Tips and Tricks to Solve Aptitude

Tips and Tricks to Solve Quantitative Aptitude Test questions on Numbers

Every placement test on quantitative aptitude will contain at least 30% questions on number systems and number series. Aptitude questions on number system form the backbone for placement preparation. You can score easily on quantitative aptitude section if you understand the basics of number system. Since the questions on number systems are simple, importance lies in acquiring the right skills to tackle these problems with speed.
Practicing problems on numbers systems not only helps in improving your speed but also provides a strong base for solving other quantitative aptitude sections like HCF and LCM, averages, percentages, time and speed, pipes and cisterns etc as well. In this tutorial let's look at how to solve number system problems easily and quickly.
Numbers are fun to learn. If you learn the concepts thoroughly you will find that solving aptitude questions on number system is a cake walk for you. There are lot of concepts involved and hence even a simple question might look a bit too complex or trickier to solve.

Important Formulas of Number System

Formulas of Number Series

1. 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
2. (1² + 2² + 3² + ..... + n²) = n ( n + 1 ) (2n + 1) / 6
3. (1³ + 2³ + 3³ + ..... + n³) = (n(n + 1)/ 2)²
4. Sum of first n odd numbers = n²
5. Sum of first n even numbers = n (n + 1)

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Mathematical Formulas

1. (a + b)(a - b) = (a² - b²)
2. (a + b)² = (a² + b² + 2ab)
3. (a - b)² = (a² + b² - 2ab)
4. (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
5. (a³ + b³) = (a + b)(a² - ab + b²)
6. (a³ - b³) = (a - b)(a² + ab + b²)
7. (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
8. When a + b + c = 0, then a³ + b³ + c³ = 3abc
9. (a + b)ⁿ = aⁿ + (ⁿC₁)a^n-1b + (ⁿC₂)a^n-2b² + … + (ⁿC_n-1)ab^n-1 + bⁿ

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Shortcuts for number divisibility check

1. A number is divisible by 2, if its unit's digit is any of 0, 2, 4, 6, 8.
2. A number is divisible by 3, if the sum of its digits is divisible by 3.
3. A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
4. A number is divisible by 5, if its unit's digit is either 0 or 5.
5. A number is divisible by 6, if it is divisible by both 2 and 3.
6. A number is divisible by 8, if the number formed by the last three digits of the given number is divisible by 8.
7. A number is divisible by 9, if the sum of its digits is divisible by 9.
8. A number is divisible by 10, if it ends with 0.
9. A number is divisible by 11, if the difference of the sum of its digits at odd places and the sum of its digits at even places, is either 0 or a number divisible by 11.
10. A number is divisible by 12, if it is divisible by both 4 and 3.
11. A number is divisible by 14, if it is divisible by 2 as well as 7.
12. Two numbers are said to be co-primes if their H.C.F. is 1. To find if a number, say y is divisible by x, find m and n such that m * n = x and m and n are co-prime numbers. If y is divisible by both m and n then it is divisible by x.

Frequently Asked Questions on Number System

• Given a number x, you will be asked to find the largest n digit number divisible by x.
• You will be given with a set of numbers (n1, n2, n3...) and asked to find how many of those numbers are divisible by a specified number x.
• Given a number series, find the sum of n terms, find n^th term etc.
• Find product of two numbers when their sum/difference and sum of their squares is given.
• Find the number when divisibility of its digits with certain numbers is given.
• Find the smallest n digit number divisible by x.
• Which of the given numbers are prime numbers.
• Number x when divided by y gives remainder r, what will be the remainder when x² is divided by y.
• Given relationship between the digits of number, find the number.
• Find result of operations (additions, subtractions, multiplications, divisions etc) on given integers. These integers can be large and the question may look difficult and time consuming. But mostly the question will map onto one of the known algebraic equations given in this first tab.

