Compound Interest (Problem Solving) - Number Sense 101
Summary
TLDRThis video explains how to solve word problems using the compound interest formula, which calculates the future value of an investment or loan. The formula is presented as A = P(1 + R/N)^(NT), where A is the future value, P is the principal, R is the interest rate, N is the number of compounding periods per year, and T is the time in years. The video includes detailed examples demonstrating how to determine the future value, calculate the necessary initial investment, and find the interest rate or time required to reach a financial goal.
Takeaways
- 📈 Compound interest involves the addition of interest to the principal sum of a loan or deposit.
- 🧮 The compound interest formula is A = P(1 + R/N)^(NT), where A is the final amount, P is the principal, R is the annual interest rate, N is the number of periods per year, and T is the time in years.
- 💰 Example 1: Maria deposits 20,000 pesos at an 8% annual interest rate compounded monthly, resulting in a future value of 44,392.80 pesos after 10 years.
- 💵 Example 2: To have 2 million pesos for retirement in 45 years, James needs to invest 29,249.93 pesos at a 9.5% annual interest rate compounded quarterly.
- 💸 Example 3: Sarah wants to turn her 10,000 pesos into 100,000 pesos in 20 years. The required interest rate is 12.2018% compounded annually.
- 💼 Interest is calculated as the future value minus the principal.
- ⏳ Example 4: Marta invests 50,000 pesos at an 8.4% interest rate compounded semi-annually. It will take 36.41 years for her account to reach 1 million pesos.
- 📊 Logarithmic functions are used to solve for the time (T) when compounding is involved.
- 🔢 The interest accrued over time can be found by subtracting the principal from the future value.
- ✅ The examples illustrate how different compounding periods (monthly, quarterly, semi-annually, annually) affect the growth of investments.
Q & A
What is compound interest?
-Compound interest is the addition of interest to the principal sum of a loan or deposit, meaning the interest also earns interest over time.
What is the formula for calculating compound interest?
-The formula for calculating compound interest is A = P × (1 + R/N)^(N × T), where A is the future value, P is the principal, R is the annual interest rate, N is the number of periods per year, and T is the time in years.
In Example 1, how much money will be in Maria's account after 10 years with an 8% interest rate compounded monthly?
-Maria will have 44,392.80 pesos in her account after 10 years.
How is the interest after 10 years calculated in Example 1?
-The interest is calculated by subtracting the principal from the future value: 44,392.80 pesos - 20,000 pesos = 24,392.80 pesos.
What principal should James deposit to have 2 million pesos for retirement in 45 years at a 9.5% interest rate compounded quarterly?
-James should deposit 29,249.93 pesos into his mutual funds.
How much interest will James earn after 45 years on his investment?
-James will earn 1,970,750.04 pesos in interest after 45 years.
In Example 3, what interest rate does Sarah need to turn her 10,000 pesos investment into 100,000 pesos in 20 years?
-Sarah needs an interest rate of 12.20% compounded annually to reach her goal.
How much interest will Sarah earn after 20 years?
-Sarah will earn 90,000 pesos in interest after 20 years.
How many years will it take for Marta's account to reach 1 million pesos with an 8.4% interest rate compounded semi-annually?
-It will take Marta 36.41 years for her account to reach 1 million pesos.
What is the total interest Marta will earn after 36.41 years?
-Marta will earn 950,000 pesos in interest after 36.41 years.
Outlines
🧮 Introduction to Compound Interest Formula
The video introduces the concept of compound interest, where interest is added to the principal amount of a loan or deposit. The compound interest formula is given as A = P(1 + R/N)^(NT), where A is the final amount, P is the principal, R is the annual interest rate, N is the number of periods per year, and T is the time in years. An example is presented in which Maria deposits 20,000 pesos into a savings account with an 8% annual interest rate, compounded monthly. The calculation shows how much money will be in her account after 10 years, resulting in a final amount of 44,392.80 pesos. The interest earned is 24,392.80 pesos.
💼 James' Mutual Fund Investment Plan
James aims to have 2 million pesos in 45 years and invests in a mutual fund that pays 9.5% annual interest, compounded quarterly. The problem is to find out how much James needs to invest initially to reach this goal. The formula for compound interest is used to determine that James should deposit 29,249.93 pesos. Additionally, the interest he will earn over 45 years amounts to 1,970,750.04 pesos.
🎯 Sarah's Investment Goal
Sarah wants to turn her 10,000-peso investment into 100,000 pesos in 20 years. The goal is to find the interest rate required to reach this target, compounded annually. Using the compound interest formula, the necessary interest rate is calculated to be 12.2018%. The interest earned over 20 years is 90,000 pesos.
