How to reduce (simplify) fraction 312/213 to lowest terms? Write it as an improper fraction, a mixed number, a decimal number, a percentage

Reduce: 312/213

Detailed calculations below:

Introduction. Fractions

A fraction consists of two numbers and a fraction bar: 312/213


The number above the bar is the numerator: 312


The number below the bar is the denominator: 213


The fraction bar means that the two numbers are dividing themselves:
312/213 = 312 ÷ 213


Divide the numerator by the denominator to get fraction's value:
Value = 312 ÷ 213


Introduction. Percent

'Percent (%)' means 'out of one hundred':


p% = p 'out of one hundred',


p% = p/100 = p ÷ 100


Note:

The fraction 100/100 = 100 ÷ 100 = 100% = 1


Multiply a number by the fraction 100/100,
... and its value doesn't change.



To reduce a fraction, divide both its numerator and denominator by their greatest common factor, GCF.

Integer numbers prime factorization:

Prime Factorization of a number: finding the prime numbers that multiply together to make that number.


312 = 23 × 3 × 13;
312 is not a prime, is a composite number;


213 = 3 × 71;
213 is not a prime, is a composite number;


* Positive integers that are only dividing by themselves and 1 are called prime numbers. A prime number has only two factors: 1 and itself.
* A composite number is a positive integer that has at least one factor (divisor) other than 1 and itself.


Calculate the greatest (highest) common factor (divisor), gcf, hcf, gcd:

Multiply all the common prime factors, by the lowest exponents (if any).


gcf, hcf, gcd (312; 213) = 3



Divide both the numerator and the denominator by their greatest common factor.

312/213 =


(23 × 3 × 13)/(3 × 71) =


((23 × 3 × 13) ÷ 3) / ((3 × 71) ÷ 3) =


(23 × 13)/71 =


104/71


The fraction is now reduced to the lowest terms.

104/71 is an improper fraction.

An improper fraction: numerator larger than denominator.


Rewrite the end result, continued below...

Rewrite the fraction:

As a mixed number (mixed fraction):

A mixed number (mixed fraction): a whole number and a proper fraction, of the same sign.


A proper fraction: numerator smaller than denominator.


104 ÷ 71 = 1 and remainder = 33 =>


104 = 1 × 71 + 33 =>


104/71 =


(1 × 71 + 33) / 71 =


(1 × 71) / 71 + 33 / 71 =


1 + 33/71 =


33/71


As a decimal number:

33/71 =


1 + 33/71 =


1 + 33 ÷ 71 ≈


1.464788732394


1.46


As a percentage:

1.464788732394 =


1.464788732394 × 100/100 =


146.478873239437/100 =


146.478873239437% ≈


146.48%


In other words:

1) Calculate fraction's value.


2) Multiply that number by 100.


3) Add the percent sign % to it.



Final answer
continued below...

Final answer:
:: written in four ways ::

As an improper fraction
(numerator larger than denominator):
312/213 = 104/71

As a mixed number (mixed fraction)
(a whole number and a proper fraction, of the same sign):
312/213 = 33/71

As a decimal number:
312/2131.4647887323941.46

As a percentage:
312/213146.48%

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Tutoring: simplifying ordinary math fractions (reducing to the lowest terms)

Steps to simplify an ordinary fraction, to reduce it to its lowest terms:

  • 1) Factor both the numerator and the denominator of the fraction into prime factors.
  • 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.
  • 3) Divide both the numerator and the denominator of the fraction by the greatest common factor, GCF (GCD).
  • In conclusion: the fraction thus obtained is called a reduced fraction or a fraction simplified to its lowest terms.
  • A fraction reduced to its lowest terms cannot be further reduced and it is called an irreducible fraction.

Read the full article >> Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples


Why reducing (simplifying) fractions to lower terms?

  • When running operations with fractions we are often required to bring them to the same denominator, for example when adding, subtracting or comparing.
  • Sometimes both the numerators and the denominators of those fractions are large numbers and doing calculations with such numbers could be difficult.
  • By simplifying (reducing) a fraction, both the numerator and denominator of a fraction are reduced to smaller values. Well, these values are much easier to work with, reducing the overall effort of working with fractions.

What is a prime number?

What is a composite number?

Prime numbers up to 1,000

Prime numbers up to 10,000

Sieve of Eratosthenes

Euclid's algorithm

Simplifying ordinary (common) math fractions (reducing to lower terms): steps to follow and examples