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The simplest way to describe how a car's power to weight ratio affects performance could be summed up in one sentence. The heavier the load, the more power it takes to move it. Ok, that might be over-simplifying it, but you get the picture. But since power to weight ratio is such a (no pun intended,) driving force for so many performance thirsty gearheads when it comes to choosing a car, we thought we'd dig a little deeper into the subject.

Calculating Power to Weight Ratio

If you're actually one of those performance thirsty gearheads, you likely already know this. But a car's power to weight ratio is the amount of power a vehicle has relative to its weight. Translation - divide the horsepower of the engine by the weight of the car and that's your power to weight ratio. For example - the Dodge Viper has a 450 hp engine to accelerate 3,320 lbs. of weight, making its power to weight ratio - .135 hp per 10 lbs.

When Performance Matters

What does that really mean? Well, for starters, it means it can go from 0 - 60 mph in 4.1 seconds. Not too shabby, right? Especially when you compare it to your mom's Ford Escort. With a 110 hp engine at 2,470 lbs. and a power to weight ratio of .045 per 10 lbs., mom's car takes 10.9 seconds to do the same.

Does that mean the car with the most massive weight and most powerful engine will perform better? It depends on how you define "performance." If you're truly a performance driven car fanatic, you probably understand that speed and acceleration matter - a lot. More horsepower = more speed. We got it. But how a car handles in different road conditions is pretty important if you're talking overall performance. A tank with a power to weight ratio equal to that of the Dodge Viper mentioned above, is not likely to take corners as smoothly. And talk about wear and tear on brakes, tires (or in this case, tracks) and other vehicle components. But in terms of sheer acceleration ability and speed - the higher your power to weight ratio, the better. That means the ideal performance car is lighter in curb weight but jacked to the hilt in horsepower. Check out the top 10 beastie cars across the globe with the best power to weight ratios out there.

1. Fahlke Larea GT1 S12 - 0.58 hp/pound (0-60 mph in 0 seconds)

2. Hennessey Venom F5 - 0.55 hp/pound (0-186 mph in less than 10 seconds)

3. T1 - 0.50 hp/pound (60 mph in 5 seconds)

4. Ultima Evolution - 0.49 hp/pound (0-60 mph in 3 seconds)

5. Ruf 9ff GT9 Vmax - 0.47 hp/pound (0-60 mph in 0 seconds)

6. Saleen S7 LM - 0.46 hp/pound (0-60 mph in 3.3 seconds)

7. Koenigsegg One:1 - 0.45 hp/pound (0-250 mph in 20 seconds)

8. Ariel Atom 500 - 0.41 hp/pound (0-60 mph in 3 seconds)

9. Lotec Sirius - 0.40 hp/pound (0-60 mph in 6 seconds)

10. Ferrari LaFerrari - 0.34 hp/pound (0-60 mph in 7 seconds)

Most of the cars that made the list come at a steep price tag but if you're talking dream cars for power to weight ratio, why not dream big? And if your dreams are all about horsepower beneath the hood, maybe you should consider a career in automotive technology. You can start by checking out J-Tech Institute. With programs in both Automotive and Diesel Technology, J-Tech is helping drive the future of the automotive industry forward. Contact us today to learn more or to schedule a campus tour of our 168,000 square foot training facility in Jacksonville, FL.

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Power to Weight Ratio and How it Affects Performance - J-Tech Institute

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The simplest way to describe how a car's power to weight ratio affects performance could be summed up in one sentence. The heavier the load, the more power it takes to move it. Ok, that might be over-simplifying it, but you get the picture.

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https://en.wikipedia.org/wiki/Tessellation

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A Tessellation (or Tiling) is when we cover a surface with a pattern of flat shapes so that there are no overlaps or gaps.

Examples:

Rectangles

Octagons and Squares

Different Pentagons

Regular Tessellations

A regular tessellation is a pattern made by repeating a regular polygon.
There are only 3 regular tessellations:
Look at a Vertex ...

A vertex is just a "corner point".

What shapes meet here?

For a regular tessellation, the pattern is identical at each vertex!

Semi-regular Tessellations

A semi-regular tessellation is made of two or more regular polygons. The pattern at each vertex must be the same!
There are only 8 semi-regular tessellations:

Question 1: For all the tessellations above, is the pattern the same at each vertex?

Question 2: One of those patterns becomes different when we make a mirror-image of it ... which one?

Other Tessellations

There are also "demiregular" tessellations, but mathematicians disagree on what they actually are!

And some people allow curved shapes (not just polygons) so we can have tessellations like these:

Curvy Shapes

Tessellation Artist

All these images were made using Tessellation Artist, with some color added using a paint program.

You can try it too - maybe you will invent a new tessellation!

World Tessellation Day

World Tessellation Day is June 17. Celebrate!

5482, 5483, 5484, 5485, 5486, 853, 854, 3368, 3369, 5487

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Learn how a pattern of shapes that fit perfectly together make a tessellation (tiling)

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Application of Geometry In Architecture 

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Application of Geometry In Architecture 

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Architecture could be said to begin with geometry. Since earliest times, builders relied on imitating natural forms—like the circular Stonehenge in Britain—and then applied mathematical principles to standardize and replicate the forms.

The Beginnings

The Greek mathematician Euclid of Alexandria is considered the first to write down all the rules related to geometry in 300 BCE. Later, in about 20 BCE, the ancient Roman architect Marcus Vitruvius penned more rules in his De Architectura, or Ten Books on Architecture. Vitruvius is responsible for all the geometry in today's built environment—at least he was the first to write down the proportions for how structures should be constructed.

Renaissance Popularity

It wasn't until centuries later, during the Renaissance, that interest in Vitruvius became popular. Cesare Cesariano (1475-1543) is considered the first architect to translate Vitruvius' work from Latin into Italian in about 1520 CE. Decades earlier, however, the Italian Renaissance artist and architect Leonardo da Vinci (1452-1519) sketched out the "Vitruvian Man" in his notebook, making da Vinci's the iconic image imprinted onto our consciousness.

The images of the Vitruvian Man are inspired by the works and writings of Vitruvius. The "man" portrayed represents the human being. The circles, squares, and ellipses that surround the figures are Vitruvian calculations of man's physical geometry. Vitruvius was the first to write his observations about the human body—that the symmetry of two eyes, two arms, two legs, and two breasts must be an inspiration of the gods.

Models of Proportion and Symmetry

Vitruvius believed that builders should always use precise ratios when constructing temples. "For without symmetry and proportion no temple can have a regular plan," Vitruvius wrote.

The symmetry and proportion in design that Vitruvius recommended in De Architectura were modeled after the human body. Vitruvius observed that all human beings are shaped according to a ratio that is astonishingly precise and uniform. For example, Vitruvius found that the human face equals one-tenth of the total body height. The foot equals one-sixth of the total body height. And so on.