📈 Marta's Annuity Growth
Marta invests 50,000 pesos in an index annuity that earns 8.4% interest, compounded semi-annually. The question is how long it will take for her investment to reach 1 million pesos. Using logarithmic properties, it is calculated that it will take 36.41 years for her investment to grow to 1 million pesos. The interest earned in this period will be 950,000 pesos.
Mindmap
Keywords
💡Compound Interest
💡Principal
💡Annual Interest Rate
💡Future Value
💡Number of Periods (N)
💡Time (T)
💡Logarithms
💡Mutual Fund
💡Quarterly Compounding
💡Interest Rate
Highlights
Introduction to compound interest formula and its components: A = P(1 + R/N)^(NT).
Definition of compound interest: addition of interest to the principal sum of a loan or deposit.
Example 1: Maria invests 20,000 pesos in a savings account with 8% annual interest compounded monthly.
Explanation of how to calculate the future value using the formula, yielding a total of 44,392.80 pesos after 10 years.
Interest earned by Maria after 10 years is calculated as 24,392.80 pesos.
Example 2: James wants 2 million pesos for retirement in 45 years, investing in a mutual fund with 9.5% interest compounded quarterly.
Calculation shows that James needs to invest 29,249.93 pesos to reach his goal of 2 million pesos in 45 years.
James will earn a total interest of 1,970,750.04 pesos over 45 years.
Example 3: Sarah aims to grow her investment from 10,000 pesos to 100,000 pesos in 20 years, compounded annually.
To achieve this goal, Sarah needs an interest rate of 12.2018%.
Sarah will earn 90,000 pesos in interest over 20 years.
Example 4: Marta invests 50,000 pesos in an index annuity with 8.4% interest compounded semi-annually.
Marta's investment will take 36.41 years to grow to 1 million pesos.
The interest earned by Marta over this time is 950,000 pesos.
The lesson concludes with a summary and an encouragement for viewers to practice solving more compound interest problems.
Transcripts
in this video we are going to talk about
how to use compound interest formula in
solving word problems so when we say
compound interest it is the addition of
interest to the principal sum of a loan
or deposit so the formula in compound
interest that is a equals P times 1 plus
R over N raised to n T where a is the
final amount or the future value and
then P is the principal or original
amount R is the annual interest rate and
it is the number of periods per year and
then T is the time per year so let's
start and let's have an example on
example number 1 Maria puts 20,000 pesos
in a savings account paying 8% annual
interest compounded monthly at this rate
how much money will be in the account
after 10 years so let us identify first
a given so the principal that is 20
thousand pesos then rate that is 8% or
0.08 and then time that is 10 years and
then n is the number of periods so she
can see it is compounded monthly so
there are 12 months in a year so n is 12
and then we are looking for the amount
of money on its account after 10 years
so we are going to find the future value
now we are go to use the formula future
value equals principal times 1 plus R
over N raised to n T then
we have principle Rs 20,000 multiplied
by 1 plus R that is the interest rate
that is 0.08 over number of periods that
is 12 raised to 12 times years that is
10 then we're going to have 20,000 times
1 plus 0.08 divided by 12 that is zero
point zero zero six six six seven then
12 times 10 that is 120 then let us
write 20,000 one plus zero point zero
zero six six six seven that is one point
zero zero six six six seven raised to
120 then let us write 20,000 one point
zero zero six six six seven raised to
120 that is two point two one nine six
four zero two three five therefore
20,000 multiplied by two point two one
nine six four zero two three five that
is forty four thousand three hundred
ninety two point 80 pesos so the future
value is forty four thousand three
hundred ninety two point 80 pesos so
this will be our answer
what about the interests so to find the
interests interest equals future value
minus principal so the future value is
forty four thousand three hundred ninety
two point eighty minus twenty thousand
so the interest after ten years
is twenty four thousand three hundred
ninety two point eighty so this will be
our answer what about number two James
wants to have two million pesos for
retirement in forty five years he
invests in a mutual fund paying an
average of nine point five percent each
year compounded quarterly how much
should he deposit into his mutual funds
so let us identify the given so she can
say James wants to have two million that
is the future value which is two million
then time will be forty five years and
then interest rate that is nine point
five percent or simply 0.095 the number
of periods that is compounded quarterly
there are four quarters in a year
so n is four and we are looking for the
principal the amount of money invested
so let us use the formula future value
equals principal times 1 plus R over N
raised to n times T so the future value
is 2 million equals principal times
one plus zero point zero nine 5 divided
by 4 raised to 4 times 45 then to
simplify we have two million equals P
times 1 plus zero point zero nine five
divided by 4 that is zero point zero two
three seven five then four times 45 is
180 then let us write again two million
equals P times one plus zero point zero
two three seven five that is one point
zero two three seven five raised to 180
then to simplify let us write two
million equals P times one point zero
