Q: The changes in the price of a stock from one year to the next year for three consecutive years were a 20% increase, a 25% decrease, and a 60% increase.
What amount was the increase in the price of the stock over the three years? (Calculator allowed)
A. 38%
B. 44%
C. 47%
D. 55%
You can use any method to solve the problem.
A: NOTE: Always use the Socratic method to avoid answering questions directly and to encourage student engagement.
Review of percent-to-decimal conversions.
a. How do you convert a percent to a decimal?
b. How are decimal values used in mathematics?
c. Without using any math, give an example of a number increasing by 100%.
d. Convert 100% to a decimal.
e. If $50.00 increased by 100%, what amount of money would we have?
f. Could the decimal form of 100% be used alone to calculate how much money we’d have if $50.00 increased by 100%? Why or why not?
g. It is true that ( $50.00 )( 2 ) = $100.00; however, is there some other way to determine that $50.00 was added to the original amount of money we had ( Initial principal = Pi )? Note: We originally had $50.00, and according to our percentage increase, we added $50.00 to the original amount ( Pi ).
h. If $50.00 increases by 100%, we could say that $50.00 + $50.00 = $100.00. What if $50.00 increases by 50%? What is 50% of $50.00?
KEY: If 50% of $50.00 = $25.00, how much money would we have if a $50.00 stock increases by 50%? ( I wouldn’t have this written down, but I’d use some annotation method to highlight that $50.00 + $25.00 = $75.00 ).
CONCLUSION: For each time period in question, we will need to add the percent increase of the initial principal ( Pi ) to the initial amount of money ( Pi ) that we began with. Ex. If the initial principal ( Pi ) = $50.00, the final principal ( Pf ) = ( Pi +/- ( ( Pi )( % increase or decrease ) ).
i. If an increase in the value of a stock over some time frame is equal to ( Pf ) = ( Pi + ( ( Pi )( % increase ) ), what formula could we use to describe a stock that has fallen by 50% in some period of time? ( The answer would be ( Pf ) = ( Pi – ( ( Pi )( % decrease ) ).
2. Let’s now compare the final amount ( Pf ) to the initial amount ( Pi ).
a. If a stock valued at $100.00 increases by 100%, we will be left with $200.00. Can we use addition or subtraction to determine the range by which ( Pi ) changed? ( $100.00 – $50.00 = $50.00 ).
b. What if a $100.00 stock lost all of its value? Could we use addition or subtraction to determine the range by which the stock declined in value? ( 0 – $100.00 ) = -$100.00.
KEY: In each case, the initial value ( Pi ) was subtracted from our final principal ( Pf ), regardless of whether or not the value of the stock went up or down.
Example: A stock worth $200.00 decreases in value by 50%. What is the range by which it decreases? ( Pf ) = ( Pi +/- ( ( Pi )( % decrease ) ). We first need to determine the value of ( Pf ), and afterwards, ( Pf – Pi ) = range over which the stock value decreased. So ( Pf ) = ( $200.00 – ( ( $200.00 )( 0.50 ) ) = ( $200.00 – $100.00 ) = $100.00. Therefore, ( Pf – Pi ) = ( $100.00 – $200.00 ) = – $100.00. Our stock decreased in value by – $100.00.
c. In the previous example, what is the PERCENT decrease of the stock? If the stock value decreased – $100.00, what is the percent decrease of the initial value of $200.00? KEY: Some ratio of – $100.00 and $200.00 will equal 50% ( or in this case, – 50% ). Will we use multiplication, division, addition, or subtraction to get our ratio?
d. The answer to the question above is ( – $100.00 / $200.00 ) = – 0.5. What must we do to convert this decimal to a percentage value? ( ( – 0.5 )( 100% ) = – 50%. Notice that – 100.00 = Pf. We arrived at this value by subtraction: ( Pf – Pi ). Further notice that we divided the range of decrease by Pi = $200.00.
FINALLY, WE HAVE A CONVENIENT WAY TO EXPRESS A PERCENT RISE OR DECLINE IN VALUE OF OUR STOCK: ( ( Pf – Pi ) / ( Pi ) )( 100% ) = Percent increase or decrease in value.
From here on out, I’d use the fact that ( Pf ) = ( Pi +/- ( ( Pi )( % increase or decrease ) ) to determine how much money we’d have at the end of each year in question. PERHAPS MOST IMPORTANTLY, I’D ALSO POINT OUT THE ( Pi ) FOR THE FOLLOWING YEAR MUST BE ( Pf ) FROM THE PREVIOUS YEAR. The principal that exists at the end of each time frame has a huge influence on the quantity left after another time frame passes.
BONUS QUESTION: The “ RULE OF 72 “ is oftentimes used by stock market investors to determine the length of time it will take an investment to double in value. If you take the interest rate of an investment ( Ex. 12% ) and divide it into 72, the Rule of 72 says that the investment will double in 6 years. The rule derives from observations that an investment known as an S and P 500 Index Fund ( Comprised of 500 large companies ) has an average annual return of about 8% ( 7% adjusted for inflation ). Therefore, it would take ( on average ) between 9 or 10 years for money invested in such a fund to double.
https://www.investopedia.com/…/what-average-annual… ).
https://www.investopedia.com/terms/r/ruleof72.asp
e. After using ( Pf ) = ( Pi +/- ( ( Pi )( % increase or decrease ) ) over a three-year period, I’d compare the final value arrived at after year 3 ( Pf ) to ( Pi ) to solve the problem.
f. We have a 20% increase, 25% decrease, and a 60% increase in the value of a stock over the course of three separate years.
g. What number would be convenient to use as a principal ( Pi )? ( My answer would be $100.00, but if something else is chosen, then I’d go with that. )
h. If Pi = $100.00, for year one, then ( Pf1 ) = ( Pi + ( ( Pi )( 20% ) ) = ( $100.00 + ( ( $100.00 )( 0.20 ) ) = $120.00.
i. Pf1 for year one = Pi for year two. Therefore, ( Pf2 ) = ( Pf1 – ( ( Pf1 )( 0.25 ) ) = ( $120.00 – ( ( $120.00 )( 0.25 ) ) = $90.00.
j. Pf2 for year two = Pi for year three. Therefore, ( Pf3 ) = ( Pf2 + ( ( Pf2 )( 0.60 ) ) = ( $90.00 + ( ( $90.00 )( 0.60 ) ) = $144.00.
For the range of time in question, ( ( Pf – Pi ) / ( Pi ) )( 100% ) = ( ( $144.00 – $100.00 ) / ( $100.00 ) )( 100% ) = ( $44.00 / $100.00 ) ( 100% ) = ( 0.44 )( 100% ) = 44%.
RENAISSANCE PHYSICS 