Use technology to find interest rates or compounding periods

Ontario Grade 11 Mathematics

Exponential Functions

Grade 11 Mathematics — B3.4

Compound-interest questions sometimes give you the starting amount, the final amount, and the time, but leave out the rate or how often interest is added. Technology can help find the missing value. You will enter a model, try or graph possible values, and check that the result fits the situation. The goal is to use technology to solve for an unknown, not to use a later-course algebra method.

What you will learn

Identify the known and unknown values in a compound-interest problem.
Use a graphing tool, spreadsheet, or calculator to estimate an unknown interest rate.
Use technology to test possible compounding periods when the period is unknown.
Check whether a technology result makes sense in the original situation.

1. Prerequisites: what the quantities mean

A principal is the starting amount of money. The amount is the money after interest has been added. Interest is money earned on savings or charged on a loan. A rate is a percent written as a decimal when used in a calculation: for example, a rate of 5% is

0.05

.

Compounding means adding interest to the account at set times. A compounding period is one such time interval. If interest is compounded monthly, there are

12

periods in one year. If it is compounded quarterly, there are

4

periods in one year.

The variable

t

represents time in years. The variable

n

represents the number of compounding periods per year. The variable

i

represents the stated annual interest rate as a decimal. In the model below, interest is compounded at the end of each period.

A=P\left(1+\frac{i}{n}\right)^{nt}

Convert a percent rate to a decimal before using it in the model.
Match the time units: $t$ is measured in years, while $n$ counts periods per year.
Use the words in the question to identify which value is unknown.

2. Plain language and the technology method

In the model,

P

is the principal and

A

is the amount after

t

years. The fraction

\frac{i}{n}

is the rate for one compounding period. The exponent

nt

counts how many periods occur over the full time.

If the rate is missing, enter the known values and leave the rate as a variable. Use a graphing calculator or graphing app to graph the amount model. Then find where that graph reaches the given final amount. The corresponding input is the rate.

Some tools let you enter an equation and find where two graphs intersect. You can graph the amount model and a horizontal line at the known final amount. Their intersection gives the rate and amount together. A spreadsheet can also test a list of candidate rates and calculate the amount for each one.

If the compounding period is missing, the unknown is

n

. Test reasonable whole-number values for

n

in the model using a calculator or spreadsheet. Compare each resulting amount with the given amount. A match identifies the period count that fits the information. If the context does not specify possible periods, report what the technology shows and state the assumption used.

A=P\left(1+\frac{i}{n}\right)^{nt}

Choose an input variable that represents the missing quantity.
Use the known final amount as the target for a graph, intersection, or table.
For a compounding-period search, test possible period counts and compare the calculated amounts.

3. Read and check a technology result

A graph gives an estimate, not always an exact decimal. Zoom in near the intersection or use a table with smaller steps to improve the estimate. A spreadsheet is useful when you want to compare several candidate values side by side.

Use a sensible viewing window. The horizontal axis should show the possible rates or periods. The vertical axis should include the target amount. If the graph does not show an intersection, adjust the window or check that the model and input values were entered correctly.

After finding a value, substitute it back into the original model. The calculated amount should be close to the given amount. Small differences may come from rounding. Also check that the answer is reasonable: a higher positive interest rate should produce a larger amount when the other values stay the same.

Set a useful graph window and refine an estimate near the target.
Round only after the technology has found a suitable value.
Substitute the result into the model to check it.

4. Application: rate or compounding period

For savings, the unknown may be the rate offered by an account. For a loan, it may be the rate implied by a starting balance and a later balance. The same model applies when the stated rate and compounding schedule are interpreted consistently.

For a missing rate, graph the model as a function of the rate and compare it with the target amount. For a missing period count, evaluate the model for possible values of

n

. Because a period count is a count of intervals, test whole-number values that make sense in the situation.

Technology does not decide whether your setup matches the question. Check that the principal, time, rate, and compounding schedule have the right meanings and units before trusting the displayed answer.

The model can be used to find either a missing rate or a missing compounding-period count.
The calculator result is meaningful only when the inputs match the question.
State the units and the meaning of the result.

Comparing candidate annual rates

Candidate annual rate	Modelled amount after 3 years, compounded monthly
4%	About CAD 1352.72
5%	About CAD 1393.80
6%	About CAD 1436.03

Worked example

Find an annual rate with a graphing tool

An account starts with CAD 1200. After 3 years of monthly compounding, it contains about CAD 1393.80. Use technology to estimate the annual interest rate.

Set up the model
The principal is $1200$ , the amount is about $1393.80$ , the time is $3$ years, and monthly compounding means $n=12$ . Enter the model with the rate as the input variable.
$A=1200\left(1+\frac{i}{12}\right)^{36}$
Graph against the target
Graph the amount model and the horizontal target line $A=1393.80$ . Use an input window that includes rates near $0.05$ . Find the intersection. Its horizontal coordinate is the estimated rate.
$A=1393.80$
Read and interpret
The intersection is near $i=0.05$ . Convert the decimal to a percent by multiplying by $100$ . The estimated stated annual rate is therefore 5%.
$0.05\times100\%=5\%$
Check the result
Put the estimated rate back into the model. The amount is about CAD 1393.80, matching the stated final amount to the nearest cent. Small display differences may be caused by rounding.
$1200\left(1+\frac{0.05}{12}\right)^{36}\approx1393.80$

Answer: The estimated annual interest rate is 5%, compounded monthly.

Check: The model with a 5% annual rate gives an amount close to CAD 1393.80 after 3 years.

Common mistakes and how to avoid them

Entering

5

as the rate instead of

0.05

.

Correction: Convert a percent to a decimal before entering it. For example, 5% is

0.05

.

Using

n=1

for monthly compounding.

Correction: Monthly compounding has

12

periods per year, so use

n=12

.

Reading the vertical coordinate of a graph intersection as the rate.

Correction: When the horizontal axis is the rate and the vertical axis is the amount, the horizontal coordinate gives the rate.

Accepting a displayed value without checking it.

Correction: Substitute the value into the original model and compare the calculated amount with the target.

Lesson summary

Use the compound-interest model to connect principal, rate, compounding schedule, time, and amount.
When the rate is unknown, graph the model against the target amount or test candidate rates with technology.
When the compounding-period count is unknown, test suitable values and compare the calculated amounts.
Check units, interpret the result, and verify it in the original model.

Check your understanding

Question 1

A question says interest is compounded quarterly. What value should be used for

n

in the model?

$4$
$12$
$3$
$1$

Show answer and explanation

4

Quarterly means four compounding periods in each year, so

n=4

.

Question 2

A graph has the annual rate on the horizontal axis and the amount on the vertical axis. Which coordinate of the intersection with the target-amount line gives the estimated rate?

The horizontal coordinate
The vertical coordinate
The sum of the coordinates
The point where the graph crosses the vertical axis

Show answer and explanation

The horizontal coordinate

The horizontal axis represents the rate, so the horizontal coordinate of the intersection is the estimated rate.

Question 3

A technology result suggests an annual rate of

0.04

. What is this rate as a percent?

4%
0.4%
40%
0.04%

Show answer and explanation

4%

Multiply the decimal by

100

to convert it to a percent:

0.04

is 4%.

Key terms

Principal: The starting amount of money in an account or loan.
Interest: Money earned on savings or charged on a loan.
Interest rate: The percent used to determine how much interest is added or charged.
Compounding: Adding interest to the amount at set times.
Compounding period: One interval at the end of which interest is added.
Intersection: A point where two graphs meet.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 11 Mathematics (MCF3M), expectation B3.4. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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