  Tips to Solve Quantitative Aptitude Test Questions on Pipes and Cisterns

• Quantitative aptitude questions on pipes and cisterns are similar to time and work problems. They are commonly asked questions for bank tests, career recruitment tests, management entrance tests and other competitive exams. Questions on pipes and cisterns can be mixed with questions on mixtures and alligations to increase the difficulty level of questions. Quantitative aptitude questions on pipes and cisterns when mixed with ratios can be a bit tricky. But a strong understanding of the concept behind solving them and practicing as much questions as possible can help you easily solve them.
  In this paper we will have a look at frequently asked top aptitude questions on pipes and cisterns, basic concepts, important formulas and tricks and shortcuts on how to solve pipes and cisterns problems easily and quickly.
  
  Frequently asked aptitude questions on pipes and cisterns include:
• Given time taken by individual pipes to empty/fill up a tank. How much does it take to fill / empty the tank if they work together.
• Given time taken by the each pipe to fill up a tank if they work alone. Also given rate in Liters/minute at which an outlet pipe can empty the tank. Find the capacity of the tank.
• Given time taken by the each pipe to fill up a tank if they work alone. After some time t min one of the pipes is closed and given tank is filled up in x hours. Find time t.
• Given time taken by the each pipe to fill up a tank if they work alone. Both the pipes are opened together and given after t minutes, a pipe is turned off. What is the total time required to fill the tank.
• Given a pipe can fill a tank in x hrs. Because of a leak at the bottom of tank, it takes y hrs to fill up the tank. If the tank is full, how much time will it take to empty the full tank.
• Given a pipe can fill the tank n times faster than another pipe. Given the time taken by them to fill up the tank when they work together. Find the time taken by them to fill up the empty tank if they function individually.
• Given time taken by each pipe to fill up an empty tank if they function separately. One of the pipes is full time functioning while 2 other pipes are open for one hour each alternately. Then find the amount of time taken to fill up the empty tank.

Core Concepts

1. A pipe which fills up the tank is known as inlet.
2. A pipe which empties the tank is known as outlet.
3. A pipe takes x hours to fill up the tank. Then 1/x parts of the tank will be filled in 1 hour.
4. A pipe takes y hours to empty the tank. Then part emptied in 1 hour = 1/y
5. Pipe A can fill a tank n times as fast as another pipe B. This means: If slower pipe B takes x min to fill up the empty tank,
   then faster pipe A takes x/n min to fill up the empty tank. If they operate together, then part of the tank that is filled up in 1 hour is (n + 1)/x

Important Formulas, Shortcuts with Explanation

Scenario 1: A tank has 2 inlet pipes A and B. Pipe A alone can fill up the tank in a hrs. Pipe B alone can fill up the tank in b hrs. How much time will it take to fill up the tank, if both pipes are opened together?

Let V be the volume of tank.
Pipe A can fill V/a parts of tank in 1hr.
Pipe B can fill V/b parts of tank in 1 hr.
If both pipes function together, let c hrs be the time taken to fill up tank.
That means, V/c parts of tank will be filled in 1 hr.
ie; V/a + V/b parts of tank will be filled in 1 hr.
V/a + V/b = V/c

c = ab/(a+b) hrs

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Scenario 2: An inlet pipe takes x hours to fill up the tank. An outlet pipe takes y hours to empty the tank. Then if both pipes are opened

1. If y > x, net part filled up in 1 hr = 1/x – 1/y
2. If x > y, net part emptied in 1 hr = 1/y – 1/x

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Scenario 3: If there are n pipes to a tank which takes p1, p2, p3, p4, .. pn hours to fill up the tank, when operating alone. Then if all pipes are opened together:

Part of the tank that is filled up in 1 hr =
Time taken to fill up the tank =

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Scenario 4: If there are n pipes to a tank which takes p1, p2, p3,p4, .. pn hours to fill up the tank, when operating alone. The tank also has an outlet pipe which takes p0 hours to empty the tank. Then if all pipes are opened together:

Part of the tank that is filled up in 1 hr = [ -ve sign implies emptying the tank]

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Scenario 5: A pipe can fill a tank in x hrs. Because of a leak at the bottom of tank, it takes y hrs to fill up the tank. If the tank is full, how much time will it take to empty the full tank?

Time take to empty the tank = xy / (y – x) hours