Simplification
Simplification is one of the most important part of Quantitative Aptitude section of any competitive exam. Today I am sharing all the techniques to solve Simplification questions quickly.
Rules of Simplification
V → Vinculum
B → Remove Brackets – in the order ( ) , { }, [ ]
O → Of
D → Division
M → Multiplication
A → Addition
S → Subtraction
| Types | Description |
| Natural Numbers: | all counting numbers ( 1,2,3,4,5….∞) |
| Whole Numbers: | natural number + zero( 0,1,2,3,4,5…∞) |
| Integers: | All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞) |
| Even & Odd Numbers : | All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞) |
| Prime Numbers: | It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞) |
| Composite Numbers: | Natural numbers which are not prime |
| Co-Prime: | Two natural number a and b are said to be co-prime if their HCF is 1. |
Divisibility
| Numbers | IF A Number | Examples |
| Divisible by 2 | End with 0,2,4,6,8 are divisible by 2 | 254,326,3546,4718 all are divisible by 2 |
| Divisible by 3 | Sum of its digits is divisible by 3 | 375,4251,78123 all are divisible by 3. [549=5+4+9][5+4+9=18]18 is divisible by 3 hence 549 is divisible by 3. |
| Divisible by 4 | Last two digit divisible by 4 | 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4. |
| Divisible by 5 | Ends with 0 or 5 | 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5. |
| Divisible by 6 | Divides by Both 2 & 3 | 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6. |
| Divisible by 8 | Last 3 digit divide by 8 | 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8. |
| Divisible by 10 | End with 0 | 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10. |
| Divisible by 11 | [Sum of its digit in odd places-Sum of its digits in even places]= 0 or multiple of 11 | Consider the number 39798847 (Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3) (23-12) 23-12=11, which is divisible by 11. So 39798847 is divisible by 11. |
Division & Remainder Rules
Suppose we divide 45 by 6
hence ,represent it as:
dividend = ( divisor✘quotient ) + remainder
or
divisior= [(dividend)-(remainder] / quotient
could be write it as
x = kq + r where (x = dividend,k = divisor,q = quotient,r = remainder)
Rules
- Modulus of a Real Number:
Modulus of a real number a is defined as
| |a| = | a, if a > 0 | |
| –a, if a < 0 |
Thus, |5| = 5 and |-5| = -(-5) = 5.
- Virnaculum (or Bar):
When an expression contains Virnaculum, before applying the ‘BODMAS’ rule, we simplify the expression under the Virnaculum.
Example:
On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder ?
Number = 342k + 47
( 18 ✘19k ) + ( 18 ✘2 ) + 11
18 ✘( 19k + 2 ) +11.
Remainder = 11
Sum Rules
(1+2+3+………+n) = 1/2 n(n+1)
(12+22+32+………+n2) = 1/6 n (n+1) (2n+1)
(13+23+33+………+n3) = 1/4 n2 (n+1)2
Questions:
Level-I:
| 1. | A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee notes. The number of notes of each denomination is equal. What is the total number of notes that he has ? | |||||||
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| 2. | There are two examinations rooms A and B. If 10 students are sent from A to B, then the number of students in each room is the same. If 20 candidates are sent from B to A, then the number of students in A is double the number of students in B. The number of students in room A is: | |||||||
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| 3. The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables together is Rs. 4000. The total price of 12 chairs and 3 tables is: | ||||||||
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| 4. If a – b = 3 and a2 + b2 = 29, find the value of ab. | ||||||||
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| 5. | The price of 2 sarees and 4 shirts is Rs. 1600. With the same Money one can buy 1 saree and 6 shirts. If one wants to buy 12 shirts, how much shall he have to pay ? | |||||||||
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| 6. | A sum of Rs. 1360 has been divided among A, B and C such that A gets of what B gets and B gets of what C gets. B’s share is: | |||||||
