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Compound Interest Maze Answer Key

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Simple and Compound Interest

Learning Outcomes

  • Summate 1-time simple interest, and simple interest over time
  • Determine APY given an interest scenario
  • Calculate compound interest

We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we get-go saving, planning for retirement, or need a loan, we need more than mathematics.

Simple Interest

Discussing involvement starts with the main, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple class, is calculated every bit a per centum of the main. For example, if you borrowed $100 from a friend and agree to repay information technology with 5% interest, and then the amount of interest y'all would pay would merely exist v% of 100: $100(0.05) = $5. The full amount you lot would repay would exist $105, the original main plus the involvement.

four rolled-up dollar bills seeming to grow out of dirt, with a miniature rake lying in between them

Simple I-fourth dimension Involvement

(1) \begin{align*}&I={{P}_{0}}r\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}r={{P}_{0}}(1+r)\\\end{align*}

  • I is the interest
  • A is the stop amount: principal plus interest

  • (2) \begin{align*}{{P}_{0}}\\\end{align*}

    is the principal (starting amount)

  • r is the interest charge per unit (in decimal form. Instance: v% = 0.05)

Examples

A friend asks to borrow $300 and agrees to repay information technology in 30 days with 3% interest. How much interest will yous earn?

Solution:

(three) \begin{align*}{{P}_{0}}\\\end{align*}

= $300

 the principal
r = 0.03 iii% rate
I = $300(0.03) = $ix. You lot volition earn $9 interest.

The post-obit video works through this example in particular.

One-time simple interest is but common for extremely brusk-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would exist earned regularly.

For case, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, ofttimes annually. Bonds take a maturity engagement, at which fourth dimension the issuer pays back the original bond value.

Exercises

Suppose your metropolis is edifice a new park, and issues bonds to enhance the coin to build it. You lot obtain a $i,000 bail that pays 5% interest annually that matures in 5 years. How much interest will you earn?
[reveal-answer q="14596″]Show Solution[/reveal-answer]
[hidden-answer a="14596″]Each year, yous would earn v% interest: $1000(0.05) = $l in interest. So over the class of five years, you would earn a total of $250 in involvement. When the bail matures, you would receive back the $ane,000 yous originally paid, leaving you lot with a total of $1,250.[/hidden-answer]

Further explanation about solving this case can be seen here.

We can generalize this idea of elementary involvement over time.

Elementary Interest over Time

(iv) \begin{align*}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align*}

  • I is the interest
  • A is the end amount: principal plus interest

  • (5) \begin{align*}{{P}_{0}}\\\end{align*}

    is the master (starting amount)

  • r is the involvement rate in decimal class
  • t is time

The units of measurement (years, months, etc.) for the time should friction match the time period for the interest rate.

Apr – Almanac Percent Charge per unit

Interest rates are usually given as an annual percentage rate (April) – the total interest that volition exist paid in the yr. If the involvement is paid in smaller time increments, the Apr will exist divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.
6\div{12}=0.5

A four% annual rate paid quarterly would be divided into four one% payments.
4\div{4}=1

Case

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses. Suppose y'all obtain a $one,000 T-note with a four% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?

Solution:

Since interest is being paid semi-annually (twice a year), the four% interest will be divided into two 2% payments.

(half dozen) \begin{align*}{{P}_{0}}\\\end{align*}

= $1000

 the principal
r = 0.02 ii% rate per half-yr
t = 8 4 years = 8 one-half-years
I = $g(0.02)(8) = $160.  You volition earn $160 interest full over the four years.

This video explains the solution.

Try It

A loan company charges $30 interest for a 1 month loan of $500. Notice the annual interest charge per unit they are charging.

Solution:

I = $xxx of interest
P_0 = $500 principal
r = unknown
t = 1 month

Using I = P_0rt, nosotros get 30 = 500·r·1. Solving, we get r = 0.06, or 6%. Since the time was monthly, this is the monthly involvement. The annual charge per unit would be 12 times this: 72% interest.

Compound Involvement

With simple interest, we were assuming that nosotros pocketed the involvement when we received information technology. In a standard bank account, any interest nosotros earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding.

a row of gold coin stacks. From left to right, they grown from one coin, to two, to four, ending with a stack of 32 coins

Suppose that we deposit $one thousand in a bank account offering 3% interest, compounded monthly. How volition our coin grow?

The three% interest is an annual per centum charge per unit (April) – the total interest to be paid during the yr. Since interest is beingness paid monthly, each month, we will earn 3% ÷ 12 = 0.25% per calendar month.

In the showtime month,

  • P0 = $one thousand
  • r = 0.0025 (0.25%)
  • I = $1000 (0.0025) = $2.l
  • A = $1000 + $ii.50 = $1002.50

In the start month, we will earn $2.50 in interest, raising our account balance to $1002.50.

