- ^{82}/_{31} - ^{33}/_{26} = ? Subtracting ordinary (common) fractions, online calculator, subtraction operation explained in detail. The answer, written in four ways. As a mixed number. As a negative improper fraction (the numerator >= the denominator). As a decimal number. As a percentage.
- ^{82}/_{31} - ^{33}/_{26} = ?
Simplify the operation
Reduce (simplify) the fractions to their lowest terms equivalents:
To fully reduce a fraction, to the lowest terms equivalent: divide the numerator and denominator by their greatest common factor, GCF.
* Why do we reduce (simplify) the fractions?
By reducing the values of the numerators and denominators of fractions, further calculations with these fractions become easier to do.
A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.
* * *
The fraction: - ^{82}/_{31} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors.
Their prime factorization:
82 = 2 × 41
31 is a prime number
GCF (2 × 41; 31) = 1
The fraction: - ^{33}/_{26} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors.
Their prime factorization:
33 = 3 × 11
26 = 2 × 13
GCF (3 × 11; 2 × 13) = 1
Rewrite the improper fractions:
An improper fraction: the value of the numerator is larger than or equal to the value of the denominator.
A proper fraction: the value of the numerator is smaller than the value of the denominator.
Each improper fraction will be rewritten as a whole number and a proper fraction, both having the same sign: divide the numerator by the denominator and write down the quotient and the remainder of the division, as shown below.
Why do we rewrite the improper fractions?
By reducing the value of the numerator of a fraction the calculations are getting easier to perform.
* * *
The fraction: - ^{82}/_{31}
- 82 ÷ 31 = - 2 and the remainder = - 20 => - 82 = - 2 × 31 - 20
- ^{82}/_{31} = ^{( - 2 × 31 - 20)}/_{31} = ^{( - 2 × 31)}/_{31} - ^{20}/_{31} = - 2 - ^{20}/_{31}
The fraction: - ^{33}/_{26}
- 33 ÷ 26 = - 1 and the remainder = - 7 => - 33 = - 1 × 26 - 7
- ^{33}/_{26} = ^{( - 1 × 26 - 7)}/_{26} = ^{( - 1 × 26)}/_{26} - ^{7}/_{26} = - 1 - ^{7}/_{26}
Rewrite the equivalent simplified operation:
- ^{82}/_{31} - ^{33}/_{26} =
- 2 - ^{20}/_{31} - 1 - ^{7}/_{26} =
- 3 - ^{20}/_{31} - ^{7}/_{26}
Perform the operation of calculating the fractions.
To add or subtract fractions we need them to have equal denominators (the same common denominator).
To calculate the fractions' operation we have to:
1) find their common denominator
2) then calculate the expanding number of each fraction
3) then build up their denominators the same by expanding the fractions to equivalent forms, which all have equal denominators (the same denominator)
* The common denominator is nothing else than the least common multiple (LCM) of the denominators of the fractions.
The LCM will be the common denominator of the fractions that we work with.
1) Find the common denominator
Calculate the LCM of the denominators:
The prime factorization of the denominators:
31 is a prime number
26 = 2 × 13
Multiply all the unique prime factors: if there are repeating prime factors we only take them once, and only the ones having the highest exponent (the highest powers).
LCM (31; 26) = 2 × 13 × 31 = 806
2) Calculate the expanding number of each fraction:
Divide the LCM by the denominator of each fraction.
- ^{20}/_{31} : 806 ÷ 31 = (2 × 13 × 31) ÷ 31 = 26
- ^{7}/_{26} : 806 ÷ 26 = (2 × 13 × 31) ÷ (2 × 13) = 31
3) Build up the fractions to the same common denominator:
Expand each fraction: multiply both its numerator and denominator by its corresponding expanding number, calculated at the step 2, above. This way all the fractions will have the same denominator.
Then keep the common denominator and work only with the numerators of the fractions.
- 3 - ^{20}/_{31} - ^{7}/_{26} =
- 3 - ^{(26 × 20)}/_{(26 × 31)} - ^{(31 × 7)}/_{(31 × 26)} =
- 3 - ^{520}/_{806} - ^{217}/_{806} =
- 3 + ^{( - 520 - 217)}/_{806} =
- 3 - ^{737}/_{806}
Fully reduce (simplify) the fraction to its lowest terms equivalent:
To fully reduce a fraction, to the lowest terms equivalent: divide the numerator and denominator by their greatest common factor, GCF.
A fully reduced (simplified) fraction is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.
- ^{737}/_{806} is already reduced to the lowest terms.
The numerator and denominator have no common prime factors.
Their prime factorization:
737 = 11 × 67
806 = 2 × 13 × 31
GCF (11 × 67; 2 × 13 × 31) = 1
Rewrite the intermediate result
As a mixed number (also called a mixed fraction):
A mixed number: a whole number and a proper fraction, both having the same sign.
