Mathematics for Competitive Exams : Study All Topics in Detail

  In this blog, we will elaborate on the key concepts of Maths for competitive exams along with helpful tips and tricks. 

General Syllabus of Maths in Competitive Exams

Here is a list of the topics which are commonly found in the Quantitative Aptitude section of Maths for Competitive Exams.

1. Percentage
2. Decimals
3. Factors and Multiples
4. Ratios
5. Geometry
6. Integers
7. Number System
8. Simplification
9. HCF and LCM
10. Ratio and Proportion
11. Problems on Age
12. Partnership
13. BODMAS
14. Average
15. Profit and Loss
16. Simple Interest
17. Compound Interest
18. Mensuration
19. Time and Work
20. Time and Distance
21. Square Root
22. Trigonometry
23. Approximation
24. Mixture and Alligation
25. Permutation and Combination
26. Boats and Streams
27. Surds and Indices
28. Pipes and Cisterns

For details study click on the above link.

Some Important questions

1. A sum borrowed under com­pound interest doubles itself in 10 years. When will it become fourfold of itself at the same rate of interest?

A. 15 years B. 20 years

C. 24 years D. 40 years

2. Find the square root of 6241 using the square root tricks.

3. A man took a loan from a bank at the rate of 12% per annum at simple interest. After 3 year he had to pay Rs. 5400 as interest only for that period. The principal amount borrowed by him was: 

(a) Rs. 2000 (b) Rs. 10,000

(c) Rs. 20,000. (d) Rs. 15,000

4. A sum of money at simple interest amounts to 1,012 in \(2\frac{1}{2}\) year and to Rs. 1067.20 in 4 years. The rate of interest per annum is : 

(a) 2.5% (b) 3%. (c) 4% (d) 5% 

5. A dishonest dealer sells goods at a 10% loss on cost price but uses 20% less weight. Compute profit or loss percentage. 

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Profit and Loss | Concept, formula, tricks and example

Introduction:- 

        Profit and Loss is an important role in competitive exams. There are questions based on profit & loss topics in almost all the competitive exams.

Here are some important statements and definitions related to profit and loss concepts. 

Cost Price (CP): 

          The price at which goods are bought is called the cost price. 

Selling Price (SP): 

          The price at which goods are sold is called the selling price.

Profit:

           When the selling price is more than the cost price, the trader makes a profit that is equal to (SP - CP).

Loss:

           When the selling price is less than the cost price, the trader makes a loss equal to (CP - SP).

Profit Percentage:  

           The value of profit, when expressed as a percent of the cost price (CP), is called profit percent.  so, Profit percent = \(\frac{(SP-CP)}{CP} \times 100\) .

Loss Percentage: 

         Loss, when expressed as a percentage of cost price, is called loss percentage. So, Loss percentage = \(\frac{(CP-SP)}{CP} \times 100\).

Marked price:

          Marked price is the price that is marked on the product.

Discount:

       If the seller can decide to give discounts to the buyer and that case selling price might be different from the marked price.

So, Discount = (Marked price) - (Selling price).

Then discount percentage = \(\frac{Discount}{Marked \ price}\times100\).

Some Important Formula and Tricks:-

A). Selling price calculation

     1).If there is a profit of P %, and Cost Price = C (given).

Then selling price (SP) = \(\frac{(100+P)}{100}\times C \).

     2). If there is a loss of L %, and Cost Price = C (given).

Then selling price (SP) =\(\frac{(100-L)}{100}\times C \).

B).  Cost price calculation
1). If there is a profit of P %, And Sale price= SP (given).
Then cost price (C ) = \(\frac{100}{(100+P)}\times SP \).
2). If there is a loss of L %,
Then cost price (C) = \(\frac{100}{(100-L)}\times SP \).

C). If the price of an item increases by r% , then the reduction in consumption so that expenditure remains the same ,is

\(\frac{r}{100+r} \times 100\%\).

D). If the price of an item decreases by r% then increase in consumption , so

expenditure on this item is same, is 

\(\frac{r}{100-r} \times 100\%\).

E). A reduction of x% in price enables a person to buy y kg more for Rs. A. Then the original price = \(\frac{x}{(100-x)\times y} \times A \).

And the reduction price = \(\frac{x}{100y} \times A \).

F). When there are two successive profits of x% and y% then the net profit percentage =\([x+y+\frac{xy}{100}]\).

G). When there is a profit of x% and loss of y% then net profit or loss percentage 

= \([x – y – \frac{xy}{100}]\).