Scientists and philosophers later discovered that the same ratio Vitruvius saw in the human body—1 to phi (Φ) or 1.618—exists in every part of nature, from swimming fish to swirling planets. Sometimes called a "golden ratio" or "divine ratio," the Vitruvian "divine proportion" has been called the building block of all life and the hidden code in architecture.

Geometry in Our Environment

"Sacred geometry," or "spiritual geometry," is the belief that numbers and patterns such as the divine ratio have sacred significance. Many mystical and spiritual practices begin with a fundamental belief in sacred geometry. Architects and designers may draw upon concepts of sacred geometry when they choose particular geometric forms to create pleasing, soul-satisfying spaces.

The following examples of geometry in the environment frequently influence architectural design.

The Body
When studied under the microscope, living cells reveal a highly ordered system of shapes and patterns. From the double helix shape of your DNA to the cornea of your eye, every part of your body follows the same predictable patterns.

Gardens
The jigsaw puzzle of life is made up of recurring shapes and numbers. Leaves, flowers, seeds, and other living things share the same spiral shapes. Pine cones and pineapples, in particular, are composed of mathematical spirals. Honeybees and other insects live structured lives that mimic these patterns. When we create a floral arrangement or walk through a labyrinth we celebrate nature’s innate forms.

Stones
Nature’s archetypes are reflected in the crystalline forms of gems and stones. Amazingly, the patterns found in your diamond engagement ring may resemble the formation of snowflakes and the shape of your own cells. The practice of stacking stones is a primitive, spiritual activity.

The Sea
Similar shapes and numbers are found beneath the sea, from the swirl of a nautilus shell to the movement of the tides. Surface waves themselves are patterned, like waves that pulse through air. Waves have mathematical properties all their own.

The Heavens
Nature’s patterns are echoed in the movement of planets and stars and the cycles of the moon. Perhaps this is why astrology lies at the heart of so many spiritual beliefs.

Music
The vibrations we call sound follow sacred, archetypal patterns. For this reason, you may find that certain sound sequences can stimulate the intellect, inspire creativity, and evoke a deep sense of joy.

The Cosmic Grid
Stonehenge, megalithic tombs, and other ancient sites stretch across the globe along underground electromagnetic tracks or ley lines. The energy grid formed by these lines suggests sacred shapes and ratios.

Theology
Best-selling author Dan Brown has made a lot of money by using the concepts of sacred geometry to weave a spell-binding tale about conspiracy and early Christianity. Brown's books are pure fiction and have been hotly criticized. But even when we dismiss The Da Vinci Code as a tall tale, we can't dismiss the importance of numbers and symbols in religious faith. Concepts of sacred geometry are expressed in the beliefs of Christians, Jews, Hindus, Muslims, and other formal religions.

From the pyramids in Egypt to the new World Trade Center tower in New York City, great architecture uses the same essential building blocks as your body and all living things. In addition, the principles of geometry are not confined to great temples and monuments. Geometry shapes all buildings, no matter how humble. Believers say that when we recognize geometric principles and build upon them, we create dwellings that comfort and inspire. Perhaps this is the idea behind the architect's conscious use of divine proportion like Le Corbusier did for the United Nations building.

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Hidden Codes: How Geometry Shapes Architecture

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Architecture begins with geometry. Find out how math relates to architecture and why some architects still believe in principles of the golden ratio.

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https://ibmathsresources.com/2018/02/27/circular-motion-modelling-a-ferris-wheel/

https://www.physicsclassroom.com/class/circles/Lesson-1/Mathematics-of-Circular-Motion

https://byjus.com/jee/circular-motion/

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September 8, 2016

Anyone who knows me well knows that I love numbers. They allow us to quantify performance and search for patterns within sport that may not be otherwise apparent. For this article, I wanted to look at the characteristics of horses and riders at the last three major international championships for show jumping (Rio 2016, WEG 2014, and London 2012).

By looking at the available data, certain trends become apparent which can give us insight into the current state of the show jumping sport at the highest level. These observations were compiled from the information provided in the FEI database, and thus the data is readily available to anyone who wants to do a similar analysis. Below I have highlighted some interesting data from the last three major games which provide insight into modern trends in show jumping.

To start, let's examine some general information about the horses and riders who competed. Across the three major championships examined, a total of 219 different riders from 54 nations competed during at least once. During these same events, 267 different horses competed. If we look at just the Olympics, the following 11 countries qualified teams for both London and Rio:

Qualifying a team for consecutive Olympics is no easy feat, so it speaks to the dominance of the above nations in being able to qualify teams for the largest competition in our sport. To do so, the nations must put the correct horse-rider combinations together at the correct times (qualifying competitions) to ensure a berth in the Olympics. Although there is always a level of chance associated with this sport, the above nations can be used as models for other nations aspiring to consistently compete teams at the top of the sport.

For individual rider characteristics, show jumping has a wide range of rider ages that compete at the highest levels with top results. For example, in London 2012, the youngest rider was 18 years old (Reed Kessler; USA) and the oldest was 65 (Ian Millar; CAN). Furthermore, equestrian is unique in that males and females compete against each other. However, there was not an equal gender split at either Olympics.

For Rio 2016, 17.3 % of riders were female, and in London 2012, 18.7 % of riders were female. This statistic is certainly one to monitor as the sport continues to develop!

The average age for a horse competing at a major championship was 11.5 years old. The percentage of horses at each age for major championships are shown in the figure below. Note that the most common ages are 10 and 11 for championship horses.

One may initially look at these ages and assume that horses peak athletically around the age of 10 or 11, and after 12 years, their ability to reach championship form is decreased. Although this may be the case, there are a number of other factors to consider:

Many horses are sold after reaching a major championship. If they are sold to another rider not competing at the championship level, the horse will never reach this level again at an 'older age'

Training-induced injuries become more prevalent as horses age

Riders tend to preserve older horses, and may perceive the championship format to be too physically taxing on their senior mounts

To further examine horse age, we can break it down by sex. Although data on the topic is limited, it is very likely that based on the sex of the horse, it will reach championship form at different times during its lifespan (due to differences in hormones, anatomy, and so on).

In the figure below, it is readily apparent that stallions were older than geldings and mares at both Olympics. One possible explanation for this is that good quality stallions may be kept for top level riders to continue to demonstrate the horse's breeding potential. Conversely, geldings and mares may be sold to younger or less competitive riders to aid in their development. Interestingly, the trend of stallions being older was not apparent in WEG 2014.

Horse Sex

You don't have to be in the horse world long to hear different experts discuss why they prefer mares, stallions or geldings. The discussion usually relates back to stories which have produced the currently held notions of these individuals. Although there may be some good reasoning for their decisions, does the sex of the horse actually matter when looking at which horses reach the top of the sport?

Below is the breakdown of stallions, geldings and mares at the last three major championships. It is clearly apparent that 1 in 2 horses at these events were geldings, while mares were the least common (1 in 5 horses are mares). Although each sex has produced dominant horses over the history of the sport, this data is worth considering when gambling on a horse to invest in for the future.

