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<p>Gerald has $\$20 \; 000$ to invest in Company $\mathrm{A}$ or Company $\mathrm{B}$.</p>
<p>Company $\mathrm{A}$ offers $3.8\%$ per annum simple interest.</p>
<p>Company $\mathrm{B}$ offers $3.5\%$ per annum compound interest compounded half-yearly.  Gerald would like to invest this amount of money for a period of $5$ years.</p>
<p>Which company should he invest in? Explain your answer.</p>

Gerald has $\$20 \; 000$ to invest in Company $\mathrm{A}$ or Company $\mathrm{B}$.

Company $\mathrm{A}$ offers $3.8\%$ per annum simple interest.

Company $\mathrm{B}$ offers $3.5\%$ per annum compound interest compounded half-yearly.  Gerald would like to invest this amount of money for a period of $5$ years.

Which company should he invest in? Explain your answer.

Compound Interest

Simple vs Compound Interest: Best Investment Choice

Standard Form and Compound Interest

<p>Faith deposited $\$12 \; 000$ in Bank $\mathrm{A}$ that pays $2\%$ per annum simple interest.  </p>
<p>She also deposited the same amount in Bank $\mathrm{B}$ that pays $1.95\%$ per annum compound interest compounded monthly.  </p>
<p>Find the total amount of money she will receive from the two banks at the end of $3$ years.</p>

Faith deposited $\$12 \; 000$ in Bank $\mathrm{A}$ that pays $2\%$ per annum simple interest.  

She also deposited the same amount in Bank $\mathrm{B}$ that pays $1.95\%$ per annum compound interest compounded monthly.  

Find the total amount of money she will receive from the two banks at the end of $3$ years.

Total Returns from Two Banks: Simple vs Compound Interest

<p>Eddy borrowed a sum of money at a compound interest of $5\%$ per annum.</p>
<p>After $3$ years, he owed a total amount of $\$2315.25.$</p>
<p>Calculate how much he borrowed. </p>

Eddy borrowed a sum of money at a compound interest of $5\%$ per annum.

After $3$ years, he owed a total amount of $\$2315.25.$

Finding the Loan Principal from Compound Interest and Final Amount

<p>A sum of $\$25000$ was loaned from a bank that charges $r\%$ interest per year, compounded yearly.  </p>
<p>If after $3$ years, an amount of $\$28121.60$ was paid up to settle the loan, calculate $r$.</p>

A sum of $\$25000$ was loaned from a bank that charges $r\%$ interest per year, compounded yearly.  

If after $3$ years, an amount of $\$28121.60$ was paid up to settle the loan, calculate $r$.

Finding the Interest Rate from a Compound Interest Loan

<p>Darius invested some money in a savings account for&nbsp;$2$years.</p>
<p>The rate of compound interest was fixed at $10\%$ per annum compounded half-yearly.</p>
<p>At the end of the $2$ years there was $\$9724.05$ in his account.</p>
<p>How much did he invest in the account?</p>

Darius invested some money in a savings account for $2$years.

The rate of compound interest was fixed at $10\%$ per annum compounded half-yearly.

At the end of the $2$ years there was $\$9724.05$ in his account.

Finding the Principal from Compound Interest (Half-Yearly)

<p>Calvin deposited $\$12 \; 000$ into each of the three banks, bank $\mathrm{A,B}$ and $\mathrm{C,}$ for&nbsp;$5$ years.</p>
<p>All three banks pay the same interest rate of $1.25\%$ per annum.  </p>
<p>Find the total amount of money he would collect if</p>
<ol class="i-list"><li><div class="list-item-paragraph">bank $\mathrm{A}$ pays simple interest.</div></li><li><div class="list-item-paragraph">bank $\mathrm{B}$ pays compound interest accumulated yearly.<br><br></div></li><li><div class="list-item-paragraph">bank $\mathrm{C}$ pays compound interest accumulated monthly.</div></li></ol>

Calvin deposited $\$12 \; 000$ into each of the three banks, bank $\mathrm{A,B}$ and $\mathrm{C,}$ for $5$ years.

All three banks pay the same interest rate of $1.25\%$ per annum.  

Find the total amount of money he would collect if

Comparing Simple and Compound Interest Across Three Banks

<p>Betty invested $\$13 \; 000$ in an investment scheme that paid $4\%$ compound interest per annum.  </p>
<p>The investment is compounded half-yearly.  </p>
<p>Calculate the value of the investment after three years.</p>

Betty invested $\$13 \; 000$ in an investment scheme that paid $4\%$ compound interest per annum.  

The investment is compounded half-yearly.  

Calculate the value of the investment after three years.

Compound Interest with Half-Yearly Compounding Over 3 Years

<p>Ariel invested $\$5000$ in an investment fund that pays a compounded interest rate of $2\%$ per annum.  </p>
<p>Find the amount of money in the fund after the end of $2$ years if this amount is compounded quarterly.</p>

Ariel invested $\$5000$ in an investment fund that pays a compounded interest rate of $2\%$ per annum.  