If negative sign occurs then loss comes, and if positive sign occurs then profit comes.

H). If A sells goods to B at a profit of x% and B sells it to C at a profit of y%. If C pays Rs P for it,then the cost price for A is 

\(\frac{100\times 100\times P}{(100+x)(100+y)}\).

I). If A sells goods to B at a loss of x% and B sells it to C at a loss of y%. If C pays Rs P for it,then the cost price for A is 

\(\frac{100\times 100\times P}{(100-x)(100-y)}\).

J). When each of the two things is sold at the same price,and a profit of p% is made on the first and a loss of L% is made on the second,then the percentage gain or loss is  

= \(\frac{100(P-L)-2PL}{(100+P)+(100-L)}\).

Note:- If profit percentage and loss percentage are equal, put P=L

Then, loss percentage= \(\frac{P^2}{100}\).

Example:-

1. A man buys a fan for Rs. 1000 and sells it at a loss of 15%. What is the selling price of the fan? 

2. If a pen cost Rs.50 after 10% discount, then what is the actual price or marked price (MP) of the pen? 

3. A dishonest dealer sells goods at a 10% loss on cost price but uses 20% less weight. Compute profit or loss percentage.

4. If selling price is doubled, the profit triples. Find the profit percent. 

5. A man buys a fan for Rs. 1000 and sells it at a loss of 15%. What is the selling price of the fan?

6. If a pen cost Rs.50 after 10% discount, then what is the actual price or marked price of the pen? 

7. A person buys a horse for 15 pounds. After one year, he sells it for 20 pounds. After one year, again he buys the same horse at 30 pounds and sells it for 40 pounds. What is the overall profit percent for that person over both the transactions? 

8. A trader sells 85 m of cloth for Rs. 8,925 at the profit of Rs. 15/m of cloth. What is the cost price of 1 m of cloth? 

9. The marked price of a shirt is Rs.1000. A shopkeeper offers 30% discount on this shirt and then again offers a 20% discount on the new price. How much will you have to pay, finally? 

How to find the Square Root of a number

    One such trick that will help you solve many numerical problems is finding the square root. As in competitive examinations often calculation based questions are asked. Here in this article, you will know how to find the square root of a number.

Definition of Square Root

The square root of a number is the number that on squaring results the given number. It is represented using this symbol "√". The square root of a number can be a rational number or an irrational number. If the square root of a number is a whole number, then it is a perfect square.

Tricks to Find Square Root

The tricks to find the square root of perfect squares is very easy to remember. We need to keep in mind the unit place digit of squares of numbers from 1 to 10. So, we find out the unit digit....

The unit digit of square number of 1 is 1.

The unit digit of square number of 2 is 4.

The unit digit of square number of 3 is 9.

The unit digit of square number of 4 is 6.

The unit digit of square number of 5 is 5.

The unit digit of square number of 6 is 6.

The unit digit of square number of 7 is 9.

The unit digit of square number of 8 is 4.

The unit digit of square number of 9 is 1.

The unit digit of square number of 10 is 0.

Hence, from the above we can figure out if any perfect square ends with the above digits at the unit place, then its square root will have the same respective number at the unit place.

For example, the square root of 49 is 7.

Square root Tricks for 3-digit Numbers

The square root of a three-digit number is always a two-digit number. Let us learn to find the square root with an example.

Example:-

Square root of 196: 

Pair the digits from the right-hand side: 1 and 96.

The unit place of 196 is 6. So, either the square root will have 4 or 6 at the unit place

Now, considering the first digit, 1, it is the square of 1. Thus, the first digit of the square root of 196 is 1.

Since 1 is the smallest number of the square. Hence, the square root of 196 will take a smaller number between 4 and 6. Thus, it will be 4.

Hence, the required square root is 14.

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Square root Tricks for 4-digit Numbers

      If the number is three-digit or four-digit, then it is difficult to find the root of these numbers, because we cannot remember the table for higher numbers. Let us find out the trick to determine the root of large numbers.

Example 1: 

 Suppose we need to find the square root of large numbers such as 3589.

Step 1: The unit digit in this number is 9, which can be a unit digit of its square root number such as 3 or 7.

Step 2: Now let us consider the first two digits that is 35 which comes between the squares of 5 and 6 because 25 < 35< 36.

Step 3: We can assume that the ten’s digit of the square root of 3589 is the lowest among the two numbers 5 and 6 is 5, and we need to find the unit digit of the square root of the number 3589.

Step 4: Now, we need to find between 53 or 57 which is the square root of 3589.

