Skip to main content

Number System Tutorial I: Integers, Fractions, Prime & Composite


How to identify if a fraction gives terminating decimal or recurring decimal

If the simplified form of a fraction consist the prime factors 2 or 5 only in its denominator, then it is terminating decimal.
OR
If the denominator of the simplified form of any fraction consist at least one factor other than 2 or 5 then the fraction gives a recurring decimal.

Which among the following fractions is/are recurring decimal?
371128371128
403160403160
3721102437211024
8431312584313125
3731431137314311

When observing the prime factors of the denominators in each fraction, it is easy to identify that the denominator of option E is a multiple of 3 and its numerator is not a multiple of 3. Therefore the prime factor 3 of the denominator should be there even after the simplification ( if it is possible).

Hence answer is 3731431137314311

We can check the prime factors of the rest of the denominators.

128=27128=27
160=25⋅5160=25⋅5
1024=2101024=210
3125=553125=55
Here all of these denominators have the prime factors 2 or 5 only.

Method of conversion

Find the corresponding fraction of 0.5¯
Let x=0.5¯Equation(1)
(1)1010x=5.5¯Equation(2)
(2)-(1)9x=5
x=59
Therefore 0.5¯=59
Find the corresponding fraction of 0.25¯
Let x=0.25¯Equation(1)
(1)100100x=25.25¯Equation(2)
(2)-(1)99x=25
x=2599
Therefore 0.25¯=2599'

Easy approach in converting Type I recurring decimal to fraction

0.ab¯=repeating group (here ab)as many 9's as the number of digits in repeating group

Examples in converting recurring decimal to fraction

0.13¯=1399
0.251¯=251999
0.1234¯=12349999

Type II group of recurring decimals

In this type the repeating digit/group of digits starts after a digit/ some digits of non-repeating digits. Eg. 0.16¯(0.1666...),0.038¯(0.03888...),0.4513¯(0.45131313...)

Method of conversion

Find the corresponding fraction of 0.16¯
Let x=0.16¯Equation(1)
Take the non-repeating digit to the left side (integral portion) of the decimal point.
(1)1010x=1.6¯Equation(2)
Take on repeating group/digit to the left side of the decimal point.
(2)10100x=16.6¯Equation(3)
(3)-(2)90x=15
x=1590=16
Therefore 0.16¯=16
Find the corresponding fraction of 0.12345¯
Let x=0.12345¯Equation(1)
Take the non-repeating digit to the left side (integral portion) of the decimal point.
(1)100100x=12.345¯Equation(2)
Take on repeating group/digit to the left side of the decimal point.
100000x=12345.345¯Equation(3)
(3)-(2)99900x=(12345-12)=12333
x=1233399900
Therefore 0.12345¯=1233399900

Easy approach in converting Type II recurring decimal to fraction

0.abcde¯=("entire decimal group" - "non-repeating decimal group")/"as many 9's as the number of repeating digits in the decimal part with as many 0's as the number of non-repeating digits in the decimal part"

Examples in converting Type II recurring decimal to fraction

0.12345¯=12345-1299900=1233399900
0.845¯=845-8990=837990

More examples

Find the corresponding fraction of 3.23¯
3.23¯=3+0.23¯
3+23-190=3+2190=3+730
=9730
Find 0.37¯+0.45¯+0.16¯
0.37¯=3490
0.45¯=4190
0.16¯=1590
0.37¯+0.45¯+0.16¯=3490+4190+1590=9090=1

Fractions

Any number which can be expressed in the form p/q, where p and q are Natural Numbers is called a fraction. Example: 12,32,5 etc. There are three types of fractions.

Proper Fractions

If the numerator of a fraction is lesser than the denominator, then it is a proper fraction. Hence the value of a proper fraction should lie in between 0 and 1. Eg. 13,25,613 etc

Improper fractions

If the numerator of a fraction is greater than or equal to its denominator then it is an improper fraction. Hence the value of any improper fraction is greater than or equal to 1. Eg. 32,107,5 etc

Mixed Fractions

Basically a mixed fraction is another expression of a corresponding improper fraction. Example: 54 is an improper fraction. I can be expressed in the following manner too.
54=4+14=1+14=114, Here 1 is the natural number part and 14 is the proper fractional part.