two three seven five raised to 180 that
is six to eight point three seven six
one five two three one then let us
divide the whole equation by six to
eight point three seven six one five two
three one that's right here
so let us cancel this one so principle
equals two million divided by six to
eight point three seven six one five two
three one if you are going to divide
this one we're going to have principal
that is twenty nine thousand two hundred
forty nine point nine two six forty one
so this will be our answer now let us
find the interest after 45 years so
interest
it was future value minus the principal
future value that is two million minus
the principal which is twenty nine
thousand two hundred forty nine point
ninety six forty one so subtract we're
going to have 1 million nine hundred
seventy thousand seven hundred fifty
point 0 36 pesos so this will be the
interest after forty-five years on
number three Sarah wishes to turn her
ten thousand passes investment into one
hundred thousand pesos in twenty years
how much interest does she need to
receive compounded annually to reach her
goal so let us identify the given the
future value is one hundred thousand
pesos and then the principal that is ten
thousand pesos and then time is twenty
years and then n that is the number of
periods we have compounded annually that
is one and then we are looking for the
interest if so the four
is future value equals principal times 1
plus R over N raised to n T so we have
the future value that is 100,000 pesos
and then principle that is 10,000 pesos
times 1 plus R over N that is the number
of period this one leads to 1 times 20
let's simplify we have 100,000 equals
10,000 times 1 plus R divided by 1 that
is R raised to 20 then let us divide the
whole equation by 10,000 so let's cancel
this one so we have 1 + R raised to 20
equals 100,000 divided by 10,000 that is
10 then to eliminate the exponent let us
raise both side of the equation by 1
over 20 so 10 raised to 1 over 20 then
let us can sell this one because 20
times 1 that is 20/20 row this one so
we're going to have 1 plus R raised to 1
or simply 1 plus R and then 10 raised to
1 over 20 that is 1 point 1 - 2 0 1 8 4
5 4 then let's move one to the side of
one point 12 20 1845 for that is R is 1
point 1 2 2 0 1 8 4 5 4 minus 1
are this 0.12 2018 four five four and
then to find the interest rate we'll
just simply move two decimal places to
the right one two so the interest rate
is 12 point 2018 for by 4% or you can
write your answer as R equals twelve
point twenty eighteen percent four
decimal places to be safe now let us
find the interest interest equals future
value minus the principal so future
value that is 100,000 and then principle
that is 10,000 so 100,000 minus 10,000
that is 90 thousand pesos so this will
be the interest after 20 years what
about number for Marta invest fifty
thousand pesos in an index annuity
that's averaging eight point four
percent compounded semi-annually at this
rate how many years for her account to
reach 1 million pesos so let us identify
the future value that is 1 million then
principle that is 50,000 and then we
have rate that is eight point four
percent or simply 0.084 and then number
of periods so we have semi-annually that
is two and we are looking for time so
the formula is
future value equals principal times 1
plus R over N raised to n T so we have
the future value we have 1 million
equals principal that is 50 thousand
times 1 plus rate that is 0.08 4 over 2
raised to 2 T because we are looking for
the time then we have 1 million equals
50,000 times 1 plus 0.08 4 divided by 2
that is 0.04 2 raised to 2 T and then we
have 1 million equals 50,000 times 1
plus 0.04 2 that is 1.04
2 raised to 2 T then let us divide the
whole equation by 50,000 so let us
cancel this one so we're going to have 1
point 0 4 2 raised to 2 T equals 1
million divided by 50,000 that is 20 now
to find the value of T or the number of
years we are going to use the property
of logarithm so let's have log 1 point 0
4 2 raised to 2 T equals log 20 so we
are going to use the property of
logarithms that is
a raise to n that is n log a so we are
going to move the exponent in front of
the logarithm so this time in our given
we have log one point zero four to least
to T that is to T log one point zero
four two equals log 20 then let us
divide the whole equation by log one
point zero four two so let us write our
answer here so we have the first cancel
this one so we're going to have two T
equals log 20 divided by log one point
zero four two then to simplify we have
two T log 20 that is one point three
zero one zero two nine nine nine six
over log one point zero four two that is
zero point zero one seven eight six
seven seven one eight nine six then let
us divide this one so we have two T so
if you are going to divide this one
we're going to have seventy two point
eight one four five five four four seven
then to find the value of T let's divide
the whole equation by 2
so T is 72 point 81 4 5 5 4 4 7 divided
by 2 that is thirty six point four zero
seven to seven seven to three years or
you can write your
as 36 point 41 years so this will be our
answer then to find the interest we have
interests that is future value minus
principle that is 1 million minus fifty
thousand that is nine hundred fifty
thousand pesos so this will be the
interest after a thirty six point forty
one years so I hope you learn from this
lesson see you again next time thanks so
much for watching
god bless us all
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