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| 7. | One-third of Rahul’s Savings in National Savings Certificate is equal to one-half of his savings in Public Provident Fund. If he has Rs. 1,50,000 as total savings, how much has he saved in Public Provident Fund ? | |||||||
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| 8. | A fires 5 shots to B’s 3 but A kills only once in 3 shots while B kills once in 2 shots. When B has missed 27 times, A has killed: | |||||||
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| 9. | Eight people are planning to share equally the cost of a rental car. If one person withdraws from the arrangement and the others share equally the entire cost of the car, then the share of each of the remaining persons increased by: | |||||||||||||||
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| 10. | To fill a tank, 25 buckets of water is required. How many buckets of water will be required to fill the same tank if the capacity of the bucket is reduced to two-fifth of its present ? | |||||||||
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Level-II:
| 1. | In a regular week, there are 5 working days and for each day, the working hours are 8. A man gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns Rs. 432 in 4 weeks, then how many hours does he work for ? | |||||||
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| 12. | Free notebooks were distributed equally among children of a class. The number of notebooks each child got was one-eighth of the number of children. Had the number of children been half, each child would have got 16 notebooks. Total how many notebooks were distributed ? | |||||||||
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| 13. | A man has some hens and cows. If the number of heads be 48 and the number of feet equals 140, then the number of hens will be: | |||||||
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| 14. |
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| 15. | David gets on the elevator at the 11th floor of a building and rides up at the rate of 57 floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the same building and rides down at the rate of 63 floors per minute. If they continue travelling at these rates, then at which floor will their paths cross ? | |||||||
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- Find the value of 1/(3+1/(3+1/(3-1/3)))
| A.) 3/10 | B.) 10/3 |
| C.) 27/89 | D.) 89/27 |
Find the value of
| A.) 3½ 99; | B.) 34/99 |
| C.) 2.131313 | D.) 3.141414 |
18.Find the value of
((0.1)3 + (0.6)3 + (0.7)3 − (0.3)(0.6)(0.7))/((0.1)2 + (0.6)2 + (0.7)2 − 0.006 − 0.42 − 0.07)
| A.) 14/10 | B.) 1.35 |
| C.) 13/10 | D.) 0 |
- Solve(0.76 × 0.76 × 0.76 − 0.008)/(0.76 × 0.76 + 0.76 × 0.2 + 0.04)
| A.) 0.56 | B.) 0.65 |
| C.) 0.54 | D.) 0.45 |
- Find the value of
| A.) 1.5 | B.) -1.5 |
| C.) 1 | D.) 0 |
Answers:
Level-I
Answer:1 Option D
Explanation:
Let number of notes of each denomination be x.
Then x + 5x + 10x = 480
16x = 480
x = 30.
Hence, total number of notes = 3x = 90.
Answer:2 Option C
Explanation:
Let the number of students in rooms A and B be x and y respectively.
Then, x – 10 = y + 10 x – y = 20 …. (i)
and x + 20 = 2(y – 20) x – 2y = -60 …. (ii)
Solving (i) and (ii) we get: x = 100 , y = 80.
The required answer A = 100.
Answer:3 Option D
Explanation:
Let the cost of a chair and that of a table be Rs. x and Rs. y respectively.
| Then, 10x = 4y or y = | 5 | x. |
| 2 |
15x + 2y = 4000
| 15x + 2 x | 5 | x = 4000 |
| 2 |
20x = 4000
x = 200.
| So, y = | 5 | x 200 | = 500. | ||
| 2 |
Hence, the cost of 12 chairs and 3 tables = 12x + 3y
= Rs. (2400 + 1500)
= Rs. 3900.
Answer:4 Option A
Explanation:
2ab = (a2 + b2) – (a – b)2
= 29 – 9 = 20
ab = 10.
Answer:5 Option B
Explanation:
Let the price of a saree and a shirt be Rs. x and Rs. y respectively.
Then, 2x + 4y = 1600 …. (i)
and x + 6y = 1600 …. (ii)
Divide equation (i) by 2, we get the below equation.
=> x + 2y = 800. — (iii)
Now subtract (iii) from (ii)
x + 6y = 1600 (-)
x + 2y = 800
—————-
4y = 800
—————-
Therefore, y = 200.
Now apply value of y in (iii)
=> x + 2 x 200 = 800
=> x + 400 = 800
Therefore x = 400
Solving (i) and (ii) we get x = 400, y = 200.
Cost of 12 shirts = Rs. (12 x 200) = Rs. 2400.