In the 2d month,

  • P0 = $1002.50
  • I = $1002.50 (0.0025) = $ii.51 (rounded)
  • A = $1002.50 + $2.51 = $1005.01

Notice that in the second month we earned more than interest than we did in the first calendar month. This is because we earned interest non only on the original $thousand we deposited, just we too earned interest on the $2.50 of interest we earned the start month. This is the cardinal advantage that compounding interest gives united states of america.

Calculating out a few more months gives the following:

Month Starting balance Interest earned Ending Residual
one k.00 ii.l 1002.50
2 1002.50 2.51 1005.01
3 1005.01 2.51 1007.52
4 1007.52 2.52 1010.04
5 1010.04 2.53 1012.57
vi 1012.57 ii.53 1015.10
7 1015.10 2.54 1017.64
8 1017.64 2.54 1020.18
9 1020.18 2.55 1022.73
x 1022.73 2.56 1025.29
xi 1025.29 ii.56 1027.85
12 1027.85 two.57 1030.42

Nosotros desire to simplify the process for computing compounding, because creating a table like the one above is time consuming. Luckily, math is practiced at giving you ways to take shortcuts. To find an equation to represent this, if Pm represents the amount of money after m months, so we could write the recursive equation:

P0 = $1000

Pthou = (1+0.0025)Pm-1

You lot probably recognize this as the recursive class of exponential growth. If non, we go through the steps to build an explicit equation for the growth in the next example.

Example

Build an explicit equation for the growth of $1000 deposited in a bank account offer 3% interest, compounded monthly.

Solution:

  • P0 = $1000
  • P­1 = 1.0025P­0 = one.0025 (grand)
  • P­2 = 1.0025P­1 = 1.0025 (1.0025 (grand)) = 1.0025 ii(1000)
  • P­3 = 1.0025P­two = 1.0025 (i.00252(1000)) = ane.00253(1000)
  • P­iv = 1.0025P­3 = 1.0025 (1.00253(chiliad)) = 1.00254(chiliad)

Observing a blueprint, we could conclude

  • Pm = (1.0025) k ($thou)

Notice that the $1000 in the equation was P0 , the starting amount. We constitute ane.0025 past adding one to the growth rate divided by 12, since nosotros were compounding 12 times per year.

Generalizing our consequence, we could write

{{P}_{m}}={{P}_{0}}{{\left(1+\frac{r}{k}\right)}^{m}}

In this formula:

  • yard is the number of compounding periods (months in our example)
  • r is the almanac interest charge per unit
  • 1000 is the number of compounds per year.

View this video for a walkthrough of the concept of compound interest.

While this formula works fine, it is more than mutual to use a formula that involves the number of years, rather than the number of compounding periods. If Due north is the number of years, then chiliad = Northward k. Making this alter gives us the standard formula for compound interest.

Compound Interest

P_{N}=P_{0}\left(1+\frac{r}{k}\right)^{Nk}

  • PNorth is the remainder in the account after N years.
  • P0 is the starting residuum of the account (also called initial deposit, or chief)
  • r is the almanac interest rate in decimal course
  • m is the number of compounding periods in one year
    • If the compounding is done annually (once a year), chiliad = 1.
    • If the compounding is washed quarterly, k = 4.
    • If the compounding is done monthly, k = 12.
    • If the compounding is done daily, k = 365.

The most of import matter to recall well-nigh using this formula is that information technology assumes that nosotros put money in the account in one case and allow it sit there earning involvement.

In the next example, we bear witness how to use the compound interest formula to find the balance on a certificate of deposit later 20 years.

Case

A document of deposit (CD) is a savings instrument that many banks offer. It commonly gives a higher interest charge per unit, merely yous cannot access your investment for a specified length of time. Suppose you eolith $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account later 20 years?

Solution:

In this example,

P0 = $3000 the initial deposit
r = 0.06 half-dozen% annual rate
1000 = 12 12 months in ane twelvemonth
N = twenty  since we're looking for how much we'll accept after xx years

So {{P}_{20}}=3000{{\left(1+\frac{0.06}{12}\right)}^{20\times12}}=\$9930.61 (round your answer to the nearest penny)

A video walkthrough of this example problem is available below.

Permit the states compare the amount of money earned from compounding against the amount you lot would earn from simple interest

Years Simple Involvement ($xv per month) 6% compounded monthly = 0.v% each month.
5 $3900 $4046.55
10 $4800 $5458.19
15 $5700 $7362.28
xx $6600 $9930.61
25 $7500 $13394.91
30 $8400 $18067.73
35 $9300 $24370.65

Line graph. Vertical axis: Account Balance ($), in increments of 5000 from 5000 to 25000. Horizontal axis: years, in increments of five, from 0 to 25. A blue dotted line shows a gradual increase over time, from roughly $2500 at year 0 to roughly $10000 at year 35. A pink dotted line shows a more dramatic increase, from roughly $2500 at year 0 to $25000 at year 35.