Studbook Registration

The average percentage of championship horses registered to different studbooks are shown in the figure below. It is interesting to note that over 50% of horses competing were Selle Francais ( SF), Belgian Warmbloods ( BWP) or Dutch Warmbloods ( KWPN). This speaks to the dominance of these breeding programs, and certainly provides riders looking for championship horses a great place to start their search for the next star.

( Studbook Key: ZANG: Zangersheide; WESTF: Westphalian; SWB: Swedish Warmblood; SF: Selle Francais; SBS: Belgian Sport Horse; OS: Oldenburg; KWPN: Dutch Warmblood; ISH: Irish Sport Horse; HOLST: Holsteiner; HANN: Hanoverian; BWP: Belgian Warmblood; AES: Anglo-European)

Last thoughts...

Of the 219 riders who competed during at least one of the last three major championships, only 18 riders competed at all three. This statistic demonstrates how difficult it is to keep top riders well mounted, and to have everything line up to allow them to compete at major games. A lot of very good riders did not meet this criteria.

Of the 267 horses who competed during the last three major championships, only 2 competed at all three: Nino des Buissonnets (SUI) and VDL VERDI TN (NED). Again, this speaks to the difficulty of maintaining a championship level horse for more than four years.

About the Author

Dr. Tim Worden has worked as a sport scientist with numerous FEI-level show jumping riders. He has a PhD in biomechanics and specializes in applying human high-performance training techniques to horses.

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What Can Numbers Teach Us About Elite Show Jumping? The Answer May Surprise You.

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Anyone who knows me well knows that I love numbers. They allow us to quantify performance and search for patterns within sport that may not be otherwise apparent. For this article, I wanted to look at the characteristics of horses and riders at the last three major international championships for show jumping (Rio 2016, WEG ...

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AI SL- Compound interest/arithmetic sequences/series- Sahar

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AI SL- Compound interest/arithmetic sequences/series- Sahar

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Written By Keerthi Kulkarni

Last Modified 07-06-2022

Applications of Compound Interest Formula: Compound interest is interest calculated on both the principal (original amount) and the interest already calculated. Every year, compound interest multiplies. We can find the applications of the compound interest formula in solving real-life problems mathematically. The various applications of compound interest formula include growth and decay, appreciation and depreciation, increase and decrease in commodity price, etc.

Compound interest keeps multiplying every year. The compound interest formula is represented as \(C.I = P{\left( {1 + \frac{{\frac{r}{n}}}{{100}}} \right)^{nt}} – P\). The compound Interest formula’s examples involve bank transactions, inflation in the profit and loss, etc. In this article, we have provided detailed information about the compound interest formulas, real-life applications of the compound interest formula, etc. Scroll down to learn more!

What is Compound Interest?

The extra or additional amount received from the borrower or paid to the lender is known as interest. We have two types of interests: simple interest and compound interest. In simple interest, the rate of interest on the principal amount remains constant throughout the given period. But, in compound interest, the interest keeps changing every year.

Compound interest is interest calculated on both principal amount and interest, compounded (multiplied) at regular intervals. The interest so far accumulated at regular intervals is added with the existing principal, and then the interest is taken for the new principal. The new principal is equalled to the sum of the initial principal and the interest calculated so far.
\(\rm{Compound}\,\rm{interest} = \rm{Interest}\,\rm{on}\,\rm{Principal} + \rm{Compounded}\,\rm{Interest}\,\rm{at}\,\rm{regular}\,\rm{intervals}\)

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Compound Interest Formula

Compound interest is the interest calculated on both principal amount and interest, compounded (multiplied) at regular intervals. Compound interest keeps multiplying every year.
The formula for calculation of the compound interest is given as follows:
\(C.I = P{\left( {1 + \frac{{\frac{r}{n}}}{{100}}} \right)^{nt}} – P\)
Here,
1. \(P-\)Initial principal amount
2. \(C.I-\)Compound interest
3. \(r-\)Rate of interest per annum
4. \(t-\)Time period in years
5. \(n-\)Frequency of the number of times the interest is compounded annually.

Applications of Compound Interest Formula Example With Solution

The applications of the compound interest formula are its uses in solving real-life problems mathematically. The various applications of the compound interest formula are listed below:

1. Compound interest not annually (monthly)

2. Growth and decay in the population

3. Increase and decrease in commodity price

4. Increase and decay in the value of the item

5. Inflation in the profit and loss

6. Bank transactions

Compounding Interest Non-Annually

In our real life, we come across many situations where we find the interest compounded non-annually (monthly, quarterly, half-yearly, sometimes daily). These are discussed here:

Calculation of Interest Compounded Semi-annually or Half-Yearly

If the principal is compounded semi-annually or half–yearly, the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}}\)
And, interest is given as:
\(C.I = P{\left( {1 + \frac{{\frac{r}{2}}}{{100}}} \right)^{2t}} – P\)

Practice Exam Questions

Calculation of Interest Compounded Quarterly

If the principal is compounded quarterly, the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}}\)
And, interest is given as:
\(C.I = P{\left( {1 + \frac{{\frac{r}{4}}}{{100}}} \right)^{4t}} – P\)

Calculation of Interest Compounded Monthly

If the principal is compounded monthly, then the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}}\)
And, interest is given as:
\(C.I = P{\left( {1 + \frac{{\frac{r}{{12}}}}{{100}}} \right)^{12t}} – P\)

Calculation of Interest Compounded Daily

If the principal is compounded daily, then the total amount is given by
\(A = P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}}\)
And, interest is given as:
\(C.I = P{\left( {1 + \frac{{\frac{r}{{365}}}}{{100}}} \right)^{365t}} – P.\)

Calculation of Growth or Decay Using Compound Interest Formula

One of the main applications of the compound interest formula is the calculation of growth of the objects or population etc. We will analyse how the compound interest formula is used to find the growth using a simple example.

A money lender has \(\rm{Rs}. 34,000\). He grows his money at \(5\%\) annual rate of interest. Now, he wants to know his total wealth at the end of three years from now. We can find his total wealth after three years (growth) using the compound interest formula as follows:

Let us assume the total wealth given today is the principal \(P = \rm{RS}. 34,000.\)
And, the growth per year is the rate of interest \(\%r = 5\%.\)
We need to find the total wealth at the end of the next three years, so consider \(n=3\) years.
The amount calculated using the compound interest formula gives his total wealth after three years.
So, the amount obtained by the compound interest formula is given by
\(A = P (1 + \frac{r}{100})^n\)
\( \Rightarrow A = 34,000{\left( {1 + \frac{5}{{100}}} \right)^3} = 34,000{\left( {\frac{{105}}{{100}}} \right)^3}\)
\(⇒ A = \rm{Rs}. 39,359.25\)
Hence, his total wealth after three years is given by \(\rm{Rs}. 39,359.25.\)

Similarly, we can find the depreciation by replacing the plus sign with the minus sign in the above formula.

