In general, this is the easiest chapter. It takes a basic understanding of percentages, ratios, and proportions, as well as the ability to do simple maths. If a question is ever asked about this chapter, it's not a hard one. But this chapter is important for most competitive exams, including the section on Data Interpretation.
SI and CI questions play a very important role in CAT exams. Practicing questions on SI and CI will enable you to save and devote time to other time consuming questions.
Before diving into the SI and CI questions, let us begin with basic definitions and subsequently to the formulae.
Important Definitions
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Simple interest: It is just a fixed percentage of the amount of money that was invested or borrowed. Some of the most important words used in the idea of interest.
Principal (P): It is the amount of money that has been deposited, loaned, etc. It is also called capital.
Interest: It is the money that the borrower has to pay back, and it is calculated based on the principal. Time: This is how long the money is lent or borrowed for.
Rate of Interest (r/R): This is the amount of interest that is added to the principal.
Amount (A): It is the loan balance plus interest
Simple interest is when the interest is calculated the same way for the whole time period only on the principal.
Compound Interest: In this case, interest is added to the total amount of the previous period, which is the sum of the principal and the interest that has been added to it so far. This means that each time we calculate an increase in the previous amount, we add interest to the total amount of the previous period.
Simple Interest
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If we are given the Principal (P), the Rate of Interest (r), and the Time Period (t), we will have
Simple Interest (SI) = \(\frac{Prt}{100}\)
Amount (A) = P + \(\frac{Prt}{100} = P (1+ \frac{rt}{100})\)
Conversion of Time Period and rate of Interest
Compound Interest (CI)
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- CI = A – P
A is the amount including interest and principal (P ) both
- Amount
A = \(P(1+\frac{r}{100})^t\)
- When the rate of interest is half-yearly
A = \(P(1+ \frac{r/2}{100})^2t\)
- When the rate of interest is quarterly
- Difference between CI and SI for two years = \(p(\frac{r}{100})^2\)
- Difference between CI and SI for three years = \(p(\frac{r}{100})^2\ (\frac{r}{100} + 3)\)
- Difference between CI and SI for n th year
- For compound interest, if r denotes the rate of interest, the change in amount over the previous year can be calculated or following.
Similarly,
Depreciation
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It is common knowledge that the prices of certain items depreciate in value over time. When the value of an item decreases in terms of currency, we say that its value is depreciating.
Where,
Vi = Initial value of the article
Vf = Final (depreciated) value of article
r = is the rate of interest by which the price of article decreases over the time period‘t’.
Calculating the Population
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It has been observed that the population of a specific locale/nation/etc. increases or decreases successively over its previous value, i.e., it increases or decreases like compound interest for money. Consequently, we use comparable formulas to calculate population.
When Population increases ,
When Population decreases,
Key concepts: This article taught us how to find the difference between SI and CI when we know the principal amount, the time period, and the rate percent. The formulas are used right away in the questions.
Previous Year Questions
Ques: Anil invests some money at a fixed rate of interest, compounded annually. If the interests accrued during the second and third year are Rs. 806.25 and & Rs.866.72, respectively, the interest accrued, in INR, during the fourth year is nearest to __ [CAT 2021]
- 929.48
- 934.65
- 931.72
- 926.84
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Ans: Option (C)
Let the principal amount be P and the interest rate be r.
Then P(1 + r)2- P(1+r) = 806.25 --------(1)
P(1+r)3- P(1+r)2= 866.72 -------(2)
Dividing (2) by (1), we get:
(P(1+r)3 - P(1+r)2)/(P(1+r)2 – P(1+r)) = 806.72/806.25
((1+r)2 - (1+r))/((1+r) – 1) = 1.075
(r2+r)/r = 1.075
r = 0.075 or 7.5%
(Interest accrued in 4th yr)/(Interest accrued in 3rd yr) = X/866.72
(P(1+r)4- P(1+r)3)/(P(1+r)3- P(1+r)2)=X/866.72
Dividing numerator and denominator by P(1 + r)2
(r2 + 2r +1 – (1+ r))/(1+ r - 1)=X/866.72
X= 1.075 x 866.72 = 931.72
Ques: Raj invested Rs.10000 in a fund. At the end of first year, he incurred a loss but his balance was more than Rs.5000. This balance, when invested for another year, grew and the percentage of growth in the second year was five times the percentage of loss in the first year. If the gain of Raj from the initial investment over the two year period is 35%, then the percentage of loss in the first year is __ [CAT 2021]
- 5
- 15
- 17
- 10
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Ans: Option (D)
In the first year, Raj invested $10,000. Assuming his expected loss was x%.