Find the amount of money in the fund after the end of $2$ years if this amount is compounded quarterly.

Compound Interest with Quarterly Compounding Over 2 Years

<p>A sum of $\$700$ is invested at simple interest and amounts to $\$952$ after $3$ years.</p>
<p>Calculate the interest rate per annum.</p>

A sum of $\$700$ is invested at simple interest and amounts to $\$952$ after $3$ years.

Calculate the Annual Interest Rate from a 3-Year Investment

<p>A sum of $\$25 \; 000$ is invested at an annual rate of $8\%$ simple interest.</p>
<p>After $9$ months, all the money is withdrew.</p>
<p>How much is this amount?</p>

A sum of $\$25 \; 000$ is invested at an annual rate of $8\%$ simple interest.

After $9$ months, all the money is withdrew.

Simple Interest on a $25,000 Investment Over 9 Months

<p>Sam invested a sum of money into a savings account at an interest rate $4\%$ per annum, compounded yearly.  <br></p>
<p>Find the sum of money he invested if there was $\$16\mathrm{}872.96$ in his account at the end of $3$ years.</p>

Sam invested a sum of money into a savings account at an interest rate $4\%$ per annum, compounded yearly.  


Find the sum of money he invested if there was $\$16\mathrm{}872.96$ in his account at the end of $3$ years.

Problem involving Compound Interest

<ol class="i-list"><li><div class="list-item-paragraph">Peter deposits $\$7680$  in a fund that pays compound interest of $3.35\%$ per annum,<br>compounded half-yearly.  <br></div><div class="list-item-paragraph">Calculate the total amount of money that Peter will have in the fund at the end of five years.</div></li><li><div class="list-item-paragraph">At the end of five years, Peter withdrew $\$6500$ from the fund and invested this<br>amount in a bank which offered simple interest rate of $4.8\%$ per annum. <br></div><div class="list-item-paragraph">Find the minimum number of full years he had to leave the money in a bank in order for it to be more than $\$12000$.</div></li></ol>

Problem involving Simple and Compound Interest

<p>John borrowed $\$8000$ from a bank which charges compound interest at a rate of $16\%$ per annum.</p>
<p>Calculate </p>
<ol class="i-list"><li><div class="list-item-paragraph">the total amount of money that he owed the bank at the end of $3$ years, </div></li><li><div class="list-item-paragraph">the interest that he paid to the bank at the end of $3$ years.</div></li></ol>

John borrowed $\$8000$ from a bank which charges compound interest at a rate of $16\%$ per annum.

<ol class="i-list"><li><div class="list-item-paragraph">Calculate the simple interest on $\$900$ at $5\%$ per annum for&nbsp;$2$ years.</div></li><li><div class="list-item-paragraph">Calculate the simple interest of $\$3000$ for&nbsp;$3$ years $5$ months at $4\%$ per annum.</div></li></ol>

Simple Interest

<p>Interest earned every year is not the same because it is calculated based on the new principal at the end of each year.</p>
<p style="text-align: center"><strong>$A=P{{(1+\frac{R}{100})}^{n}}$</strong></p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/public/question-images/pasted/1765862227391-n9k4rz341j.png" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>So, total amount will be$A$and compound interest will be $A-P$.</p>
<p style="margin-left: 1rem">If <strong>"compound interest"</strong>, <strong>"compounded"</strong> are NOT stated, then interest is taken to be <strong>simple.</strong></p>
<p style="margin-top: 1.5rem">Let’s see what will the interest be on <strong>$\$1500$ for&nbsp;$4$ years</strong> at given that <strong>$5\%$ per annum</strong> if it is compounded yearly, monthly, half-yearly and quarterly.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: left; --cell-justify: flex-start"></th><th style="text-align: center; --cell-justify: center"><p><strong>Compounded</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>No of periods</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Interest rate for each period</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Amount at the end of 4 years</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Interest earned</strong></p></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start; vertical-align: top"><p>$1$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>yearly</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>Period (yearly)</p><p>$n=4$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$5\%$</p><p>$R=5$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A = 1500\left(1 + \frac{5}{100}\right)^4}$</p><p>${A = 1823.26}$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A - P}$<br>= $\$323.26$</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start; vertical-align: top"><p>$2$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>monthly</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$4$ years $\times 12$months</p><p>$=48$months</p><p>$n=48$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$1$ year $\to 5\%$</p><p>$1$ month $\to \frac{5}{12}\%$</p><p>$R=\frac{5}{12}$</p><p>&nbsp;</p><p>&nbsp;</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A = 1500\left(1 + \frac{5/12}{100}\right)^{48}}$</p><p>$A$ = $\$1831.34$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A - P}$<br>= $\$331.34$</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start; vertical-align: top"><p>$3$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>half yearly</p><p>6 months</p><p><strong>bi-annually</strong></p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p><p>&nbsp;</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$4$years $\times 2$</p><p>=$8$ halves of a year</p><p>$n=8$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$1$ year $\to 5\%$</p><p>$\frac{1}{2}$ year $\to \frac{5}{2}\%$</p><p>$R=\frac{5}{2}$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A = 1500\left(1 + \frac{5/2}{100}\right)^{8}}$</p><p>$A$ = $\$1827.60$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A - P}$<br>= $\$327.60$</p></td></tr><tr><td style="text-align: left; --cell-justify: flex-start; vertical-align: top"><p>$4$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>quarterly</p><p>$1$ year $\to 4$quarters</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$4$years</p><p>$=4\times 4$ quarters</p><p>$=16$</p><p>$n=16$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>$1$ year $\to 5\%$</p><p>$\frac{1}{4}$ year $\to \frac{5}{4}\%$</p><p>$R=\frac{5}{4}$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A = 1500\left(1 + \frac{5/4}{100}\right)^{16}}$</p><p>$A$ = $\$1829.83$</p></td><td style="text-align: center; --cell-justify: center; vertical-align: top"><p>${A - P}$<br>= $\$329.83$</p></td></tr></tbody></table></div></div>

Interest earned every year is not the same because it is calculated based on the new principal at the end of each year.

So, total amount will be$A$and compound interest will be $A-P$.

If **"compound interest"**, **"compounded"** are NOT stated, then interest is taken to be **simple.**

Let’s see what will the interest be on **$\$1500$ for $4$ years** at given that **$5\%$ per annum** if it is compounded yearly, monthly, half-yearly and quarterly.

<p><strong>Interest rate for each period</strong></p>

<p><strong>Amount at the end of 4 years</strong></p>

<p>Interest earned every year is the same because it is calculated based on the original principal.</p>
<p style="text-align: center">${I = \frac{PRT}{100}}$</p>
<p>${I}$: interest, ${P}$: principal, $R\%$: interest rate per <strong>year,</strong> ${T}$: number of <strong>years</strong></p>
<p>So, total amount will be ${P + I}$.</p>

Interest earned every year is the same because it is calculated based on the original principal.

${I}$: interest, ${P}$: principal, $R\%$: interest rate per **year,** ${T}$: number of **years**

Recall what is a Simple Interest

<p>The mass of one atom of nitrogen is $2.33\times {{10}^{-11}}$ picograms. ($1$ pico $={{10}^{-12}}$)</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express this mass in grams, giving your answer in standard form.</div></li><li><div class="list-item-paragraph">A molecule of nitrogen dioxide ${\left(\mathrm{NO}_2\right)}$ contains one atom of nitrogen and two atoms of oxygen.
</div><div class="list-item-paragraph">Given that the mass of one atom of oxygen is $2.66\times {{10}^{-23}}$grams, find the mass of one molecule of nitrogen dioxide.
</div><div class="list-item-paragraph">Give your answer in grams in standard form. </div></li></ol>

The mass of one atom of nitrogen is $2.33\times {{10}^{-11}}$ picograms. ($1$ pico $={{10}^{-12}}$)

Standard Form

Standard Form in Atomic Mass Calculations

<ol class="i-list"><li><div class="list-item-paragraph">The mass of the Sun is $1.99\times {{10}^{27}}$ tonnes.
</div><div class="list-item-paragraph">Write this mass, in kilograms, giving your answer in standard form. ($1$ tonne $=1000$kg)</div></li><li><div class="list-item-paragraph">The radius of a micro-organism is $0.000\,054\,65$cm.
</div><div class="list-item-paragraph">Express $0.000\,054\,65$cm in standard form, giving your answer correct to $3$ significant figures.</div></li></ol>

Standard Form in Measurement Problems

<p>The Earth is approximately $150$ gigametres away from the Sun.</p>
<ol class="i-list"><li><div class="list-item-paragraph">Express $150$ gigametres in kilometres, giving your answer in standard form</div></li><li><div class="list-item-paragraph">Light take $3\frac{1}{3}$ nanoseconds to travel $1$ metre. Calculate the time light takes to travel from the Sun to the Earth, giving your answer in seconds, in standard form.</div></li></ol>

The Earth is approximately $150$ gigametres away from the Sun.

Standard Form in Astronomy

<p>The storage capacity of an external hard disk is $3$ terabytes (TB). ($1$ tera $={{10}^{12}}$)<br></p>
<p>If the size of a song clip is $5.5$ megabytes (MB), ($1$ mega $={{10}^{6}}$) how many songs can be stored in the external hard disk?<br></p>
<p>Give your answer in standard form, correct to $3$ significant figures.</p>

The storage capacity of an external hard disk is $3$ terabytes (TB). ($1$ tera $={{10}^{12}}$)


If the size of a song clip is $5.5$ megabytes (MB), ($1$ mega $={{10}^{6}}$) how many songs can be stored in the external hard disk?


Give your answer in standard form, correct to $3$ significant figures.

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