Equally you can see, over a long menstruation of time, compounding makes a large difference in the account rest. You lot may recognize this as the deviation between linear growth and exponential growth.

Evaluating exponents on the Desmos calculator

When we need to calculate something like 5^3 information technology is easy enough to simply multiply 5\cdot{5}\cdot{5}=125.  But when we need to calculate something similar 1.005^{240}, it would be very tedious to calculate this by multiplying 1.005 by itself 240 times!  So to make things easier, nosotros tin can harness the power of our scientific calculators.  In this class, nosotros are using the Desmos calculator.  If you just want to square a number, the primal is aii .  If you want to raise a number to another power, you employ the key ab on the main menu.

To evaluate 1.005^{240} we'd blazon i.005 ab 240 .  Endeavor it out – yous should get the answer in the figure below:

This figure shows how an exponent is evaluated in Desmos.

Well-nigh scientific calculators have a button for exponents.  If you are non using the Desmos calculator, it is typically either labeled like:

^ , y^x ,   or x^y .

Instance

You know that you lot will need $twoscore,000 for your child'southward education in eighteen years. If your account earns 4% compounded quarterly, how much would you demand to eolith now to reach your goal?

Solution:

In this case, we're looking for P0 .

r = 0.04 four%
yard = four four quarters in 1 year
North = 18 Since nosotros know the residual in eighteen years
Pxviii = $40,000 The amount we have in eighteen years

In this example, we're going to have to fix up the equation, and solve for P0 .

(7) \begin{align*}&40000={{P}_{0}}{{\left(1+\frac{0.04}{4}\right)}^{18\times4}}\\&40000={{P}_{0}}(2.0471)\\&{{P}_{0}}=\frac{40000}{2.0471}=\$19539.84 \\\end{align*}

So you would need to deposit $nineteen,539.84 now to have $40,000 in 18 years.

Rounding

If you are not inputting your unabridged formula into Desmos and choose to do it section past section, Information technology is important to be very careful near rounding when calculating things with exponents. In general, yous want to keep as many decimals during calculations equally you can. Exist sure to keep at least 3 meaning digits (numbers after any leading zeros). Rounding 0.00012345 to 0.000123 will usually give you a "shut plenty" reply, but keeping more digits is always ameliorate.

Instance

To see why not over-rounding is and then of import, if you choose non to enter your formula all at once into Desmos, suppose you were investing $chiliad at 5% interest compounded monthly for 30 years.

P0 = $1000 the initial deposit
r = 0.05 five%
k = 12 12 months in 1 year
North = 30 since we're looking for the corporeality after 30 years

If we first compute r/k, we find 0.05/12 = 0.00416666666667

Hither is the effect of rounding this to different values:

 

r/yard rounded to:

Gives P­30­ to be: Error
0.004 $4208.59 $259.15
0.0042 $4521.45 $53.71
0.00417 $4473.09 $5.35
0.004167 $4468.28 $0.54
0.0041667 $4467.lxxx $0.06
no rounding $4467.74

If you're working in a bank, of course you wouldn't round at all. For our purposes, the reply we got by rounding to 0.00417, iii significant digits, is shut enough – $5 off of $4500 isn't too bad. Certainly keeping that fourth decimal place wouldn't take injure.

View the following for a demonstration of this case.

Using your  Desmos calculator

In many cases, yous can avoid rounding completely by how you enter things in your figurer. For example, in the example above, nosotros needed to summate {{P}_{30}}=1000{{\left(1+\frac{0.05}{12}\right)}^{12\times30}}

Nosotros can rapidly calculate this on the Desmos Calculator by putting in the formula all at one time:

To enter this into the calculator, type in the following:

thousand * (1 + .05/12) ab (12 * thirty)

Annotation: ab is in the offset row, second cavalcade of the principal card in a higher place. At present you tin can circular your final answer to the nearest cent.

Attributions

This chapter contains material taken from Math in Society (on OpenTextBookStore) by David Lippman, and is used under a CC Attribution-Share Alike iii.0 Us (CC BY-SA 3.0 US) license.

This chapter contains material taken from of Math for the Liberal Arts (on Lumen Learning) by Lumen Learning, and is used under a CC Past: Attribution license.

Compound Interest Maze Answer Key,

Source: https://granite.pressbooks.pub/math502/chapter/simple-and-compound-interest/

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