After one year, the quantity will be 10,000 times (1 - x/100). = 10000 - 100*x.
Given that the remaining balance was greater than Rs 5000, x is less than 50 percent.
When Raj invested this amount in the second year, he earned five times the return of the first year.
The amount after two years is therefore: (10000 - 100x) (1 + (5 - x)/100.
Over the course of two years, Raj's income increased by a total of 35%, which is equal to Rs 13,500.
Consequently, the total quantity is Rs 13,500.
(10000 - 100x)(1 + (5 - x)/100) = 13,500
(100 – 5.x)(100 – x)= 13,500
10000 – 100*x +500*x – 5x2 = 13500
5x2 – 400x + 3500=0
Solving the equation the roots are:
x = 10. x= 70.
Since x < 50, x = 10 percent.
Ques: Bank A offers 6% interest rate per annum compounded half-yearly. Bank B and Bank Coffer simple interest but the annual interest rate offered by Bank C is twice that of Bank B. Raju invests a certain amount in Bank B for a certain period and Rupa invests 7 10,000 in Bank C for twice that period. The interest that would accrue to Raju during that period is equal to the interest that would have accrued had he invested the same amount in Bank A for one year. The interest accrued,in INR. to Rupa is __ [CAT 2021]
- 3436
- 2436
- 2346
- 1436
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Ans: Option (B)
Bank A: 6% p.a. half yearly (CI)
Bank B: x% p.a (at Simple Interest)
Bank C: 2x% p.a (at Simple Interest)
Let Raju invest Rs P in bank B for t years. Hence, Rupa invests Rs 10,000 in bank C for 2t years.
Now.
P (x/100) t = P (1 + 3/100)2 - P
(x/100) t = 1.0609 - 1
(x/100) t = 0.0609
We need to calculate
SI = 10000 x 2t x (2x/100)= 40000 (x/100) t
= 40000 x 0.0609 = Rs.2436
Ques: Veeru invested Rs 10000 at 5% simple annual interest, and exactly after two years, Joy invested Rs 8000 at 10% simple annual interest. How many years after Veer's investment, will their balances, i.e., principal plus accumulated interest, be equal? [CAT 2020]
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Ans: Let both sums add up to the same amount after n years.
Then.
1000(1+ 5 (n+2)/100)=800(1+10n/100)
ie.1.5 = 15n/100
→ n=10
Hence 12 years after veeru invested their balances will be equal.
Ques: For the same principal amount, the compound interest for two years at 5% per annum exceeds the simple interest for three years at 3% per annum by Rs1125. Then the principal amount in rupees is __ [CAT 2020]
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Ans: Let P be the principal amount
According to the question,
(P.(1+ 5/100)2- P) - ((Px3x3)/100)=1125
=> P(0.1025-0.09) = 1125
=> P= 90,000
Ques: A person invested a certain amount of money at 10% annual interest, compounded half-yearly. After one and a half years, the interest and principal together became Rs 18522. The amount, in rupees, that the person had invested is __ [CAT 2020]
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Ans: Let Sum equal to P
According to the question,
P(1+10/200)3 = 18522
→ P=18522x(20/21)3=16000
Ques: John borrowed Rs. 210000 from a bank at an interest rate of 10% per annum, compounded annually. The loan was repaid in two equal instalments, the first after one year and the second after another year. The first instalment was interest of one year plus part of the principal amount, while the second was the rest of the principle amount plus due interest thereon. Then each instalment, in , is __ [CAT 2018]
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Ans: Principal = Rs.210000
Rate of interest = 10%
No of Instalments = 2
According to the question,
Principal = [x/(1+r/100) + x/(1+r/100)2]
2,10,000=[x/(1+10/100) + x/(1+10/100)2]
2,10,000=10x/11+100x/121
2,10,000=(110x+100x)/121
(2,10,000)(121)=210x
X=1,21,000
So, each instalment= Rs. 1,21,000
Ques: In the Garbar Jhala, Aminabad a shopkeeper first raises the price of a Jewellery by x% then he decreases the new price by x%. After one such up down cycle, the price of a Jewellery decreased by Rs. 21025. After a second updown cycle the jewellery was sold for Rs. 484416. What was the original price of the jewellery. [CAT 2018]
- Rs. 500000
- Rs. 600625
- Rs. 525625
- Rs. 526000
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Ans: Option (C)
Let P be the original price.
