Percentages, Interest and More

I am sure you have heard of the phenomenal “power of compounding.” Elders love to preach about the wisdom of starting a savings account at the age of 25 (when you don’t have any money left from your pay check after your not-so-sensible Vuitton/Gucci/Chanel escapades!) rather than at the age of 40 (when you have 2 mortgages, 2 kids and a high maintenance Cadillac). Let’s crunch some numbers to see if they are right.

Starting at age 25, if you put $200 every month for 40 years at 10% per annum, you will have more than 1.25 million at the age of 65.

Starting at age 40, if you put $200 every month for 25 years at 10% per annum, you will have only $265,000 at the age of 65.
You might have reservations about the fact that in the first case, you are investing more, after all! It’s not just about the longer time period. So look at it another way.

If you invest $25,000 at age 25 at 10% per annum, you will have $1.13 million at age 65.

But if you invest $25,000 at age 40 at 10% per annum, you will have $271,000 at age 65.

In the first case, you invest for less than double the time (for 40 years as compared to 25 years in the second case) but you get four times the return (1.31 million as compared to 271,000)… It seems that elders might have been right about this after all. (Now I wish I had listened to my dad!)

Since compounding has a powerful influence on finances, it is something you will come across often during your MBA studies. So GMAT likes to test you on it too. Let’s get crackin’ then.

As I mentioned in my previous post, compound Interest is an important application of successive percentage changes. GMAT tests you on simple and compound interest and sometimes, may even test you on the relation between the two. So let’s look at both of them one by one.

When you say that the rate of simple interest is 10% per annum, it means that you earn 10% on your original principal every year. Say I deposit $1000 for 4 years at 10% simple interest per annum. Amount at the end of one year is simply 1000*(11/10) = $1100 i.e. I earn $100 in a year. Since it is simple interest, every year I will earn the same amount i.e. $100. So total simple interest earned will be $100*4 = $400. If you observe carefully, we have calculated total simple interest using the following concept:

\(Simple \ Interest = \frac{Interest \ Rate}{100} *Principal * No. \ of \ years\)

which is exactly what the formula for calculation of simple interest gives us.

Now let’s go on to compound interest. Compounding means successive percentage changes. It means that a sum of money increases by a certain percentage in a year. At the end of the year, the interest earned is combined with the principal and next year, interest is earned on this combined amount.

Say I deposit $1000 for 4 years at 10% compound interest per annum. Amount at the end of one year is simply 1000*(11/10) = $1100 i.e. I earn $100 in a year. Till now it is just like simple interest. But from next year on, we will earn on this extra $100 that we earned this year too. Amount at the end of 2nd year = 1100*(11/10) = 1210. Amount at the end of third year will be 1210*(11/10) and so on… As you can see, this is just 1000*(11/10)(11/10)(11/10)(11/10) or

\(A = P(1 + r)^i\)

(“i” is the number of time periods and r is the percentage rate of interest)

which is nothing but the ‘amount in case of compound interest’ formula

How does knowing this help us? GMAT tests your ingenuity and conceptual understanding. I will give below two examples where you will see how knowing this helps.

Example 1:
A bank launched a new financial instrument called VDeposit. A VDeposit offers you variable rate of compound interest in accordance with the current market rate. Ethan deposited $8000 in a VDeposit. If he gets interest rates of 10% in the first two years and 12.5% in the third year, what is the total amount at the end of 3 years?

As you can see, solving it using the standard formula is slightly cumbersome since we would have to use it twice. I would rather view it as:

\(8000*(\frac{11}{10})(\frac{11}{10})(\frac{9}{8}) = 1000*(\frac{11}{10})*(\frac{11}{10})*(9)\)

Notice that even though 12.5% compound interest was offered in the third year, we can still cancel off the 8 of 8000 with 8 of 12.5% increase when we view the calculation this way.

Amount = $10,890

Example 2:
Mark deposited $D in a scheme offering 5% simple interest per annum. Tetha deposited $D in a scheme offering 5% compound interest per annum. At the end of second year, Tetha had earned a total of $2.50 more than Mark. What is the value of D?

Till the end of first year, simple interest and compound interest cases are exactly the same. The difference comes in at the end of second year when compound interest offers interest on previous year’s interest too. $2.50 is 5% interest earned in the second year on first year’s interest.

\(2.5 = (\frac{5}{100}) * I\)

I = $50

So interest earned in the first year is $50, which is 5% of the deposited amount D

\(50 = (\frac{5}{100})*D\)

D = $1000

In these and many more case, it pays to understand the concept of simple and compound interest.

Now, I will leave you with a question:

In the case of yearly compound interest, the ratio of amounts at the end of the 20th year to the amount at the end of the 22nd year is 0.81. What is the rate of interest?

Here I will take a topic I briefly introduced in the previous post on percentages. Let me start with the question I posted there.

What does a 20% sale with an additional 25% off on the $85 sweater that you have your eye on mean to you?

