percentage of 57.1% for the game because he made 8 baskets on 14 shots and Shaq had 7 baskets on 14 shots which was only 50%. pg 184 What are the 4 categories of results for positive & negative tests? -
  2. True Positives 2. False Positives 3. True Negatives 4. False Negatives pg. 186 True Positive - an actual positive, malignant, result for a malignant tumor False Positive - A benign tumor gets a positive, malignant, result in which the results suggests their tumor is malignant. True Negative - Identifies benign tumors as benign. False Negative - The result is negative even though the women actually have cancer. pg. Ratio - Dividing two values to find the ratio of the two quantities. The ratio of $80,000 and $20,000 is $80,000 divided by $20,000 which equals 4 to 1. Notice the units of dollars cancelled each other out so the ratio is just a number. Because comparisons only make sense when compared with quantities of the same units, they always end up without units.

Accumulated balance after Y years, also known as future value, or, FV P = - Starting Principal also known as present value, or, PV APR = - Annual Percentage Rate - must be written as a decimal in the formula. When interest is compounded just once a year. Y = - Number of years n = - Number of times it compounds; Bi-annually (every two years like bi-weekly is every two weeks), annually, Semi-annually (half a year), quarterly, monthly or daily. APY - Annual percentage Yield. The actual percentage by which a balance increases in 1 year. It's equal to the APR if interest is compounded annually & does NOT depend on the starting principal. Also known as the effective yield or just the yeild. (If no value is given just plug one in to find it) Compound Interest formula for more than one time a year - A= P (1+APR/n) ex (n*Y) Installment Loan - A loan you pay off with equal regular payments. Also called an Amortized loan. The portions of installment loan payments going toward principal and toward interest vary as the loan is paid down. Early in the loan most of it goes towards interest & gradually decreases as the portion toward the principal gradually increases. Installment Loan formula - PMT= P * (APR/n)

[1-(1+APR/n) ex (-nY)] To find payments on an installment loan -

  1. Use the loan payment formula 2. Multiply the monthly payment by the loan term in months to find the total payments 3. Subtract the principal from the total payments to find the total interest. Credit cards - interest charges operate like compound interest in reverse. Mortgage - One of the most popular types of installment loans designed specifically to buy a home. Down payment - Amount of money you must pay up front to be given a mortgage, usually 10%-20% of the purchase price Closing costs - Are fees you must pay in order to be given a loan. Two types: 1. Direct fees for appraisal, checking credit, etc. Usually a set dollar amount. 2. Fees charged as points where each point is 1% of the total loan amount. Many are divided into two categories of an "origination fee" and "discount points". per & 'to' = - a fraction bar; compare LA population to NV population is LA:NV or LA/NV. Use this method for seemingly unrelated values only, not a changing value. Denominator of a ratio is always - the Reference Value Periodic payment - An agreement that calls for periodic payments on the loan amount until it's paid in full.