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Applications of Compound Interest Formula – Growth or Increase

We know that growth per year is called the rate of growth. We can find the growth or total growth (where the rate of growth for n years is constant) using the compound interest formula as given below:
Final number \(= \rm{initial}\,\rm{number} \times (1 + \frac {r}{100})^3\)
Let the rate of growth is different for different years. Let \(r_1\%\) is the rate of growth for the first year and \(r_2\%\) is the rate of growth for the second year. Then, the total growth after two years is given by using the compound interest formula as follows:

Final value \(= \rm{initial}\,\rm{value} \times (1 + \frac {r_1}{100})(1 + \frac {r_2}{100})\)

Applications of Compound Interest Formula – Decay or Decrease or Depreciation

We know that the decay or decrease of the value per year is called the rate of decay. We can find the total value (where the rate of decay for \(n\) years is constant) by using the compound interest formula as shown below:

Value after \(n\) years \(= \rm{present}\,\rm{value}\times (1 – \frac {r}{100})^n\)
Let the rate of decay is different for different years. Let \(r_1\%\) is the rate of decay for the first year and \(r_2\%\) is the rate of decay for the second year. Then, the total value after two years is given by using the compound interest formula as follows:
Value after two years \(= \rm{Present}\,\rm{value} \times (1 – \frac {r_1}{100})(1 – \frac {r_2}{100}).\)

Solved Examples – Applications of Compound Interest Formula

Q.1. The present population of town \(A\) is \(10,000\). If the population of town \(A\) was increased by \(10\%\) every year. Find the population of town \(A\) after three years.
Ans: Given the initial population of town \(A = 10000\).
Rate of growth \(r = 10\%\)
Let \(n = 3\) years
The population of town \(A\) is found by using the compound interest formula as follows:
\(\rm{Final}\,\rm{population} = \rm{intial}\,\rm{population} × (1 + \frac{r}{100})^n\)
\(⟹ \rm{Final}\,\rm{population}\,\rm{after}\,\rm{three}\,\rm{years} = 10000 × (1 + \frac{10}{100})^3 = 13310\)
Hence, the total population of town \(A\) after three years is \(13310\).

Q.2. The count of a certain breed of bacteria was found to increase at the rate of \(2\%\) per hour. Find the bacteria at the end of \(2\) hours if the count was initially \(600000\).
Ans: Given, the initial population \((P)=600000\), rate \((R) = 2\%\).
Since the population of bacteria increases at the rate of \(2\%\) per hour, we use the formula
\(A = P(1 + \frac{R}{100})^n\)
Thus, the population at the end of \(2\) hours \(= 600000 (1 + \frac{2}{100})^2\)
\(= 600000 (1 + 0.02)^2\)
\(= 624240\)
Hence, the count of bacteria after two hours is \(624240\).

Q.3. A town had \(10,000\) residents in \(2010\). Its population decays at a rate of \(10\%\) per annum. What will be its total population in \(2015\)?
Ans: The population of the town decreases by \(10\%\) every year. Thus, it has a new population every year.
The population for the next year is calculated on the current year population. For the decrease, we have the formula
\(A = P (1 – \frac {r}{100})^n\)
Therefore, the population at the end of \(5\) years
\(= 10000(1 – \frac{10}{100})^5\)
\(= 10000(1 – 0.1)^5\)
\( = 5904.9\)
\( = 5904\) (Approx).

Q.4. A sum of \(\rm{Rs}.5000\) is borrowed at the rate is \(8\%\). The amount was compounded monthly. Find the compound interest for two years?
Ans: Given, principal \((P) = \rm{Rs}.5000\)
Rate of interest \((r) = 8\%\)
Time period \((T) = 2\) years
Compound interest after two years is
\(= \rm{Principal}(1 + \frac{r}{12 × 100})^{12 × \rm{time}} – \rm{Principal}\)
\( = 5000(1 + \frac{8}{1200})^{12 × 2} – 5000\)
\( = 5000 ( 1 + \frac{2}{300})^{24} – 5000\)
\( = 5000 × 1.1712 – 5000\)
\( = 5856 – 5000 = \rm{Rs}.856\)
The monthly compound interest for two years is \(\rm{Rs}.856\)

Q.5. Calculate the compound interest to be paid on a loan of \(\rm{Rs}.2000\) for \(\frac{3}{2}\) years at \(10\%\) per annum compounded half-yearly?
Ans: Principal, \(P = \rm{Rs}.2000\),
Time, \(T = \frac{3}{2}\rm{years} = 3 × \frac{1}{2}\rm{years} = 3\,\rm{half}\,\rm{years}\).
Rate, \(r = \frac{10\%}{2} = 5\%,\,2\)
Amount, \(A\) can be given as:
\(A = P (1 + \frac{r}{100})^n\)
\(⟹ A = 2000 × (1 + \frac{5}{100})^3 = 2000 ×(\frac{21}{20})^3 = \rm{Rs}.2315.25\)
So, \(CI = A – P = \rm{Rs}.2315.25 – \rm{Rs}.2000 = \rm{Rs}.315.\)

Summary:

This article discussed what compound interest is and the formula used to calculate compound interest. We have studied the applications of the compound interest formula with the help of solved examples.
We have studied the growth and decay, increasing and decreasing the value or population using the compound interest formula.

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Frequently Asked Questions

We have provided some frequently asked questions on Applications of Compound Interest Formula here:

Q.1. What is the use of the compound interest formula, and who benefits from this?
Ans: The compound interest is used to get more money as interest. The investors benefit from the compound interest since the interest received is on the principal plus on the interest they already earned.

Q.2. How do you calculate the compound interest if the interest is compounded every month?
Ans: If the principal is compounded monthly (\(12\) months in a year), the interest is given by
\({\text{Compound}\;\text{interest}} = {\text{Principal}}{\left( {1 + \frac{{\frac{{\text{r}}}{{12}}}}{{100}}} \right)^{12 \times {\text{time}}}} – {\text{Principal}}.\)

Q.3. What is the compound interest formula used for the growth or increase?
Ans: \({\text{Final}\;\text{growth}} = {\text{initial}\;\text{growth}} \times {\left( {1 + \frac{r}{{100}}} \right)^n}\)

Q.4. What is the main disadvantage of the compound interest?
Ans: The main disadvantage of compound interest is that it is more expensive, and we need to pay more as an interest to the lender if we have taken some amount from the lender.

Q.5. What are the applications of the compound interest formula?
Ans: Some of the applications of the compound interest formula are
1. Growth and decay
2. Appreciation and depreciation
3. Increase or decrease the value

Now you are provided with all the necessary information on the applications of compound interest formula and we hope this detailed article is helpful to you. If you have any queries regarding this article, please ping us through the comment section below and we will get back to you as soon as possible.