Decrease in value of P after one cycle = P(x/100)2 =21025 -------(i)
Again the final value after second cycle
P(1+x/100)(1 - x/100)(1+x/100)(1 - x/100)=484416
P(1- (x/100)2)=484416------- (ii)
Dividing Eq. (ii) by Eq (i), we get
[1- (x/100)2]2/(x/100)2= 484416/21025=2304/100
1- (x/100)2/(x/100)2= Sq.root(2304/100)=48/10
Let x/100=k, then (1- k)2/k =48/10
10k2-48k-10=0
5k2-k-10=0
k=5 or k= -(1/5)
x=20%
Hence, P(x/100)2 =21025
P = 525625
Ques: Alex invested his savings in two parts. The simple interest earned on the first part at 15% per annum for 4 years is the same as the simple interest earned on the second part at 12% per annum for 3 years. Then, the percentage of his savings invested in the first part is __ [CAT 2022]
- 40%
- 62.5%
- 60%
- 37.5%
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Ans: Option (D)
Solution:-
Let the amount invested in first and second parts is F and S respectively.
(Fx15x4)/100 =(Sx12x3)/100
F/S = 3/5
-% amount invested in first part = 3/(3+5)x100%= 37.5%
Ques: Pinto invests one-fifth of his capital at 6%, one-third at 10% and the remaining at 1%, each rate being simple interest per annum. Then, the minimum number of years required for the cumulative interest income from these investments to equal or exceed his initial capital is __ [CAT 2022]
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Ans: Let the total investment be Rs. 1500.
300 is invested at 6% = Interest/year = Rs. 18
500 is invested at 10% = Interest/year = Rs. 50
700 is invested at 1% = Interest/year = Rs. 7
Total interest received/year = 18 + 50 + 7 = Rs. 75
Time required to receive Rs. 1500 as interest = 1500/75 = 20 years.
Ques: Nitu has an initial capital of Rs. 20,000. Out of this, she invests Rs. 8,000 at 5.5% in bank A, Rs.5,000 at 5.6% in bank B and the remaining amount at x% in bank C, each rate being simple interest per annum. Her combined annual interest income from these investments is equal to 5% of the initial capital. If she had invested her entire initial capital in bank C alone, then her annual interest income, in rupees, would have been __ [CAT 2022]
- 800
- 700
- 900
- 1000
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Ans: Option (A)
When Rs. 8000 is invested at 5.5%, it earns Rs. 440 in interest each year.
When Rs. 5,000 is invested at 5.6%, it earns Rs. 280 in interest each year.
When Rs. 7,000 is invested at x%, it earns Rs. 7,000 x % in interest each year.
Overall, 20,000 is invested which earns 5% yearly interest = 5% of 20000 = 1000
440 + 280 + 70x = 1000
70x = 1000 - 720 = 280
x = 4%
: If she had put all of her money in bank C from the start, her annual interest income would be 4% of $20,000, or Rs. 800.
Ques: Amala, Bina, and Gouri invest money in the ratio 3 : 4 : 5 in fixed deposits having respective annual interest rates in the ratio 6 : 5 : 4. What is their total interest income (in Rs) after a vear, if Bina's interest income exceeds Amala's by Rs 250? [CAT 2019]
- 7000
- 7250
- 6350
- 6000
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Ans: Option (B)
Let the amount invested by Amala, Bina and Gouri be 3x, 4x and 5x respectivelv.