Step 5: Since the ten’s digit is 5 and the next number is 6, we need to multiply both the numbers like 5 x 6 = 30 and since 30 is less than 35.

Step 6: Square root of 3589 will be the bigger number between 53 and 57 i.e. 57.

Therefore, \(\sqrt{3589}\) = 57.

Example 2:

 Let us have a look at another example, the square root of 7056.

Here is the step by step method:

Now, consider the unit digit that is 6. Which all numbers have the unit digit 6 on their square roots. That are 4 and 6.

Now let’s consider the first two digits that is 70 which comes between the squares of 8 and 9 because of 64< 70 <81

We can assume that the ten’s digit of the square root of the 7056 is the lowest among the two numbers that is 8

Now, we need to find the unit digit of the square root of the number 7056. For that, we need between 84 and 86 which is the square root of 7056

Since the ten’s digit is 8 and the next number is 9, we need to multiply both the numbers like 8 x 9 = 72 and since 72 is bigger than 70

Square root of 7056 will be the lesser number between 84 and 86 that is 84

Therefore, \(\sqrt{7056} = 84\).

Square Root of 5-Digit Numbers

To find the square root of a 5-digit number, follow the below steps with the help of an example.

Example: Find the square root of 40401.

Step 1: Pair the digits from right to left, 404 and 01.

Step 2: Check the unit digit of the number and compare it with the above table. Here, the unit digit is 1, thus possible unit digits for the square root of 40401 is 1 and 9. But the square root of 1 is 1, thus unit digit will have 1.

Step 3: Now, the first three digits are 404. It will lie between 400 < 404 < 441. Hence, the square root of first three digit will be near to 20.

Step 4: Hence, the required square root is 201

Note:- similar way we can find out the square root of 6 digit number and so on.

Example:- 

1. Find the square root of 6241 using the square root tricks.

2. Find the square root of 735305  using the square root tricks.

Compound Interest for all Competitive Exam

 

Introduction:-

    "Compound Interest" is the one of the most important topic in quantitative aptitude. In "competitive exams" the most important thing is time management, if you know how to manage your time then you can do well in competitive exams. So here we are providing "shortcut tricks" on Compound Interest for your help.

First we know the following terms:- 

Principal (P): 

         The original sum of money loaned or deposited. Also known as capital.

Interest (I): 

         The amount of money that you pay to borrow money or the amount of money that you earn on a deposit.

Time (T): 

          The duration for which the money is borrowed or deposited. The duration does not necessarily have to be years. The duration can be semi-annual, quarterly or any which may be deemed fit.

Rate of Interest (R): 

            The percent of interest that you pay for money borrowed, or earn for money deposited.

         Read Simple Interest on click here

Compound Interest:- 

    Compound interest is an interest when you earn interest on both the money you have saved and the interest you earn.

Formula:- 

A). If the principal is P, time is t and rate of interest per annum is R% , also amount is A.

Then we get, 

1. Amount A = P\([1+\frac{R}{100}]^t\).  When interest compounded annually.

And Compound Interest over the time t is 

CI= A - P = P\([(1+\frac{R}{100})^t-1]\).

2. If interest compounded semi- annually, Then the amount is A= P\([1+\frac{R}{2×100}]^{2t}\). 

3. If interest compounded quarterly, Then the amount is A= P\([1+\frac{R}{4×100}]^{4t}\). 

4. If interest is compounded annually and time is in fraction say 2 \(\frac{3}{5}\) years.

Then amount = P\([1+\frac{R}{100}]^2[1+\frac{\frac{3}{5} R}{100}]\).

B).  If population of a city is P and it increases by R % annually, then population after n years is given by: = P\([1+\frac{R}{100}]^t\).

C).  If population of a city is P and it decreases by R % annually, then the population after n years is given by: 

=  P\([1-\frac{R}{100}]^t\).

D). When the rates of interest are different for different years, say R₁, R₂, R₃ percent for the first, second and third year, respectively, then 

Amount= P\([1+\frac{R_1}{100}][1+\frac{R_2}{100}][1+\frac{R_3}{100}]\).

E).  If difference between compound interest and simple interest is given for:

1) Two years

C.I. – S.I. = P\([\frac{R}{100}]^2\).

2) Three years

C.I. – S.I. =P\([\frac{R}{100}]^2[\frac{ (300 + R) }{100}]\).

F). A sum of money placed at compound interest becomes x time in ‘a’ years and y times in ‘b’ years. These two sums can be related by the following formula: 

\([x]^{\frac{1}{a}}=[y]^{\frac{1}{b}}\).