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Written By Keerthi Kulkarni Last Modified 07-06-2022

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Applications of Compound Interest Formula: Definition, Uses

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Know the different applications of compound interest formula in detail. Learn about compound interest formulas, examples, etc., at Embibe.

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Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more.

The following examples share how probability is used in 10 real-life situations on a regular basis.

Example 1: Weather Forecasting

Perhaps the most common real life example of using probability is weather forecasting.

Probability is used by weather forecasters to assess how likely it is that there will be rain, snow, clouds, etc. on a given day in a certain area.

Forecasters will regularly say things like "there is an 80% chance of rain today between 2PM and 5PM" to indicate that there's a high likelihood of rain during certain hours.

Example 2: Sports Betting

Probability is heavily used by sports betting companies to determine the odds they should set for certain teams to win certain games.

For example, a sports betting company may look at the current record of two teams and determine that team A has a 90% probability of winning while team B has just a 10% probability of winning.

Based on these probabilities, the company would offer a higher payout for people who bet on team B to win since it's highly unlikely that team B will actually win.

Political forecasters use probability to predict the chances that certain candidates will win various elections.

For example, a forecaster might say that candidate A has a 60% chance of winning, candidate B has a 20% chance of winning, candidate C has a 10% chance of winning, etc. to give voters an idea of how likely it is that each candidate will win.

Note: A real-life example of a site that uses probability to perform political forecasting is FiveThirtyEight.

Example 4: Sales Forecasting

Many retail companies use probability to predict the chances that they'll sell a certain amount of goods in a given day, week, or month.

This allows the companies to predict how much inventory they'll need. For example, a company might use a forecasting model that tells them the probability of selling at least 100 products on a certain day is 90%.

This means they'll need to make sure they have at least 100 products on hand to sell (or preferably more) so they don't run out.

Example 5: Health Insurance

Health insurance companies often use probability to determine how likely it is that certain individuals will spend a certain amount on healthcare each year.

For example, a company might use factors like age, existing medical conditions, current health status, etc. to determine that there's a 90% probability that a certain individual will spend $10,000 or more on healthcare in a given year.

Individuals who are likely to spend more on healthcare will be charged higher premiums because the insurance company knows that they'll be more expensive to insure.

Example 6: Grocery Store Staffing

Grocery stores often use probability to determine how many workers they should schedule to work on a given day.

For example, a grocery store may use a model that tells them there is a 75% chance that they'll have more than 800 customers come into the store on a given day.

Based on this probability, they'll schedule a certain amount of workers to be at the store on that day to handle that many customers.

Example 7: Natural Disasters

The environmental departments of countries often use probability to determine how likely it is that a natural disaster like a hurricane, tornado, earthquake, etc. will strike the country in a given year.

If the probability is quite high, then the department will make decisions about housing, resource allocation, etc. that will minimize the effects done by the natural disaster.

Ordinary people use probability every day when they decide to drive somewhere.

Based on the time of day, location in the city, weather conditions, etc. we all tend to make probability predictions about how bad traffic will be during a certain time.

For example, if you think there's a 90% probability that traffic will be heavy from 4PM to 5:30PM in your area then you may decide to wait to drive somewhere during that time.

Investors use probability to assess how likely it is that a certain investment will pay off.

For example, a given investor might determine that there is a 1% chance that the stock of company A will increase 100x during the upcoming year.

Based on this probability, the investor will decide how much of their net worth to invest in the stock.

Probability is routinely used by anyone who plays card games on a regular basis.

For example, professional poker players use probability to determine how likely it is that a certain hand of cards will win and this informs them on how much they should bet.

If a player knows that there is a high probability that they will win a certain hand based on their cards, they will be more likely to bet more money.

Conversely, if they think the probability that they'll win is low then they may bet significantly less money.

The following tutorials provide additional information about probability:

Examples of Using Conditional Probability in Real Life
Probability vs. Proportion: What's the Difference?
Probability vs. Likelihood: What's the Difference?
Law of Total Probability: Definition & Examples

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10 Examples of Using Probability in Real Life - Statology

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Probability is used in all types of areas in real life including weather forecasting, sports betting, investing, and more. The following examples share how probability is used in 10 real-life situations on a regular basis. Perhaps the most common real life example of using probability is weather forecasting.

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Pearson Product Moment Correlations (PMCC)- Reem

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Pearson Product Moment Correlations (PMCC)- Reem

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Pearson Product-Moment Correlation - When you should run this test, the range of values the coefficient can take and how to measure strength of association.

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Understand when to use the Pearson product-moment correlation, what range of values its coefficient can take and how to measure strength of association.

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https://sciencing.com/how-is-calculus-used-in-economics-13593453.html

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https://www.restaurant-website-builder.com/restaurant-app-statistics-infographic

https://www.klipfolio.com/blog/tips-to-manage-restaurant-data

https://www.investopedia.com/terms/s/statistics.asp#:~:text=Statistics%20is%20a%20branch%20of,linear%20algebra%2C%20and%20probability%20theory.

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People Want Convenient Ways to Order Food

As tech-savvy Millennials start having kids, they don't have as much time to eat in restaurants. That's one reason that food delivery is expected to grow 12% per year over the next 5 years. Even without kids, 57% of Millennials say that they prefer food delivery so they can watch TV at home without getting interrupted.

Simply having apps for small restaurants and chains, however, won't guarantee that more people choose your business. Restaurant mobile app features need to make ordering as easy as possible.

Your customers will want to see options like:

A mobile-ready menu that helps them browse and choose foods.

Quick checkout that lets them choose between cash, credit card, and other payment options.

A delivery time estimate that helps them prepare for the food's arrival.

A customer reward program that tracks their purchases and automatically offers rewards.

You can get started by using the GloriaFood branded app. This white label restaurant ordering app lets you upload pictures, menus, and logos so you can better brand your business to customers.

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Restaurant App Statistics: Why Your Restaurant Needs An App

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Restaurant App Statistics: Discover how restaurant apps are reshaping the US restaurant industry & why more customers prefer to use a custom mobile app for restaurant ordering instead of a third party food portal. Why you need a branded restaurant app to sell more >>

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AI SL- Pearson Product Moment Correlation (PPMC)- Reem

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AI SL- Pearson Product Moment Correlation (PPMC)- Reem

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https://statistics.laerd.com/statistical-guides/pearson-correlation-coefficient-statistical-guide.php

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The correlation coefficient is the measurement of the correlation between two variables. Pearson correlation formula is used to see how the two sets of data are co-related. The linear dependency between the data set is checked using the Pearson correlation coefficient. It is also known by the name of the Pearson product-moment correlation coefficient. The value of the Pearson correlation coefficient product lies between -1 to +1. If the correlation coefficient is zero, then the data is said to be not related. A value of +1 indicates that the data are positively correlated and a value of -1 indicates a negative correlation.

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Book a Free Trial Class

What Is Pearson Correlation Formula?

The Pearson correlation coefficient is symbolised by the letter “r”. RephraseThe Pearson correlation formula for the coefficient r is given by:

\(r=\frac{n\left(\sum x y\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n \sum x^{2}-\left(\sum x\right)^{2}\right]\left[n \sum y^{2}-\left(\sum y\right)^{2}\right]}}\)

Where,
\(r=\) Pearson correlation coefficient
\(x=\) Values in the first set of data \(y=\) Values in the second set of data \(n=\) Total number of values

Let's solve a few solved examples based on the Pearson correlation formula.

Solved Examples Using Pearson Correlation Formula

Example 1: A survey was conducted in your city. Given is the following sample data containing a person's age and their corresponding income. Find out whether the increase in age has an effect on income using the correlation coefficient formula. (Use \(\frac{1}{\sqrt{181}}\) as 0.074 and \(\frac{1}{\sqrt{209}}\) as 0.07)

Age | 25 | 30 | 36 | 43
Income | 30000 | 44000 | 52000 | 7000

Solution:

To simplify the calculation, we divide y by 1000.

Age (xi) | Income/1000 (yi/1000) | \(x_i - \bar{x}\) | \(y_i - \bar{y}\) | \((x_i - \bar{x})^2\) | \((y_i - \bar{y})^2\) | \((x_i - \bar{x})(y_i - \bar{y})\)
25 | 30 | -8.5 | -19 | 72.25 | 361 | 161.5
30 | 44 | -3.5 | -5 | 12.25 | 25 | 17.5
36 | 52 | 2.5 | 3 | 6.25 | 9 | 7.5
43 | 70 | 9.5 | 21 | 90.25 | 441 | 199.5
\(\bar{x} = 33.5\) | \(\bar{y} = 49\) | | | \(\Sigma (x_i - \bar{x})^2 = 181\) | \(\Sigma (y_i - \bar{y})^2 = 836\) | \(\Sigma(x_i-\bar{x})(y_i - \bar{y}) = 386\)

Pearson correlation coefficient for sample = \(\dfrac{\Sigma (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\Sigma (x_i - \bar{x})^2 \Sigma (y_i - \bar{y})^2 }}\) = \(\dfrac{386}{\sqrt{181}\sqrt{836}}\) = \(\dfrac{193}{\sqrt{181}\sqrt{209}}\) = 0.99

Answer: Yes, with the increase in age a person's income increases as well, since the Pearson correlation coefficient between age and income is very close to 1.

Example 2: Marks obtained by 5 students in algebra and trigonometry as given below:

\(\begin{array}{|c|c|c|c|c|c|} \hline \text { Science } & 16 & 15 & 12 & 10 & 8 \\ \hline \text { Geometry } & 11 & 18 & 10 & 20 & 17 \\ \hline \end{array}\)
Calculate the Pearson correlation coefficient.

Solution:
Construct the following table:

The formula for Pearson correlation coefficient is:
\(r=\frac{n\left(\sum x y\right)-\left(\sum x\right)\left(\sum y\right)}{\sqrt{\left[n \sum x^{2}-\left(\sum x\right)^{2}\right]\left[n \sum y^{2}-\left(\sum y\right)^{2}\right]}}\) \(r=\frac{5 \times 902-61 \times 76}{ \left.\sqrt{\left[5 \times 789(61)^{2} \| 5 \times 1234-(76)^{2}\right.}\right]}\) \(r=-0.424\)

Answer: r = -0.424

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Pearson Correlation Formula - Learn the Pearson Correlation Formula using Solved Examples

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The value of the Pearson correlation coefficient product lies between -1 to +1. If the correlation coefficient is zero, then the data is said to be not related.

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https://www.statisticshowto.com/probability-in-real-life/

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Probability main index >

Key takeways:

Probability has thousands of everyday uses, from weather forecasting to credit scores.

Probability distributions help to forecast power failures and network outages.

Without probability, any form of gambling wouldn't exist.

Probability isn't just about flipping coins and counting cards in a deck: it is used in a wide range of real life areas, from insurance to meteorology and politics to economic forecasting.

Examples of Probability in Real Life

Probability Trees

Probability trees help to map out probabilities. As an example, the Federal government used this probability tree to assess the probability of another eruption on Mount Pinatubo in the Philippines:

Minimum spanning trees are a type of probability tree used for network designs like cable networks. They are also used for a diverse number of real life applications including real-time face tracking in video streams. For more examples, see: Tree Diagram Real Life Example.

Conditional Probability

Conditional probability can be used to find the probability a presidential re-election campaign will be successful: the re-election of a president depends upon metrics like popularity ratings, the voting preference of voters, and the success of social media campaigns.

Meteorologists use conditional probability to make weather forecasts. For example, your local weather might state you have a 50% chance of rain today. This is conditional on many factors including a cold front coming to your area, rain cloud formation and wind direction.

Predictive Analytics

Predictive analytics predicts the probability of future events based on past behavior. A few real life examples:

Credit scores: a consumer's past and current financial behavior determines the probability they will make timely payments in the future.

Sports: Billy Bean (made famous by Brad Pitt in the movie Moneyball) used predictive analytics to turn his low-performing Major League Baseball Team into one of the highest performing teams.

Power plants use predictive analytics to reduce outages and maintenance costs.

Probability Distributions and Their Real Life Uses

The binomial distribution, which models the probability of "Successes" and "Failures", is often presented as coin tosses or other trivial examples. However, the disribution is used in many real life situations. For example, let's say a new drug is found to help cure the common cold. It's either going to cure a cold (a "Success") or it isn't (a "Failure"). Another, more practical example: If you hunt for a parking space in a busy downtown area, you're either going to find a space, or you are not. If this is something you do regularly, it could be helpful to know the probability (i.e., if it's even worth trying to find a space on a given day).

Computer hardware failures can be modeled by the exponential distribution or Weibull distribution. Thes are memoryless (independent of their histories).

Fun Uses of Probability in Real Life

Probability in everyday life isn't just reserved for practical purposes. It can also be used for fun! For example, you can use probability to:

Permutations and combinations have many uses in gambling, including the need for state lotteries to calculate the number of ways lottery balls can be chosen.

Probability in Real Life: Articles

Related Articles

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Probability main index > Key takeways: Probability has thousands of everyday uses, from weather forecasting to credit scores. Probability distributions help to forecast power failures and network outages. Without probability, any form of gambling wouldn't exist.

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AA SL 1.4 Financial Applications of Geometric Sequences and Series

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AA SL 1.4 Financial Applications of Geometric Sequences and Series

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https://courses.lumenlearning.com/wmopen-mathforliberalarts/chapter/introduction-accounts-that-gain-interest/

https://www.pettyson.co.uk/about-us/our-blog/525-mortgage-types-explained

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Learning Outcomes

Calculate the balance on an annuity after a specific amount of time

Discern between compound interest, annuity, and payout annuity given a finance scenario

Use the loan formula to calculate loan payments, loan balance, or interest accrued on a loan

Determine which equation to use for a given scenario

Solve a financial application for time

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. In this section, we will explore the math behind specific kinds of accounts that gain interest over time, like retirement accounts. We will also explore how mortgages and car loans, called installment loans, are calculated.

Savings Annuities

For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we save for the future by depositing a smaller amount of money from each paycheck into the bank. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA plans are examples of savings annuities.

An annuity can be described recursively in a fairly simple way. Recall that basic compound interest follows from the relationship

[latex]{{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}[/latex]

For a savings annuity, we simply need to add a deposit, d, to the account with each compounding period:

[latex]{{P}_{m}}=\left(1+\frac{r}{k}\right){{P}_{m-1}}+d[/latex]

Taking this equation from recursive form to explicit form is a bit trickier than with compound interest. It will be easiest to see by working with an example rather than working in general.

Example

Suppose we will deposit $100 each month into an account paying 6% interest. We assume that the account is compounded with the same frequency as we make deposits unless stated otherwise. Write an explicit formula that represents this scenario.

Generalizing this result, we get the savings annuity formula.

Annuity Formula

[latex]P_{N}=\frac{d\left(\left(1+\frac{r}{k}\right)^{Nk}-1\right)}{\left(\frac{r}{k}\right)}[/latex]

PN is the balance in the account after N years.

d is the regular deposit (the amount you deposit each year, each month, etc.)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.

For example, if the compounding frequency isn’t stated:

If you make your deposits every month, use monthly compounding, k = 12.

If you make your deposits every year, use yearly compounding, k = 1.

If you make your deposits every quarter, use quarterly compounding, k = 4.

Etc.

When do you use this?

Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest.

Compound interest assumes that you put money in the account once and let it sit there earning interest.

Compound interest: One deposit

Annuity: Many deposits.

Examples

A traditional individual retirement account (IRA) is a special type of retirement account in which the money you invest is exempt from income taxes until you withdraw it. If you deposit $100 each month into an IRA earning 6% interest, how much will you have in the account after 20 years?

This example is explained in detail here.

Try It

A conservative investment account pays 3% interest. If you deposit $5 a day into this account, how much will you have after 10 years? How much is from interest?

Try It

Financial planners typically recommend that you have a certain amount of savings upon retirement.  If you know the future value of the account, you can solve for the monthly contribution amount that will give you the desired result. In the next example, we will show you how this works.

Example

You want to have $200,000 in your account when you retire in 30 years. Your retirement account earns 8% interest. How much do you need to deposit each month to meet your retirement goal?

View the solving of this problem in the following video.

Try It

Solving For Time

We can solve the annuities formula for time, like we did the compounding interest formula, by using logarithms. In the next example we will work through how this is done.

Example

If you invest $100 each month into an account earning 3% compounded monthly, how long will it take the account to grow to $10,000?

This example is demonstrated here:

Payout Annuities

Removing Money from Annuities

In the last section you learned about annuities. In an annuity, you start with nothing, put money into an account on a regular basis, and end up with money in your account.

In this section, we will learn about a variation called a Payout Annuity. With a payout annuity, you start with money in the account, and pull money out of the account on a regular basis. Any remaining money in the account earns interest. After a fixed amount of time, the account will end up empty.

Payout annuities are typically used after retirement. Perhaps you have saved $500,000 for retirement, and want to take money out of the account each month to live on. You want the money to last you 20 years. This is a payout annuity. The formula is derived in a similar way as we did for savings annuities. The details are omitted here.

Payout Annuity Formula

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

P₀ is the balance in the account at the beginning (starting amount, or principal).

d is the regular withdrawal (the amount you take out each year, each month, etc.)

r is the annual interest rate (in decimal form. Example: 5% = 0.05)

k is the number of compounding periods in one year.

N is the number of years we plan to take withdrawals

Like with annuities, the compounding frequency is not always explicitly given, but is determined by how often you take the withdrawals.

When do you use this?

Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.

Compound interest: One deposit

Annuity: Many deposits.

Payout Annuity: Many withdrawals

Example

After retiring, you want to be able to take $1000 every month for a total of 20 years from your retirement account. The account earns 6% interest. How much will you need in your account when you retire?

View more about this problem in this video.

Try It

Evaluating negative exponents on your calculator

With these problems, you need to raise numbers to negative powers.  Most calculators have a separate button for negating a number that is different than the subtraction button.  Some calculators label this (-) , some with +/- .  The button is often near the = key or the decimal point.

If your calculator displays operations on it (typically a calculator with multiline display), to calculate 1.005-240 you’d type something like:  1.005 ^ (-) 240

If your calculator only shows one value at a time, then usually you hit the (-) key after a number to negate it, so you’d hit: 1.005 yx 240 (-)  =

Give it a try – you should get 1.005-240 = 0.302096

Example

You know you will have $500,000 in your account when you retire. You want to be able to take monthly withdrawals from the account for a total of 30 years. Your retirement account earns 8% interest. How much will you be able to withdraw each month?

A detailed walkthrough of this example can be viewed here.

Try It

Try It

A donor gives $100,000 to a university, and specifies that it is to be used to give annual scholarships for the next 20 years. If the university can earn 4% interest, how much can they give in scholarships each year?

Loans

Conventional Loans

In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called amortized loans or installment loans). Examples include auto loans and home mortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front.

One great thing about loans is that they use exactly the same formula as a payout annuity. To see why, imagine that you had $10,000 invested at a bank, and started taking out payments while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and imagine that you are acting as the bank, and a car lender is acting as you. The car lender invests $10,000 in you. Since you’re acting as the bank, you pay interest. The car lender takes payments until the balance is zero.

Loans Formula

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

P₀ is the balance in the account at the beginning (the principal, or amount of the loan).

d is your loan payment (your monthly payment, annual payment, etc)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

N is the length of the loan, in years.

Like before, the compounding frequency is not always explicitly given, but is determined by how often you make payments.

When do you use this?

The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.

Compound interest: One deposit

Annuity: Many deposits

Payout Annuity: Many withdrawals

Loans: Many payments

Example

You can afford $200 per month as a car payment. If you can get an auto loan at 3% interest for 60 months (5 years), how expensive of a car can you afford? In other words, what amount loan can you pay off with $200 per month?

Details of this example are examined in this video.

Try It

Example

You want to take out a $140,000 mortgage (home loan). The interest rate on the loan is 6%, and the loan is for 30 years. How much will your monthly payments be?

View more about this example here.

Try It

Try It

Janine bought $3,000 of new furniture on credit. Because her credit score isn’t very good, the store is charging her a fairly high interest rate on the loan: 16%. If she agreed to pay off the furniture over 2 years, how much will she have to pay each month?

