Word Problems
(Simple & Compound Rates)
Many students from business background may know some formulas to calculate simple rate & compound rates. We’ll need to use only those formulas to calculate these rates. First let’s learn what are simple rates and compound rate and the difference between them.
Simple Rate:
Let’s suppose you invest PKR 100 in a bank, that pays you 10% simple annual interest. After one year, your gross amount, which is sum of initial investment and interest earned, would be PKR 110. After another year, your gross amount would be PKR 120 and so on. Notice that the 10% simple rate is always applied on initial investment, which is also know as principle amount, that is PKR 100 here. As 10% of 100 is 10, so we’ll only need to add 10 to our bank balance every year to see the gross amount. This is simple rate. Mathematically, in simple rate scenario,
Gross Amount = npr + p
The gross amount is actually the closing balance or we can say the final balance after changes/adjustments.
And
Interest Amount = npr
The interest amount is the increment amount from initial investment. In other words, interest amount is the different between gross amount and initial investment i.e {interest amount = (npr + p) – p = npr}
where,
n = number of years
p = principle amount or initial investment
r = rate of interest
In simple rate, in order to see the gross amount after two years, as we know
Gross Amount = npr + p
⇒ = 2 × (100) × (0.10) + 100
⇒ = 20 + 100
⇒ = 120 Answer —————– (eq. 1)
Compound Rate:
Let’s suppose again you invest PKR 100 in a bank, that pays you 10% compounded annually. After one year, your gross amount, would be PKR 110. But after another year, your gross amount would not be PKR 120, rather it would add interest amount to be 10% of PKR 110, rather than PKR 100, in second year. So here your gross amount would be PKR 121 after 2 years. This is compound rate. In compound rates, the interest is accumulated annually, semi-annually, quarterly, or even monthly as the case may be. Now, let’s make analysis of these different scenarios in compound rate.
1) Compounded Annually:
2) Compounded Semi-annually:
3) Compounded Quarterly:
4) Compounded Monthly:
Mathematically, in compound rate scenario,
⇒ Gross Amount = 100(1 + 0.10⁄1)2 × 1
⇒ = 100(1.10)2
⇒ = 100 (1.21)
⇒ = 121 Answer —————– (eq. 2)
Let’s take an harder scenario of compound rate.
Suppose, every hour, some bacteria in a container quadruple. If initially there were 1000 bacteria in the container, how many bacteria present after 4 hours?
Solution:
Note that in question, it was nowhere mentioned the word ‘rate’ neither words like ‘simple’ or ‘compound’ exist. Rather you see word quadruple, which means 4 times. In Quantitative Part-3, you have learned the relationship between times and percent change. So we’ll first convert times to percent change, which would be the rate at which bacteria are increasing every hour.
Important Note: Remember that if the rate of population increase or decrease is given, although it doesn’t mentioned whether it’s simple rate or compound rate, it will always be compound rate.
Because for instance initially 2 bacteria were there and rate is 100% hourly, this will give in total 4 bacteria in one hour, 16 bacteria in two hours, 32 bacteria in 3 hours, and so on…
As the rate of population growth is 100% i.e twice, so every next hour the population of bacteria would be double the population in previous hour. In simple, not only the two initial bacteria giving birth but also the other bacteria are doing so to enhance population.
Now, by looking at Quantitative Part-3, you saw that the bacteria population growth rate must be 300% (as it’s 4 times, so 4 times means 300% increase). Now let’s use the formula of compound rate to solve the problem as follows:
As we know,
Using equation of compound rate:
⇒ Gross Amount = 100(1 + 0.10⁄1)2 × 1
⇒ = 100(1.10)2
⇒ = 100 (1.21)
⇒ = 121 Answer
Now, let’s do an important scenario, that usually come in exams.
Due to some sever disease, the population of City L was decreased by 25%. If the average annual growth rate of the population of City L is 10%, minimum how many years would it take for the population of City L to exceed it’s population just before the population victimization by the disease?
Solution:
As I mentioned early that in case of population growth rate, always its a case of compound rate even if not mentioned. So let’s use the compound rate formula to solve this problem.
Firstly, we need to assume a population before the disease attack, and let’s suppose it was 100 (for our ease of calculation). Now, after 25% decline, it must remain 75. Now question is that after minimum how many years would the population of 75 would exceed its population 100, if population growth rate is 10%?
According to the given condition, we have
principle amount = 75
Rate = 10% = 0.10
Number of years = n (that we need to find)
Rate is compounded annually, because it’s mentioned annual rate of 10%
Now, as we know that
Using equation of compound rate:
⇒ 100 < 75(1 + 0.10⁄1)n
⇒100⁄75 < (1.10)n
⇒ 4⁄3 < (1.1)n
⇒ 4 × 1⁄3 < (1.1)n
⇒ 4 × 0.333… < (1.1)n (As 1⁄3 ≈ 0.333…)
⇒ 1.333… < (1.1)n
Now, multiply 1.1 again and again until result exceed 1.333…, and just count the number of times (in which 1.1 is multiplied) at the moment when the result exceed 1.333…
Again multiplying 1.1 without calculator is a challenge, but it’s not hard. You must be smart. You’ll learn how to multiply such things and even tough values quickly and accurately in ‘Arithmetic’ topic later on. Here just we do this calculation.
As you have learn in Study Plan for Beginners, that eliminating decimal point ‘.’ from the decimal ‘1.1’ would require to write 10 in the denominator.
So we can say
1.1 = 11⁄10
Now multiply 11⁄10 again and again with itself such that numerator multiply with numerator and denominator multiply with denominator.
You’ll see after multiplying 3 times, the result would be 1331⁄1000 = 1.331 which is still less than 1.333…; So don’t multiply 1.1 further, as with 4th times multiply, it would definitely exceed 1.333…
Therefore,
⇒ n = 4 years Answer
Difference between Simple & Compound rates:
Now, from (eq. 1) and (eq. 2), you can see that compound rate of return are more beneficial for an investor than simple rate of return.
Not only in profit, but also in case of loss, compound rate always beneficial.
Suppose a loss of 10% with initial investment(i.e principle amount) of PKR 100. After first year, both simple and compound rates would incur PKR 10 as loss. But in second year, simple rate incur same amount of loss (as here rate would be charged on principle amount); while in second year, compound rate would incur PKR 9 loss (as here the rate for second year would be charged on the closing balance of first year which is the beginning balance of second year and that is PKR 90 after deduction of loss PKR 10 incur in first year). Therefore, after two years, simple rate would give remaining balance (i.e gross amount) PKR 80. While compound rate would give gross amount PKR 81.
Important Note: Remember in whole GAT quantitative section, you’ll always ask to solve question with solvable solution without calculator. In simple words, you’ll never encounter some complex values that require calculator to solve.
As this topic is neither difficult, nor comprehensive, so it’s practice is only in full practice tests.