Comparison of fractions

Comparison of different types of fractions is a basic requirement in Data Interpretation. Some direct questions from the comparison concept also can be expected in your exam. Hence you must be familiar with different methods for the comparison of fractions.

Fraction comparison using cross multiplication

Let's look at example to see how to compare two fractions quickly using cross multiplication.

Which is greater, 37 or 716 ?

For the comparison of any two fractions, it is quite easy to apply the cross multiplication method. Here, multiply the numerator of the first fraction with the denominator of the second fraction. This product is representing the first fraction. ie. 3×16=48

Similarly multiply the numerator of the second fraction with the denominator of the first fraction and this product is representing the second fraction. ie.7×7=49

Here 49>48, therefore 716>37.
Pages
1
2
3

Comments

Popular posts from this blog

IBPS Clerk 2020 Notification Out for 1557 Vacancies

IBPS (Institute of Banking Personnel Selection) conducts a common recruitment process (CRP) every year to the recruitment of clerical cadre in multiple banks all over the country. All public sector banks use CRP as a base to fill the vacancies for this post. The Recruitment Notification for IBPS Clerk 2020 exam has been released on 1st September 2020. IBPS is conducting Clerk exam for the 10th year now and hence named IBPS Clerk CRP X. The IBPS Clerk CRP exam is conducted on two levels- preliminary exam and mains. Students qualifying in both these exams are thus selected for the post. Here, we are discussing about the exam notification, application process, salary, syllabus, exam pattern, vacancy, eligibility criteria, selection process and other details of the exam. IBPS Clerk 2020 Notification IBPS has released the Official Notification for IBPS Clerk 2020 Exam on the 1st day of September 2020 to recruit 1557+ vacancies of clerical posts in 11 public sector banks. The Exam dates have...

Short Tricks on Number System

Number system is an important topic for upcoming SSC & Railways Exams. Here, we are going to help you with Basic Concepts & Short Tricks on Number System in Quant Section. We will be providing you with details of the topic to make the Quant Section and calculation easier for you all to understand. We hope you all will like the post. Important Formulas of Number System Formulas of Number Series 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2 (12 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6 (13 + 23 + 33 + ..... + n3) = (n(n + 1)/ 2)2 Sum of first n odd numbers = n2 Sum of first n even numbers = n (n + 1)Mathematical Formulas (a + b)(a - b) = (a2 - b2) (a + b)2 = (a2 + b2 + 2ab) (a - b)2 = (a2 + b2 - 2ab) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) (a3 + b3) = (a + b)(a2 - ab + b2) (a3 - b3) = (a - b)(a2 + ab + b2) (a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac) When a + b + c = 0, then a3 + b3 + c3 = 3abc (a + b)n = an + (nC1)an-1b + (nC2)an-2b2 + … + (nCn-1)abn...

10 Math Games in 10 Minutes or Less

  Math games bring out kids’ natural love of numbers. As students transition into the new school year, help them sharpen their number skills with some of these fun and effective games. 5 Minutes 1. Simon Says, “Geometry!” Ramp up this traditional game by having kids illustrate the following geometric terms using only their arms: parallel and perpendicular lines; acute, right, and obtuse angles; and 0-, 90-, and 180-degree angles. Challenge: Increase the pace of the commands and see if your students can keep up! 2. Round the Block Have students stand in a square. Give one of them a ball and a math challenge that requires a list of responses, such as counting by twos or naming shapes that have right angles. Before the student answers, he passes the ball to the person next to him. Children pass the ball around the square as quickly as they can, and the student must give the answer before the ball comes back to him. Challenge: When the correct answer is given, the child who has the bal...