Calculating the Balance

With loans, it is often desirable to determine what the remaining loan balance will be after some number of years. For example, if you purchase a home and plan to sell it in five years, you might want to know how much of the loan balance you will have paid off and how much you have to pay from the sale.

To determine the remaining loan balance after some number of years, we first need to know the loan payments, if we don’t already know them. Remember that only a portion of your loan payments go towards the loan balance; a portion is going to go towards interest. For example, if your payments were $1,000 a month, after a year you will not have paid off $12,000 of the loan balance.

To determine the remaining loan balance, we can think “how much loan will these loan payments be able to pay off in the remaining time on the loan?”

Example

If a mortgage at a 6% interest rate has payments of $1,000 a month, how much will the loan balance be 10 years from the end the loan?

This example is explained in the following video:

Oftentimes answering remaining balance questions requires two steps:

1. Calculating the monthly payments on the loan

2. Calculating the remaining loan balance based on the remaining time on the loan

Example

A couple purchases a home with a $180,000 mortgage at 4% for 30 years with monthly payments. What will the remaining balance on their mortgage be after 5 years?

More explanation of this example is available here:

Solving for Time

Recall that we have used logarithms to solve for time, since it is an exponent in interest calculations. We can apply the same idea to finding how long it will take to pay off a loan.

Try It

Joel is considering putting a $1,000 laptop purchase on his credit card, which has an interest rate of 12% compounded monthly. How long will it take him to pay off the purchase if he makes payments of $30 a month?

FYI

Home loans are typically paid off through an amortization process, amortization refers to paying off a debt (often from a loan or mortgage) over time through regular payments. An amortization schedule is a table detailing each periodic payment on an amortizing loan as generated by an amortization calculator.

If you want to know more, click on the link below to view the website “How is an Amortization Schedule Calculated?” by MyAmortizationChart.com. This website provides a brief overlook of Amortization Schedules.

How is an Amortization Schedule Calculated?

Which Formula to Use?

Now that we have surveyed the basic kinds of finance calculations that are used, it may not always be obvious which one to use when you are given a problem to solve. Here are some hints on deciding which equation to use, based on the wording of the problem.

Loans

The easiest types of problems to identify are loans.  Loan problems almost always include words like loan, amortize (the fancy word for loans), finance (i.e. a car), or mortgage (a home loan). Look for words like monthly or annual payment.

The loan formula assumes that you make loan payments on a regular schedule (every month, year, quarter, etc.) and are paying interest on the loan.

Loans Formula

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

P₀ is the balance in the account at the beginning (the principal, or amount of the loan).

d is your loan payment (your monthly payment, annual payment, etc)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

N is the length of the loan, in years.

Interest-Bearing Accounts

Accounts that gain interest fall into two main categories.  The first is on where you put money in an account once and let it sit, the other is where you make regular payments or withdrawals from the account as in a retirement account.

Interest

If you’re letting the money sit in the account with nothing but interest changing the balance, then you’re looking at a compound interest problem. Look for words like compounded, or APY. Compound interest assumes that you put money in the account once and let it sit there earning interest.

COMPOUND INTEREST

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

PN is the balance in the account after N years.

P0 is the starting balance of the account (also called initial deposit, or principal)

r is the annual interest rate in decimal form

k is the number of compounding periods in one year

If the compounding is done annually (once a year), k = 1.

If the compounding is done quarterly, k = 4.

If the compounding is done monthly, k = 12.

If the compounding is done daily, k = 365.

The exception would be bonds and other investments where the interest is not reinvested; in those cases you’re looking at simple interest.

SIMPLE INTEREST OVER TIME

[latex]\begin{align}&I={{P}_{0}}rt\\&A={{P}_{0}}+I={{P}_{0}}+{{P}_{0}}rt={{P}_{0}}(1+rt)\\\end{align}[/latex]

I is the interest

A is the end amount: principal plus interest

[latex]\begin{align}{{P}_{0}}\\\end{align}[/latex] is the principal (starting amount)

r is the interest rate in decimal form

t is time

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

Annuities

If you’re putting money into the account on a regular basis (monthly/annually/quarterly) then you’re looking at a basic annuity problem.  Basic annuities are when you are saving money.  Usually in an annuity problem, your account starts empty, and has money in the future. Annuities assume that you put money in the account on a regular schedule (every month, year, quarter, etc.) and let it sit there earning interest.

ANNUITY FORMULA

[latex]P_{N}=\frac{d\left(\left(1+\frac{r}{k}\right)^{Nk}-1\right)}{\left(\frac{r}{k}\right)}[/latex]

PN is the balance in the account after N years.

d is the regular deposit (the amount you deposit each year, each month, etc.)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same number of compounds in a year as there are deposits made in a year.

If you’re pulling money out of the account on a regular basis, then you’re looking at a payout annuity problem.  Payout annuities are used for things like retirement income, where you start with money in your account, pull money out on a regular basis, and your account ends up empty in the future. Payout annuities assume that you take money from the account on a regular schedule (every month, year, quarter, etc.) and let the rest sit there earning interest.

PAYOUT ANNUITY FORMULA

[latex]P_{0}=\frac{d\left(1-\left(1+\frac{r}{k}\right)^{-Nk}\right)}{\left(\frac{r}{k}\right)}[/latex]

P₀ is the balance in the account at the beginning (starting amount, or principal).

d is the regular withdrawal (the amount you take out each year, each month, etc.)

r is the annual interest rate (in decimal form. Example: 5% = 0.05)

k is the number of compounding periods in one year.

N is the number of years we plan to take withdrawals

Remember, the most important part of answering any kind of question, money or otherwise, is first to correctly identify what the question is really asking, and then determine what approach will best allow you to solve the problem.

Try It

For each of the following scenarios, determine if it is a compound interest problem, a savings annuity problem, a payout annuity problem, or a loans problem. Then solve each problem.

1. Marcy received an inheritance of $20,000, and invested it at 6% interest. She is going to use it for college, withdrawing money for tuition and expenses each quarter. How much can she take out each quarter if she has 3 years of school left?

2. Paul wants to buy a new car. Rather than take out a loan, he decides to save $200 a month in an account earning 3% interest compounded monthly. How much will he have saved up after 3 years?

3. Keisha is managing investments for a non-profit company.       They want to invest some money in an account earning 5% interest compounded annually with the goal to have $30,000 in the account in 6 years. How much should Keisha deposit into the account?

4. Miao is going to finance new office equipment at a 2% rate over a 4 year term. If she can afford monthly payments of $100, how much new equipment can she buy?

5. How much would you need to save every month in an account earning 4% interest to have $5,000 saved up in two years?

In the following video, we present more examples of how to use the language in the question to determine which type of equation to use to solve a finance problem.

In the next video example, we show how to solve a finance problem that has two stages, the first stage is a savings problem, and the second stage is a withdrawal problem.

Try It

Try It

Try It

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Annuities and Loans | Mathematics for the Liberal Arts