It means a big rebate. Let’s see how much:

If you reduce 85 by 20%, it becomes \(85 * \frac{4}{5} = $68\). Now, you reduce it again by 25% and it becomes \(68 *(\frac{3}{4}) = $51\)

Notice that a 20% discount and then a 25% discount is not equal to a 45% discount (\(85 * \frac{55}{100} = $46.75\)). It is less than 45% but to the imperceptive, oblivious customer, it registers as 45% (Now you know why the retailers use the strategy of marking down by 20% and then giving an ‘additional’ 25% later!). The difference arises because the 20% discount was given on $85 but the 25% discount was given on $68. 25% of 68 is definitely less than 25% of 85 and hence the overall percentage decrease is less than 45%.

This is called successive percentage change – a number is changed by some percentage and then the new number is changed by another percentage. Both the percentage changes are not applied to the same original number.

The most popular example of successive percentage change is population change. Let us look at an example to understand this.

Example 1:

A city’s population was 10,000 at the end of 2008. In 2009, it increased by 10% and in 2010, it decreased by 18.18%. What was the city’s population at the end of 2010?

Solution:

Population at the end of 2008 = 10,000

Population at the end of 2009 \(= 10,000 * (\frac{11}{10}) = 11,000\)

Population at the end of 2010 \(= 11,000 * (\frac{9}{11}) = 9000\)

Simply put, population at the end of 2010 \(= 10,000 * (\frac{11}{10}) * (\frac{9}{11}) = 9000\)

It is best to do the calculations in a single step because you do not need to calculate the intermediate population values. Besides, there is a good possibility that factors will get canceled out and hence, you will need to do fewer calculations.

Obviously, there is no limit to the number of successive percentage changes that can be made to a number. The approach remains unchanged in any case. Let me elaborate with another example:

Example 2:

Six months back, the cost of an air ticket from Detroit to San Francisco was $400. Four months back, the fares increased by 12.5%. Last month, the fares increased by 25% and yesterday, the airlines again increased the fares by 11.11%. What is the price of a Detroit to San Francisco ticket today?

Solution:

Price of a ticket today \(= 400 * (\frac{9}{8}) * (\frac{5}{4}) * (\frac{10}{9}) = $625\)

This is much faster than finding the ticket price at every price change which would need the following steps:

Price of a ticket four months back \(= 400 * (\frac{9}{8}) = $450\)

Price of a ticket last month back \(= 450 * (\frac{5}{4}) = $562.5\)

and finally, price of a ticket today \(= 562.5 * (\frac{10}{9}) = $625\)

So, in case you do not need the intermediate values, do not calculate them.

When there are only two successive percentage changes, we can derive a formula. In some cases, the formula makes the solution very simple.

When a number, N, changes by x% and then changes again by y%, we do the following to find the new number:

New number \(= N * (1 + \frac{x}{100}) * (1 + \frac{y}{100})\)

Now, \((1 + \frac{x}{100}) * (1 + \frac{y}{100}) = 1 + \frac{x}{100} +\frac{y}{100} + \frac{xy}{10000}\)

If we say that \(x + y + \frac{xy}{100} = z\), then \((1 + \frac{x}{100}) * (1 + \frac{y}{100}) = 1 + \frac{z}{100}\)

Here, z is the effective percentage change when a number is changed successively by two percentage changes. Let’s take another example to see the formula in action:

Example 3:

A city’s population was 10,000 at the end of 2008. In 2009, it increased by 20% and in 2010, it decreased by 10%. What was the city’s population at the end of 2010?

Solution:

x% = 20%

y% = – 10% (Notice the negative sign here because this is a decrease)

Effective percentage change \(= x + y + \frac{xy}{100} = 20 + (– 10) + \frac{20*(-10)}{100} = 8\%\)

Population at the end of 2010 \(= 10,000 * (\frac{108}{100}) = 10800\)

Note: When the percentage is a decrease, a negative sign is used as shown above.

This formula is used only when there are two successive percentage changes and the percentages are easy to work with e.g. 15% and 25%, -10% and – 30% etc.

With more than two successive percentage changes or trickier percentage values e.g. 11.11% and 18.18%, 9.09% and 6.25% etc, stick to the method shown above.

A major application of successive percentage changes in GMAT is the MarkUp-Discount-Profit questions. We will take that topic next week but I will leave you with a question to ponder upon:

If a retailer marks up his goods by 40% and then offers a discount of 10%, what is his profit%?

Mark Up, Discount and Profit questions confuse a lot of people. But, actually, most of them are absolute sitters — very easy to solve — a free ride! How? We will just see. Let me begin with the previous post’s question.

Question: If a retailer marks up an article by 40% and then offers a discount of 10%, what is his percentage profit?

Let us say the retailer buys the article for $100 ($100 is his cost price of the item). He marks it up by 40% i.e. increases his cost price by 40% (100 * 140/100) and puts a tag of $140 on the article. Now, the article remains unsold and he puts it on sale – 10% off everything. So the article marked at $140, gets $14 off and sells at $126 (because 140 * 9/10 = 126). This $126 is the selling price of the article. To re-cap, we obtained this selling price in the following way:

\(Cost \ Price * (1 + Mark \ Up\%) * (1 – Discount\%) = 100 * (1 + \frac{40}{100}) * (1 – \frac{10}{100}) = 126 = Selling \ Price\)

The profit made on the item is $26 (obtained by subtracting 100, the retaile’s cost price, from 126, the retailer’s selling price).