Answer:6 Option C
Explanation:
Let C’s share = Rs. x
| Then, B’s share = Rs. | x | , A’s share = Rs. | 2 | x | x | = Rs. | x | ||
| 4 | 3 | 4 | 6 |
| x | + | x | + x = 1360 | |
| 6 | 4 |
| 17x | = 1360 | |
| 12 |
| x = | 1360 x 12 | = Rs. 960 |
| 17 |
| Hence, B’s share = Rs. | 960 | = Rs. 240. | ||
Answer:7 Option C
Explanation:
Let savings in N.S.C and P.P.F. be Rs. x and Rs. (150000 – x) respectively. Then,
| 1 | x = | 1 | (150000 – x) |
| 3 | 2 |
| x | + | x | = 75000 | |
| 3 | 2 |
| 5x | = 75000 | |
| 6 |
| x = | 75000 x 6 | = 90000 |
| 5 |
Savings in Public Provident Fund = Rs. (150000 – 90000) = Rs. 60000
Answer:8 Option A
Explanation:
Let the total number of shots be x. Then,
| Shots fired by A = | 5 | x |
| 8 |
| Shots fired by B = | 3 | x |
| 8 |
| Killing shots by A = | 1 | of | 5 | x | = | 5 | x |
| 3 | 8 | 24 |
| Shots missed by B = | 1 | of | 3 | x | = | 3 | x |
| 2 | 8 | 16 |
| 3x | = 27 or x = | 27 x 16 | = 144. | |||
| 16 | 3 |
| Birds killed by A = | 5x | = | 5 | x 144 | = 30. | ||
| 24 | 24 |
Answer:9 Option A
Explanation:
| Original share of 1 person = | 1 |
| 8 |
| New share of 1 person = | 1 |
| 7 |
| Increase = | 1 | – | 1 | = | 1 | ||
| 7 | 8 | 56 |
| Required fraction = | (1/56) | = | 1 | x | 8 | = | 1 | ||
| (1/8) | 56 | 1 | 7 |
Answer:10 Option C
Explanation:
Let the capacity of 1 bucket = x.
Then, the capacity of tank = 25x.
| New capacity of bucket = | 2 | x |
| 5 |
| Required number of buckets = | 25x |
| (2x/5) |
| = | 25x | x | 5 | |
| 2x |
| = | 125 |
| 2 |
= 62.5
Level-II:
Answer:11 Option B
Explanation:
Suppose the man works overtime for x hours.
Now, working hours in 4 weeks = (5 x 8 x 4) = 160.
160 x 2.40 + x x 3.20 = 432
3.20x = 432 – 384 = 48
x = 15.
Hence, total hours of work = (160 + 15) = 175.
Answer:12 Option C
Explanation:
Let total number of children be x.
| Then, x x | 1 | x = | x | x 16 x = 64. |
| 8 | 2 |
| Number of notebooks = | 1 | x2 = | 1 | x 64 x 64 | = 512 | ||
Answer:13 Option D
Explanation:
Let the number of hens be x and the number of cows be y.
Then, x + y = 48 …. (i)
and 2x + 4y = 140 x + 2y = 70 …. (ii)
Solving (i) and (ii) we get: x = 26, y = 22.
The required answer = 26.
Answer:14 Option B
Explanation:
| Given exp. = | (a + b)2 – (a – b)2 |
| ab |
| = | 4ab |
| ab |
= 4 (where a = 469, b = 174.)
Answer:15 Option C
Explanation:
Suppose their paths cross after x minutes.
Then, 11 + 57x = 51 – 63x 120x = 40
| x = | 1 |
| 3 |
| Number of floors covered by David in (1/3) min. = | 1 | x 57 | = 19. | ||
| 3 |
So, their paths cross at (11 +19) i.e., 30th floor.
Answer:16 Option ‘C’
Explanation:
1/[3 + (1/(3+1/(3 – 1/3)))]
=> 1/[3 + 1/(3 + 1/(8/3))]
=> 1/[3 + 1/(3 + 3/8)]
=> 1/[3 + 8/27]
=> 1/(89/27)
=> 27/89
Answer:17 Option ‘D’
Explanation:
6/9 + 7/9 + 9/9 + 69/99
2/3 + 7/9 + 1 + 69/99
(66 + 77 + 99 + 69)/99
311/99 => 3.141414
Answer:18 Option ‘A’
Explanation:
((0.1)3 + (0.6)3 + (0.7)3 − (0.3)(0.6)(0.7))/((0.1)2 + (0.6)2 + (0.7)2 − 0.006 − 0.42 − 0.07)
=> (0.1 + 0.6 + 0.7)3/(0.1 + 0.6 + 0.7)2
=> 0.1 + 0.6 + 0.7 => 1.4 = 14/10
Answer:19 Option ‘A’
Answer:20 Option ‘D’
11/30 − [1/6 + 1/5 + [7/12 − 7/12]]
11/30 − [1/6 + 1/5 + [0]]
11/30 − [(5 + 6)/30]
11/30 − 11/30 = 0.,
Simplification is the act of making something simpler or easier to understand. It can be applied to a wide range of topics, from complex mathematical equations to everyday tasks.
There are many different ways to simplify something. One common approach is to break it down into smaller, more manageable pieces. This can be done by identifying the key Elements of the problem and then focusing on each one individually. Another approach is to look for ways to eliminate unnecessary details. This can be done by asking yourself whether each detail is essential to understanding the overall concept.