Also, let the respective rates be 6r. 5r and 4r.
Respective ratio of simple interest is (3x x 6r) : (4x x 5r) : (5x x 4r) = 18xr : 20xr :20xr
Bina's interest income exceeds Amala's by Rs 250.
20xr - 18xr = 250.
.: xr = 125.
Total interest income = 18xr + 20xr + 20xr = 58xr = 58 x 125 = Rs. 7250.
Ques: A person invested a total amount of Rs 15 lakh. A part of it was invested in a fixed deposit earning 6% annual interest, and the remaining amount was invested in two other deposits in the ratio 2 : 1, earning annual interest at the rates of 4% and 3%, respectively. If the total annual interest income is Rs 76,000 then the amount (in Rs lakh) invested in the fixed deposit was__ [CAT 2019]
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Ans: Since the amount invested is in the ratio 2 : 1, we can assume the amount to be 2x and x respectively.
Amount invested in fixed deposit = 15L - 3x (where L is lakhs)
Simple interest earned on fixed deposit = [(15L - 3x) x (6/100) x 1] ------ (1)
Simple interest earned on 2x principle = 2x. (4/100) x 1 ------ (2)
Simple interest earned on x principle = x.(3/100) x 1 ------- (3)
(1) + (2) + (3) = 76000
Solving we get; x = 2L.
So, amount invested in fixed deposit = 15L - 3x = 15L - 6L = 9L.
Ques: Amal invests Rs 12000 at 8% interest, compounded annually, and Rs 10000 at 6% interest, compounded semi-annually, both investments being for one year. Bimal invests his money at 7.5% simple interest for one year. If Amal and Bimal get the same amount of interest, then the amount, in Rupees, invested by Bimal is __ [CAT 2019]
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Ans: First investment of Anmol = 12,000 x (1.08) = 12.960
Interest on his investment is 12,960 – 12,000 = 960.
The amount on the second investment of Anmol = 10,000 x (1.03)2 = 10,609
So the Interest on this investment is 10,609 - 10,000 = 609
So the total interest on these returns = 960 + 609 = 1,569.
Bimal has to get this as Simple Interest by investing X rupees at 7.5%
That means. X x 0.075 = 1,569
X = 20,920
So, Bimal has to invest 20,920 rupees.
Ques: Gopal borrows Rs. X from Ankit at 8% annual interest. He then adds Rs. Y of his own money and lends Rs. X+Y to Ishan at 10% annual interest. At the end of the year, after returning Ankit's dues, the net interest retained by Gopal is the same as that accrued to Ankit. On the other hand, had Gopal lent Rs. X+2Y to Ishan at 10%, then the net interest retained by him would have increased by Rs. 150. If all interests are compounded annually, then find the value of X + Y. [CAT 2019]
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Ans: The only difference between the two situations is that Gopal gave Ishan Rs.Y more in the second situation than he did in the first.
So, the extra interest that Gopal kept is equal to 0.1Y.
Therefore 0.1.Y = 150 or Y = 1500.
Now, Ankit lent Gopal Rs. X at 8% interest.
Therefore the interest retained by Ankit = Rs.0.08X.
Gopal lent Ishan Rs. (X + 1500) at 10% interest.
The interest retained by Gopal = 0.1X + 150 - 0.08X = 0.02X + 150
08X = 0.02X + 1500 or X = 2500.
X + Y = 4000.
How to Prepare SI and CI Questions for CAT?
- Go through the formulae thoroughly provided in the previous section of the article.
- The intricate points are the use of per annum and half yearly interest rates during the calculation of Simple interest and compound interest.
- Keeping track of time while solving will increase the speed of solving and eventually help you save time during the exams.
- Calculation of Depreciation and installments are often asked during exams and students get confused while solving.
- Practicing without calculators will give you an edge while interpreting the rate of interest during attempting mocks.
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