How to completely reduce (simplify) the fraction 340/9 to the lowest terms (its simplest form, the smallest possible numerator and denominator)? Write the result as an improper fraction, as a mixed number, as a decimal number and as a percentage
Reduce (simplify) the fraction: 340/9
Detailed calculations below:
Fractions, brief introduction
A fraction consists of two numbers and the fraction bar: 340/9
The number above the fraction bar is the numerator: 340
The number under the fraction bar is the denominator: 9
The fraction bar means that the two numbers divide themselves:
340/9 = 340 ÷ 9
Divide the numerator by the denominator to get the value of the fraction:
The value = 340 ÷ 9
Percentages, short introduction
'Percent, p%' means 'p 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 will not change.
To completely reduce (simplify) a fraction to its lowest terms, divide the numerator and the denominator of the fraction by their greatest (highest) common factor (divisor), gcf (hcf, gcd)
To calculate the greatest (highest) common factor (divisor), gcf (hcf, gcd), we perform the prime factorization of the two numbers.
The prime factorization of the two numbers:
The prime factorization of a number: finding the prime numbers that multiply together to make that number.
340 = 22 × 5 × 17
340 is not a prime number but a composite one.
9 = 32
9 is not a prime number but a composite one.
* The natural numbers that are only divisible by 1 and themselves are called prime numbers. A prime number has exactly two factors: 1 and itself.
* A composite number is a natural number that has at least one other factor than 1 and itself.
Calculate the greatest (highest) common factor (divisor), gcf (hcf, gcd):
Multiply all the common prime factors, taken by their smallest exponents (powers).
But the two numbers have no common prime factors:
gcf, hcf, gcd (340; 9) = 1
Coprime numbers (prime to each other, relatively prime)
The fraction cannot be reduced (simplified)
The numerator and denominator of the fraction are relatively prime numbers (coprime, prime to each other).
The only common factor (divisor) of the two numbers is 1.
340/9 is an improper fraction.
An improper fraction: the numerator is greater than or equal to the denominator.
Rewrite the fraction:
As a mixed number (mixed fraction):
A mixed number: an integer and a proper fraction that have the same sign.
The original improper fraction is obtained by adding the integer number and the proper fraction.
A proper fraction: the numerator is smaller than the denominator.
340 ÷ 9 = 37 and Remainder = 7 =>
340 = 37 × 9 + 7 =>
340/9 =
(37 × 9 + 7) / 9 =
(37 × 9) / 9 + 7 / 9 =
37 + 7/9 =
37 7/9
As a decimal number:
Divide the numerator of the fraction by its denominator.
37 7/9 =
37 + 7/9 =
37 + 7 ÷ 9 ≈
37.777777777778 ≈
37.78
As a percentage:
Multiply the fraction's value by the fraction 100/100
37.777777777778 =
37.777777777778 × 100/100 =
3,777.777777777778/100 =
3,777.777777777778% ≈
3,777.78%
In other words:
1) Calculate the value of the fraction.
2) Multiply this number by 100.
3) Add the percent sign, %.
The final answer
This continues below...
The final answer:
:: Written in four ways ::
As an improper fraction:
340/9 = 340/9
As a mixed number (mixed fraction):
340/9 = 37 7/9
As a decimal number:
340/9 ≈ 37.777777777778 ≈ 37.78
As a percentage:
340/9 ≈ 3,777.78%
Other similar operations with fractions reducing (simplification):
The latest 5 fractions that have been fully reduced (simplified) to their lowest terms (to their simplest form, the smallest possible numerator and denominator)
Online Calculator: Completely reduce (simplify) fraction to the lowest terms
To fully reduce (simplify) fractions to the lowest terms:
Divide the numerator and denominator of the fraction by their greatest (highest) common factor (divisor), gcf (hcf, gcd).
Result written as a proper or an improper fraction, as a mixed number, as a decimal or an integer and as a percentage.
Tutoring: simplifying fractions - reducing them to the lowest terms
Steps to simplify a fraction, to reduce it to its lowest terms:
- A fraction fully simplified, a fraction reduced to its lowest terms is a fraction that can no longer be simplified, it has been reduced to its simplest equivalent fraction, the one having the smallest numerator and denominator possible - prime to each other.
- 1) Run the prime factorization of both the numerator and the denominator of the fraction.
- 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 their greatest common factor, GCF (GCD).
- The fraction thus obtained is called a reduced fraction or a fraction simplified to its lowest terms.
- A fraction reduced to its lowest terms may no longer be reduced and it is called an irreducible fraction.
Example: reduce the fraction 315/1,155 as much as possible, simplify it to the lowest terms.
1) Run the prime factorization of both the numerator and the denominator of the fraction.
- The numerator of the fraction is 315, its breaking down into prime factors is:
315 = 3 × 3 × 5 × 7 = 32 × 5 × 7 - The denominator of the fraction is 1,155, its breaking down into prime factors is:
1,155 = 3 × 5 × 7 × 11. 2) Calculate the greatest common factor, GCF (or the greatest common divisor, GCD) of the fraction's numerator and denominator.
- The greatest common factor, gcf (315; 1,155), is calculated by multiplying all the common prime factors of the numerator and the denominator, taken by their lowest powers (their lowest exponents):
- GCF (315; 1,155) = (32 × 5 × 7; 3 × 5 × 7 × 11) = 3 × 5 × 7 = 105
3) Divide both the numerator and the denominator of the fraction by their greatest common factor, GCF (GCD).
- The numerator and denominator of the fraction are divided by their greatest common factor, GCF:
315/1,155 =
(32 × 5 × 7)/(3 × 5 × 7 × 11) =
((32 × 5 × 7) ÷ (3 × 5 × 7)) / ((3 × 5 × 7 × 11) ÷ (3 × 5 × 7)) =
3/11 - The fraction thus obtained is called a fraction reduced to the lowest terms.
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 reducing (simplifying) a fraction, both its numerator and denominator are reduced to smaller values, much easier to work with, which will decrease the overall effort of working with that fraction.
Some articles on the prime numbers