He got a profit % of \((\frac{26}{100}) * 100 = 26\%\) (Profit/Cost Price x 100)

Or we can say that \(Cost \ Price * (1 + Profit\%) = 100 * (1 + \frac{26}{100}) = 126 = Selling \ Price\)

The italicized parts above show the two ways in which you can reach the selling price: using mark-up and discount or using profit. The same thing is depicted in the diagram below:



Therefore, \(Cost \ Price * (1 + Mark \ Up\%) * (1 – Discount\%)= Cost \ Price * (1 + Profit\%)\)

Or

\((1 + Mark \ Up\%) * (1 – Discount\%)= (1 + Profit\%)\)

Look at the numbers here: Mark Up: 40%, Discount: 10%, Profit: 26% (Not 30% that we might expect because 40% – 10% = 30%)

Why? Because the discount offered was on $140, not on $100. So a bigger amount of $14 was reduced from the price. Hence the profit decreased. This leads us to an extremely important question in percentages – What is the base? 100 was increased by 40% but the new number 140 was decreased by 10%. So in the two cases, the bases were different. Hence, you cannot simple subtract 10 from 40 and hope to get the Profit %. Also, mind you, almost certainly, 30% will be one of the answer choices, albeit incorrect. (The GMAT doesn’t forego even the smallest opportunity of tricking you into making a mistake!)

Let’s see this concept in action on a tricky third party question:

A dealer offers a cash discount of 20%. Further, a customer bargains and receives 20 articles for the price of 15 articles. The dealer still makes a profit of 20%. How much percent above the cost price were his articles marked?

a) 100%
b) 80%
c) 75%
d) 66+2/3%
e) 50%

This question involves two discounts:
1. the straight 20% off
2. discount offered by selling 20 articles for the price of 15 – a discount of cost price of 5 articles on 20 articles i.e. a discount of 5/20 = 25%

Using the formula given above:

\((1 + \frac{m}{100})(1 – \frac{20}{100})(1 – \frac{25}{100}) = (1 + \frac{20}{100})\)
m = 100

Therefore, the mark up was 100%.

Answer (A) This question is discussed HERE.

Note: The two discounts are successive percentage discounts.

Another application of successive percentage changes is the concept of compounding. But more on that, in the next post.

Compound Interest Problems from our Special Questions Directory.

Theory on Percent and Interest Problems

DS Percent and Interest Problems
PS Percent and Interest Problems

is this This is R% of principal + R% of total interest accumulated in Year2

This is 275 + R% of (275 + 275 + R% of 275) = 176

because as per the formula, I + R%of(I+I+R%of I)

Dear Bunuel, I also do not understand why we need R % of 275 for the first year and not just 275 as supposed according to the formula? Plus, I also think that 176 must include the interests compounded from the first, second, and third years like it is done in the formula, and if they are actually included then I do not understand why it is mentioned like "R% of total interest accumulated in Year2" and not like "R% of total interest accumulated in Year3" . Could you please elaborate on this. Thanks!

avinash R1 Gayluk its R% of 275(275 is interest earned at the end of year 1) + R% of interest accumulated in year 2{which is
275(interest earned in year 1) + 275(interest yearned in year 2 on principle) + R% of 275 ( here R% of 275 is interest earned on the year 1 interest since its compounded)}

Gayluk

I was also confused by the equation so I decided to make an overview of the interest earned each year with the compound interest rate
1 Year --> 275
2 Year --> 550 + 275*X
3 Year --> 825 + 275*X + (550+275*X)*X

We know that 1001 should be total of compound interest earned, so:

825 + 275X + (550+275X)*X = 1101

This simplifies to:
--> 275X^2 + 825X - 176 = 0

In the case of yearly compound interest, the ratio of amounts at the end of the 20th year to the amount at the end of the 22nd year is 0.81. What is the rate of interest?
A=initial amount
R=interest rate

{A*(1+R/100)^20}/{A*(1+R/100)^22}=0.81
1/(1+R/100)^2=0.81
1/(1+R/100)=0.9 (since we are talking about a money is increasing, we can ignore -0.9)
100/(100+R)=9/10
(100+R)/100=1+R/100=10/9=1+1/9
R/100=1/9
R=11.1%

is my calculation right?

Posted from my mobile device

Could someone please confirm if this is the correct methodology for this question?

I would do it this way by applying logic:

Let the amount at the end of 20th year is P

Amount at the end of 22nd year would be P *(1+r/100)*(1+r/100)

Ratio of these two is given as 0.81 = 81/100

(1+r/100)^2 = (10/9)^2

(1+r/100) = 10/9

r/100 = 10/9-1 = 1/9

Therefore r = 100/9 = 11.11%

Posted from my mobile device

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