G). If an amount of money grows up to Rs x in t years and up to Rs y in (t+1) years on compound interest, then 

\(R\%=\frac{(y-x)\times 100}{x}\).

H). A sum at a rate of interest compounded yearly becomes Rs. \(A_1\) in n years and Rs. \(A_2\) in (n + 1) years, then 

\(P=A_1(\frac{A_1}{A_2})^n\).

I). If a certain sum becomes x times of itself in t years, the rate of compound interest will be equal to 

\(r=100[x^{\frac{1}{t}}-1]\).

J). If the compound interest on a certain sum for two years is CI and simple interest for two years is SI ,then rate of interest per annum is

\(r\%=2(\frac{CI-SI}{SI})\times 100\).

Some Important Example:- 

 1. If the amount is 2.25 times of the sum after 2 years at com­pound interest (compound annu­ally) , the rate of interest per an­num is :

A. 25%   B. 30%     C. 45%      D. 50%

2.  A sum borrowed under com­pound interest doubles itself in 10 years. When will it become fourfold of itself at the same rate of interest?

A. 15 years               B. 20 years

C. 24 years               D. 40 years

3. Bharat took a loan of Rs. 20000 to purchase one LCD TV set from a finance company. He promised to make the payment after three years. The company charges compound interest at the rate of 10% per annum for the same. But, suddenly the company announces the rate of interest as 15% per annum for the last one year of the loan period. What extra amount does Rohit have to pay due to this announcement of the new rate of interest?

A) Rs. 7830       B) Rs. 4410

C) Rs. 6620       D) None of these

4. The difference between C.I. and S.I. on a certain sum at 10 % per annum for 2 years is Rs. 530. Find the sum.

A). 53000     B). 57500    C). 69800   D). 28090 

5. Rs. 39030 is divided between ‘a’ and ‘b’ in such a way that the amount given to ‘a’ on C.I. in 7 years is equal to the amount given to ‘b’ on C.I. in 9 years. Find the part of ‘a’. If the rate of interest is 4%.

(A) 20200.  (B) 20900.  (C) 20280.  (D) 20100

6. A sum of Rs. 2000 amounts to Rs. 4000 in two years at compound interest. In how many years does the same amount become Rs. 8000.

(A) 2.    (B) 4.    (C) 6.      (D) 8 

7. The compound interest on a certain sum at \(\frac{50}{3}\% \) for 3 years is Rs. 127. Find simple interest on same sum for same period and rate.

A). Rs. 205        B). Rs. 175    

C). Rs. 152         D). Rs. 108

8. A sum of money is accumulating at compound interest at a certain rate of interest. If simple interest instead of compound were reckoned, the interest for first two years would be diminished by Rs 20 and that for the first three years by 61. Find the sum.

A). Rs 7000          B). Rs 47405

C). Rs 45305         D). Rs 8000

9. The difference between C.I. and S.I. accrued on an amount of Rs. 20,000 in 2 years was Rs. 392. Find the rate of interest per annum.

A). 11.5 %    B). 13 %     C). 14 %     D). 12 %

10. There is 60% increase in an amount in 6 years at simple interest. What will be the compound interest of Rs. 12,000 after 3 years at the same rate?

A). Rs. 2160         B). Rs. 3120

C). Rs. 3972          D). Rs. 6240

11. What is the difference between the compound interests on Rs. 5000 for \(1\frac{1}{2}\) years at 4% per annum compounded yearly and half-yearly?

A). Rs. 2.04           B). Rs. 3.06

C). Rs. 4.80            D). Rs. 8.30

12.  If the simple interest on a sum of money for 2 years at 5% per annum is Rs. 50, what is the compound interest on the same at the same rate and for the same time?

A). Rs. 51.25            B). Rs. 52  

C). Rs. 54.25            D). Rs. 60

13. The compound interest on a certain sum for 2 years at 10% per annum is Rs. 525. The simple interest on the same sum for double the time at half the rate percent per annum is:

A). Rs. 400            B). Rs. 500    

C). Rs. 600            D). Rs. 800

14. The amount on ₹ 25000 in 2 years at annually compound interest. if the rate for the successive years be 4 % and 5 % per annum respectively is 

A) ₹ 28500             B) ₹ 30000

C) ₹ 26800              D) ₹ 27300

15. A certain amount of money earns ₹ 540 as simple interest in 3 years. If it earns compound interest of ₹ 376.20 at the same rate of interest in 2 years, find the principal (in Rupees). 

A) 2100     B) 2000     C) 1600     D) 1800