A proper fraction: the value of the numerator is smaller than the value of the denominator.
- 3 - ^{737}/_{806} = - 3 ^{737}/_{806}
As a negative improper fraction:
(the numerator >= the denominator)
An improper fraction: the value of the numerator is larger than or equal to the value of the denominator.
- 3 - ^{737}/_{806} =
^{( - 3 × 806)}/_{806} - ^{737}/_{806} =
^{( - 3 × 806 - 737)}/_{806} =
- ^{3,155}/_{806}
As a decimal number:
Simply divide the numerator by the denominator, without a remainder, as shown below:
- 3 - ^{737}/_{806} =
- 3 - 737 ÷ 806 ≈
- 3.914392059553 ≈
- 3.91
As a percentage:
A percentage value p% is equal to the fraction: ^{p}/_{100}, for any decimal number p. So, we need to change the form of the number calculated above, to show a denominator of 100.
To do that, multiply the number by the fraction ^{100}/_{100}.
The value of the fraction ^{100}/_{100} = 1, so by multiplying the number by this fraction the result is not changing, only the form.
- 3.914392059553 =
- 3.914392059553 × ^{100}/_{100} =
^{( - 3.914392059553 × 100)}/_{100} =
^{ - 391.439205955335}/_{100} ≈
- 391.439205955335% ≈
- 391.44%
The final answer:
:: written in four ways ::
As a mixed number (also called a mixed fraction):
- ^{82}/_{31} - ^{33}/_{26} = - 3 ^{737}/_{806}
As a negative improper fraction:
(the numerator >= the denominator)
- ^{82}/_{31} - ^{33}/_{26} = - ^{3,155}/_{806}
As a decimal number:
- ^{82}/_{31} - ^{33}/_{26} ≈ - 3.91
As a percentage:
- ^{82}/_{31} - ^{33}/_{26} ≈ - 391.44%
How are the numbers written: comma ',' used as a thousands separator; point '.' as a decimal separator; numbers rounded off to max. 12 decimals (if the case). The symbols used: '/' fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.
Other similar operations:
Add common ordinary fractions, online calculator:
Fractions additions: the latest fractions added by the users
How to: Adding ordinary (simple, common) fractions. Steps.
There are two cases regarding the denominators when we add ordinary fractions:
- A. the fractions have like denominators;
- B. the fractions have unlike denominators.
A. How to add ordinary fractions that have like denominators?
- Simply add the numerators of the fractions.
- The denominator of the resulting fraction will be the common denominator of the fractions.
- Reduce the resulting fraction.
An example of adding ordinary fractions that have like denominators, with explanations
^{3}/_{18} + ^{4}/_{18} + ^{5}/_{18} = ^{(3 + 4 + 5)}/_{18} = ^{12}/_{18};
- We simply added the numerators of the fractions: 3 + 4 + 5 = 12;
- The denominator of the resulting fraction is: 18;
The resulting fraction is being reduced: ^{12}/_{18} = ^{(12 ÷ 6)}/_{(18 ÷ 6)} = ^{2}/_{3}.
B. To add fractions with different denominators (unlike denominators), build up the fractions to the same denominator. How is it done?
1. Reduce the fractions to the lowest terms (simplifying).
- Factor the numerator and the denominator of each fraction down to prime factors (prime factorization).
- Calculate GCF, the greatest common factor (also called GCD, greatest common divisor, HCF, greatest common factor) of each fraction's numerator and denominator.
- GCF is the product of all the unique common prime factors of the numerator and the denominator, taken by the lowest exponents.
- Divide the numerator and the denominator of each fraction by their greatest common factor, GCF - after this operation the fraction is reduced to its lowest terms equivalent.
2. Calculate the least common multiple, LCM, of all the fractions' new denominators:
- LCM is going to be the common denominator of the added fractions.
- Factor all the new denominators of the reduced fractions (run the prime factorization).
- The least common multiple, LCM, is the product of all the unique prime factors of the denominators, taken by the largest exponents.
3. Calculate each fraction's expanding number:
- The expanding number is the non-zero number that will be used to multiply both the numerator and the denominator of each fraction, in order to build all the fractions up to the same common denominator.
- Divide the least common multiple, LCM, calculated above, by each fraction's denominator, in order to calculate each fraction's expanding number.
4. Expand each fraction:
- Multiply each fraction's both numerator and denominator by expanding number.
- At this point, fractions are built up to the same denominator.
5. Add the fractions:
- In order to add all the fractions simply add all the fractions' numerators.
- The end fraction will have as a denominator the least common multiple, LCM, calculated above.
6. Reduce the end fraction to the lowest terms, if needed.
More on ordinary (common) fractions / theory: