{"id":2461,"date":"2023-05-10T16:57:10","date_gmt":"2023-05-10T16:57:10","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=2461"},"modified":"2025-08-27T00:36:35","modified_gmt":"2025-08-27T00:36:35","slug":"simple-and-compound-interest-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/simple-and-compound-interest-learn-it-4\/","title":{"raw":"Simple and Compound Interest: Learn It 4","rendered":"Simple and Compound Interest: Learn It 4"},"content":{"raw":"

Compound Interest vs Simple Interest<\/h2>\r\n

Let us compare the amount of money earned from compounding against the amount you would earn from simple interest<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Years<\/td>\r\nSimple Interest\r\n\r\n

[latex]$15[\/latex] per month<\/p>\r\n<\/td>\r\n

[latex]6\\%[\/latex] Annual Rate Compounded Monthly\r\n\r\n

[latex]0.5\\%[\/latex] each month<\/p>\r\n<\/td>\r\n<\/tr>\r\n

[latex]0[\/latex]<\/td>\r\n[latex]$3,000[\/latex]<\/td>\r\n[latex]$3,000[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]5[\/latex]<\/td>\r\n[latex]$3,900[\/latex]<\/td>\r\n[latex]$4,046.55[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]10[\/latex]<\/td>\r\n[latex]$4,800[\/latex]<\/td>\r\n[latex]$5,458.19[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]15[\/latex]<\/td>\r\n[latex]$5,700[\/latex]<\/td>\r\n[latex]$7,362.28[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]20[\/latex]<\/td>\r\n[latex]$6,600[\/latex]<\/td>\r\n[latex]$9,930.61[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]25[\/latex]<\/td>\r\n[latex]$7,500[\/latex]<\/td>\r\n[latex]$13,394.91[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]30[\/latex]<\/td>\r\n[latex]$8,400[\/latex]<\/td>\r\n[latex]$18,067.73[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]35[\/latex]<\/td>\r\n[latex]$9,300[\/latex]<\/td>\r\n[latex]$24,370.65[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

\r\n[caption id=\"attachment_6218\" align=\"aligncenter\" width=\"500\"]\"Line Figure 1. Compounding can cause exponential growth in the account balance over time[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.<\/p>\r\n

\r\n

Linear growth vs exponential growth<\/strong><\/p>\r\n

Recall that linear growth<\/strong> increases at a constant rate. A graph of linear growth will describe a straight line between any two points on the graph. The graph changes by the same additive amount per unit of input.<\/p>\r\n

For example, a bank account that grows at [latex]$5[\/latex] per year experiences linear growth.<\/p>\r\n

Exponential growth<\/strong> describes a quantity growing at a rate proportional to itself for each unit of input. The graph changes by a multiple of its current value per unit of input. The graph will describe a quickly rising curve.<\/p>\r\n

For example, an account that grows at [latex]2\\%[\/latex] per year experiences exponential growth.<\/p>\r\n<\/section>\r\n

Using a Calculator to Solve Compound Interest<\/h2>\r\n

Utilizing a calculator to compute compound interest can streamline the task considerably, but it's important to approach this tool with a few critical pointers in mind. These include understanding the functions on your calculator, the importance of precision in your inputs, and the nuances of rounding numbers.\u00a0<\/p>\r\n

\r\n

How to: Evaluate exponents on the calculator<\/strong><\/p>\r\n

When we need to calculate something like [latex]5^3[\/latex] it is easy enough to just multiply [latex]5\\cdot{5}\\cdot{5}=125[\/latex].\u00a0 But when we need to calculate something like [latex]1.005^{240}[\/latex], it would be very tedious to calculate this by multiplying [latex]1.005[\/latex] by itself [latex]240[\/latex] times!\u00a0 So to make things easier, we can harness the power of our scientific calculators.<\/p>\r\n

Most scientific calculators have a button for exponents.\u00a0 It is typically either labeled like:<\/p>\r\n

^ , \u00a0 [latex]y^x[\/latex] ,\u00a0\u00a0 or [latex]x^y[\/latex] .<\/p>\r\n

To evaluate [latex]1.005^{240}[\/latex] we'd type [latex]1.005[\/latex] ^ [latex]240[\/latex], or [latex]1.005 \\space{y^{x}}\\space 240[\/latex].\u00a0 Try it out - you should get something around [latex]3.3102044758[\/latex].<\/p>\r\n<\/section>\r\n

\r\n

It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding [latex]0.00012345[\/latex] to [latex]0.000123[\/latex] will usually give you a \u201cclose enough\u201d answer, but keeping more digits is always better.<\/p>\r\n<\/section>\r\n

To see why not over-rounding is so important, suppose you were investing [latex]$1000[\/latex] at [latex]5\\%[\/latex] interest compounded monthly for [latex]30[\/latex] years.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]P_0 = $1000[\/latex]<\/td>\r\nthe initial deposit<\/td>\r\n<\/tr>\r\n
[latex]r = 0.05[\/latex]<\/td>\r\n[latex]5\\%[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]n = 12[\/latex]<\/td>\r\n[latex]12[\/latex] months in [latex]1[\/latex] year<\/td>\r\n<\/tr>\r\n
[latex]t = 30[\/latex]<\/td>\r\nsince we\u2019re looking for the amount after [latex]30[\/latex] years<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

If we first compute [latex]\\frac{r}{n}[\/latex], we find [latex]\\frac{0.05}{12} = 0.00416666666667[\/latex].<\/p>\r\n

Here is the effect of rounding this to different values:<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
[latex]\\frac{r}{n}[\/latex] rounded to:<\/strong><\/td>\r\nGives [latex]P_{30}[\/latex] to be:<\/strong><\/td>\r\nError<\/strong><\/td>\r\n<\/tr>\r\n
[latex]0.004[\/latex]<\/td>\r\n[latex]$4208.59[\/latex]<\/td>\r\n[latex]$259.15[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.0042[\/latex]<\/td>\r\n[latex]$4521.45[\/latex]<\/td>\r\n[latex]$53.71[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.00417[\/latex]<\/td>\r\n[latex]$4473.09[\/latex]<\/td>\r\n[latex]$5.35[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.004167[\/latex]<\/td>\r\n[latex]$4468.28[\/latex]<\/td>\r\n[latex]$0.54[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]0.0041667[\/latex]<\/td>\r\n[latex]$4467.80[\/latex]<\/td>\r\n[latex]$0.06[\/latex]<\/td>\r\n<\/tr>\r\n
no rounding<\/td>\r\n[latex]$4467.74[\/latex]<\/td>\r\n\u00a0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

If you\u2019re working in a bank, of course you wouldn\u2019t round at all. For our purposes, the answer we got by rounding to [latex]0.00417[\/latex], three significant digits, is close enough - [latex]$5[\/latex] off isn\u2019t too bad. Certainly keeping that fourth decimal place wouldn\u2019t have hurt.<\/p>\r\n<\/section>\r\n

In many cases, you can avoid rounding completely by how you enter things in your calculator.<\/p>\r\n

\r\n

For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{12\\times30}}[\/latex]<\/p>\r\n

We can quickly calculate [latex]12\u00d730 = 360[\/latex], giving [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{360}}[\/latex].<\/p>\r\n

Now we can use the calculator.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Type this<\/strong><\/td>\r\nCalculator shows<\/strong><\/td>\r\n<\/tr>\r\n
[latex]0.05 \u00f7 12 =[\/latex] .<\/td>\r\n[latex]0.00416666666667[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]+ 1 =[\/latex] .<\/td>\r\n[latex]1.00416666666667[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]y^x 360 =[\/latex] .<\/td>\r\n[latex]4.46774431400613[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]\u00d7 1000 =[\/latex] .<\/td>\r\n[latex]4467.74431400613[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

The previous steps were assuming you have a \u201cone operation at a time\u201d calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:<\/p>\r\n

[latex]1000 \u00d7 \u00a0( 1 + 0.05 \u00f7 12 ) y^x 360 =[\/latex]<\/center><\/section>","rendered":"

Compound Interest vs Simple Interest<\/h2>\n

Let us compare the amount of money earned from compounding against the amount you would earn from simple interest<\/p>\n\n\n\n\n\n\n\n\n\n\n\n
Years<\/td>\nSimple Interest<\/p>\n

[latex]$15[\/latex] per month<\/p>\n<\/td>\n

[latex]6\\%[\/latex] Annual Rate Compounded Monthly<\/p>\n

[latex]0.5\\%[\/latex] each month<\/p>\n<\/td>\n<\/tr>\n

[latex]0[\/latex]<\/td>\n[latex]$3,000[\/latex]<\/td>\n[latex]$3,000[\/latex]<\/td>\n<\/tr>\n
[latex]5[\/latex]<\/td>\n[latex]$3,900[\/latex]<\/td>\n[latex]$4,046.55[\/latex]<\/td>\n<\/tr>\n
[latex]10[\/latex]<\/td>\n[latex]$4,800[\/latex]<\/td>\n[latex]$5,458.19[\/latex]<\/td>\n<\/tr>\n
[latex]15[\/latex]<\/td>\n[latex]$5,700[\/latex]<\/td>\n[latex]$7,362.28[\/latex]<\/td>\n<\/tr>\n
[latex]20[\/latex]<\/td>\n[latex]$6,600[\/latex]<\/td>\n[latex]$9,930.61[\/latex]<\/td>\n<\/tr>\n
[latex]25[\/latex]<\/td>\n[latex]$7,500[\/latex]<\/td>\n[latex]$13,394.91[\/latex]<\/td>\n<\/tr>\n
[latex]30[\/latex]<\/td>\n[latex]$8,400[\/latex]<\/td>\n[latex]$18,067.73[\/latex]<\/td>\n<\/tr>\n
[latex]35[\/latex]<\/td>\n[latex]$9,300[\/latex]<\/td>\n[latex]$24,370.65[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

\n
\"Line
Figure 1. Compounding can cause exponential growth in the account balance over time<\/figcaption><\/figure>\n<\/div>\n

 <\/p>\n

As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth.<\/p>\n

\n

Linear growth vs exponential growth<\/strong><\/p>\n

Recall that linear growth<\/strong> increases at a constant rate. A graph of linear growth will describe a straight line between any two points on the graph. The graph changes by the same additive amount per unit of input.<\/p>\n

For example, a bank account that grows at [latex]$5[\/latex] per year experiences linear growth.<\/p>\n

Exponential growth<\/strong> describes a quantity growing at a rate proportional to itself for each unit of input. The graph changes by a multiple of its current value per unit of input. The graph will describe a quickly rising curve.<\/p>\n

For example, an account that grows at [latex]2\\%[\/latex] per year experiences exponential growth.<\/p>\n<\/section>\n

Using a Calculator to Solve Compound Interest<\/h2>\n

Utilizing a calculator to compute compound interest can streamline the task considerably, but it’s important to approach this tool with a few critical pointers in mind. These include understanding the functions on your calculator, the importance of precision in your inputs, and the nuances of rounding numbers.\u00a0<\/p>\n

\n

How to: Evaluate exponents on the calculator<\/strong><\/p>\n

When we need to calculate something like [latex]5^3[\/latex] it is easy enough to just multiply [latex]5\\cdot{5}\\cdot{5}=125[\/latex].\u00a0 But when we need to calculate something like [latex]1.005^{240}[\/latex], it would be very tedious to calculate this by multiplying [latex]1.005[\/latex] by itself [latex]240[\/latex] times!\u00a0 So to make things easier, we can harness the power of our scientific calculators.<\/p>\n

Most scientific calculators have a button for exponents.\u00a0 It is typically either labeled like:<\/p>\n

^ , \u00a0 [latex]y^x[\/latex] ,\u00a0\u00a0 or [latex]x^y[\/latex] .<\/p>\n

To evaluate [latex]1.005^{240}[\/latex] we’d type [latex]1.005[\/latex] ^ [latex]240[\/latex], or [latex]1.005 \\space{y^{x}}\\space 240[\/latex].\u00a0 Try it out – you should get something around [latex]3.3102044758[\/latex].<\/p>\n<\/section>\n

\n

It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to keep at least 3 significant digits (numbers after any leading zeros). Rounding [latex]0.00012345[\/latex] to [latex]0.000123[\/latex] will usually give you a \u201cclose enough\u201d answer, but keeping more digits is always better.<\/p>\n<\/section>\n

To see why not over-rounding is so important, suppose you were investing [latex]$1000[\/latex] at [latex]5\\%[\/latex] interest compounded monthly for [latex]30[\/latex] years.<\/p>\n\n\n\n\n\n\n
[latex]P_0 = $1000[\/latex]<\/td>\nthe initial deposit<\/td>\n<\/tr>\n
[latex]r = 0.05[\/latex]<\/td>\n[latex]5\\%[\/latex]<\/td>\n<\/tr>\n
[latex]n = 12[\/latex]<\/td>\n[latex]12[\/latex] months in [latex]1[\/latex] year<\/td>\n<\/tr>\n
[latex]t = 30[\/latex]<\/td>\nsince we\u2019re looking for the amount after [latex]30[\/latex] years<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

If we first compute [latex]\\frac{r}{n}[\/latex], we find [latex]\\frac{0.05}{12} = 0.00416666666667[\/latex].<\/p>\n

Here is the effect of rounding this to different values:<\/p>\n\n\n\n\n\n\n\n\n\n
[latex]\\frac{r}{n}[\/latex] rounded to:<\/strong><\/td>\nGives [latex]P_{30}[\/latex] to be:<\/strong><\/td>\nError<\/strong><\/td>\n<\/tr>\n
[latex]0.004[\/latex]<\/td>\n[latex]$4208.59[\/latex]<\/td>\n[latex]$259.15[\/latex]<\/td>\n<\/tr>\n
[latex]0.0042[\/latex]<\/td>\n[latex]$4521.45[\/latex]<\/td>\n[latex]$53.71[\/latex]<\/td>\n<\/tr>\n
[latex]0.00417[\/latex]<\/td>\n[latex]$4473.09[\/latex]<\/td>\n[latex]$5.35[\/latex]<\/td>\n<\/tr>\n
[latex]0.004167[\/latex]<\/td>\n[latex]$4468.28[\/latex]<\/td>\n[latex]$0.54[\/latex]<\/td>\n<\/tr>\n
[latex]0.0041667[\/latex]<\/td>\n[latex]$4467.80[\/latex]<\/td>\n[latex]$0.06[\/latex]<\/td>\n<\/tr>\n
no rounding<\/td>\n[latex]$4467.74[\/latex]<\/td>\n\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

If you\u2019re working in a bank, of course you wouldn\u2019t round at all. For our purposes, the answer we got by rounding to [latex]0.00417[\/latex], three significant digits, is close enough – [latex]$5[\/latex] off isn\u2019t too bad. Certainly keeping that fourth decimal place wouldn\u2019t have hurt.<\/p>\n<\/section>\n

In many cases, you can avoid rounding completely by how you enter things in your calculator.<\/p>\n

\n

For example, in the example above, we needed to calculate [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{12\\times30}}[\/latex]<\/p>\n

We can quickly calculate [latex]12\u00d730 = 360[\/latex], giving [latex]{{P}_{30}}=1000{{\\left(1+\\frac{0.05}{12}\\right)}^{360}}[\/latex].<\/p>\n

Now we can use the calculator.<\/p>\n\n\n\n\n\n\n\n
Type this<\/strong><\/td>\nCalculator shows<\/strong><\/td>\n<\/tr>\n
[latex]0.05 \u00f7 12 =[\/latex] .<\/td>\n[latex]0.00416666666667[\/latex]<\/td>\n<\/tr>\n
[latex]+ 1 =[\/latex] .<\/td>\n[latex]1.00416666666667[\/latex]<\/td>\n<\/tr>\n
[latex]y^x 360 =[\/latex] .<\/td>\n[latex]4.46774431400613[\/latex]<\/td>\n<\/tr>\n
[latex]\u00d7 1000 =[\/latex] .<\/td>\n[latex]4467.74431400613[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

 <\/p>\n

The previous steps were assuming you have a \u201cone operation at a time\u201d calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter:<\/p>\n

[latex]1000 \u00d7 \u00a0( 1 + 0.05 \u00f7 12 ) y^x 360 =[\/latex]<\/div>\n<\/section>\n","protected":false},"author":15,"menu_order":22,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":89,"module-header":"learn_it","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2461"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/users\/15"}],"version-history":[{"count":41,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2461\/revisions"}],"predecessor-version":[{"id":15746,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2461\/revisions\/15746"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/parts\/89"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapters\/2461\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/media?parent=2461"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/pressbooks\/v2\/chapter-type?post=2461"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/contributor?post=2461"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/wp-json\/wp\/v2\/license?post=2461"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}