Simplification can be a powerful tool for improving Communication and understanding. By making information more concise and easy to grasp, you can help people to better understand your ideas and your work.
Here are some specific examples of how simplification can be used in different contexts:
- In mathematics, simplification can be used to solve complex equations. For example, the equation $x^2+y^2=1$ can be simplified to $x=y=\pm\sqrt{1}$.
- In everyday life, simplification can be used to make tasks easier to complete. For example, you can simplify the task of cooking dinner by using pre-made ingredients or by following a simple recipe.
- In Software engineering, simplification can be used to create more user-friendly interfaces. For example, you can simplify a website by using clear and concise language, and by providing helpful links and instructions.
Simplification can be a valuable tool for improving communication and understanding. By making information more concise and easy to grasp, you can help people to better understand your ideas and your work.
Here are some additional tips for simplifying information:
- Use plain language. Avoid using jargon or technical terms that your audience may not understand.
- Use visuals. Images and diagrams can be a great way to simplify complex concepts.
- Be concise. Get to the point quickly and avoid unnecessary details.
- Use examples. Examples can help people to understand abstract concepts.
- Be organized. Use headings, subheadings, and bullet points to make your information easy to follow.
- Proofread your work. Make sure that your information is accurate and free of errors.
Simplification is an important skill that can be applied to a wide range of topics. By following the tips above, you can improve your ability to simplify information and make it more accessible to your audience.
What is the difference between a summary and a simplification?
A summary is a brief statement or account of the main points of something. A simplification is a version of something that has been made simpler or easier to understand.
What are the benefits of simplification?
Simplification can make information easier to understand, remember, and use. It can also help to reduce Stress and anxiety.
What are the challenges of simplification?
Simplification can sometimes lead to loss of detail or accuracy. It can also be difficult to simplify complex information without making it too simplistic.
How can I simplify information?
There are a number of ways to simplify information. One way is to break it down into smaller chunks. Another way is to use plain language and avoid jargon. You can also use visuals, such as charts and graphs, to help illustrate your points.
What are some examples of simplification?
Some examples of simplification include:
- Writing a summary of a book.
- Creating a simplified version of a complex instruction manual.
- Developing a training program that is easy to understand and follow.
What are some common mistakes to avoid when simplifying information?
Some common mistakes to avoid when simplifying information include:
- Leaving out important details.
- Using jargon or technical language.
- Making the information too simplistic.
What are some Resources for Learning more about simplification?
There are a number of resources available for learning more about simplification. Some of these resources include:
- Books on simplification, such as “The Art of Clear Thinking” by Rolf Dobelli.
- Websites on simplification, such as the website of the Plain Language Association International.
- Courses on simplification, such as the course “Simplification: The Art of Making Complex Information Easy to Understand” offered by the University of California, Berkeley.
Sure, here are some MCQs without mentioning the topic SIMPLIFICATION:
What is the value of $2^3$?
(A) 8
(B) 16
(C) 32
(D) 64What is the value of $3^2$?
(A) 9
(B) 16
(C) 25
(D) 36What is the value of $4^2$?
(A) 16
(B) 36
(C) 64
(D) 100What is the value of $5^2$?
(A) 25
(B) 50
(C) 125
(D) 225What is the value of $6^2$?
(A) 36
(B) 72
(C) 108
(D) 144What is the value of $7^2$?
(A) 49
(B) 98
(C) 147
(D) 196What is the value of $8^2$?
(A) 64
(B) 128
(C) 192
(D) 256What is the value of $9^2$?
(A) 81
(B) 162
(C) 243
(D) 324What is the value of $10^2$?
(A) 100
(B) 200
(C) 300
(D) 400What is the value of $11^2$?
(A) 121
(B) 132
(C) 143
(D) 154What is the value of $12^2$?
(A) 144
(B) 168
(C) 192
(D) 216What is the value of $13^2$?
(A) 169
(B) 196
(C) 225
(D) 256What is the value of $14^2$?
(A) 196
(B) 225
(C) 256
(D) 289What is the value of $15^2$?
(A) 225
(B) 256
(C) 289
(D) 324What is the value of $16^2$?
(A) 256
(B) 289
(C) 324
(D) 361What is the value of $17^2$?
(A) 289
(B) 324
(C) 361
(D) 396What is the value of $18^2$?
(A) 324
(B) 361
(C) 396
(D) 435What is the value of $19^2$?
(A) 361
(B) 396
(C) 435
(D) 476What is the value of $20^2$?
(A) 400
(B) 441
(C) 484
(D) 529What is the value of $21^2$?
(A) 441
(B) 484
(C) 529
(D) 576