Compound Interest (Problem Solving) - Number Sense 101

Summary

This 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

00:00

🧮 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.

05:00

💼 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.

10:04

🎯 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.

15:06

📈 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.

Keywords

💡Compound Interest

Compound interest refers to the process where interest is added to the principal of a deposit or loan so that the added interest also earns interest from that point onward. In the video, the compound interest formula is used to solve problems where interest is compounded at regular intervals, such as monthly or quarterly.

💡Principal

The principal is the initial sum of money invested or loaned before any interest is applied. For example, in the first problem in the video, Maria’s principal is 20,000 pesos, which is the starting amount she deposits into her savings account.

💡Annual Interest Rate

The annual interest rate is the percentage of the principal that is paid as interest each year. In the video, this rate is crucial for calculating future values, such as when Maria earns 8% annual interest on her savings account.

💡Future Value

Future value represents the amount of money in the account after interest has been applied over time. It includes both the principal and the accumulated interest. In the first example, Maria's future value after 10 years is 44,392.80 pesos.

💡Number of Periods (N)

This refers to how often interest is compounded in a year. In the video, periods can be monthly, quarterly, or annually. For example, when interest is compounded monthly, N is 12, meaning interest is applied 12 times per year.

💡Time (T)

Time refers to the number of years for which the investment or loan is compounded. In the video, the problems commonly involve time periods such as 10 years for Maria’s savings or 45 years for James' retirement investment.

💡Logarithms

Logarithms are mathematical operations used to solve for time or interest rate when values are compounded over time. In the video, logarithms are applied in problem 4 to find how long it will take for Marta’s investment to grow to 1 million pesos.

💡Mutual Fund

A mutual fund is an investment vehicle made up of a pool of money collected from many investors. In the video, James invests in a mutual fund with a 9.5% annual return, which helps him save for his retirement in 45 years.

💡Quarterly Compounding

This is the process of applying interest four times a year, or every quarter. In the video, James' investment is compounded quarterly, meaning interest is added four times each year to maximize growth.

💡Interest Rate

The interest rate is the percentage at which interest is applied to the principal. It plays a crucial role in determining how much the principal will grow over time. For instance, in the third problem, Sarah needs a 12.20% interest rate to grow her 10,000 pesos to 100,000 pesos over 20 years.

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