- ⁴²⁷/₂₁₁ - ¹⁹⁷/₃₄₄ - ²²¹/₃₅₈ + ²³²/₃₈₄ + ²¹⁶/_6,614 + ³⁶¹/₂₁₇ + ²²¹/₄₁₉ - ²⁵⁶/₄₇₉ + 278 = ? Adding Up Common (Ordinary) Fractions, Online Calculator. Addition Operation Explained Step by Step

• To reduce a fraction to its 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 fraction that was reduced (simplified) to the lowest terms is one with the smallest possible numerator and denominator, one that can no longer be reduced, and it is called an irreducible fraction.

• The numerator and denominator have no common prime factors.
• Their prime factorization:
427 = 7 × 61
• 211 is a prime number
• GCF (7 × 61; 211) = 1

• The numerator and denominator have no common prime factors.
• Their prime factorization:
197 is a prime number
• 344 = 2³ × 43
• GCF (197; 2³ × 43) = 1

• The numerator and denominator have no common prime factors.
• Their prime factorization:
221 = 13 × 17
• 358 = 2 × 179
• GCF (13 × 17; 2 × 179) = 1

• The prime factorizations of the numerator and denominator:
• 232 = 2³ × 29
• 384 = 2⁷ × 3
• Multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).
• GCF (232; 384) = 2³ = 8

• Yet another method to simplify the fraction:

• Without calculating GCF, factor the numerator and denominator and cross out all the common prime factors.

• ²³²/₃₈₄ = ^{(23 × 29)}/_{(2⁷ × 3)} = ^{((23 × 29) ÷ 23 )}/_{((2⁷ × 3) ÷ 2³ )} = ²⁹/₄₈

• 216 = 2³ × 3³
• 6,614 = 2 × 3,307
• GCF (216; 6,614) = 2

• We could have simplified the fraction without calculating the GCF. Just factor the numerator and denominator and cross out the common prime factors.

• ²¹⁶/_6,614 = ^{(23 × 33)}/_{(2 × 3,307)} = ^{((23 × 33) ÷ 2)}/_{((2 × 3,307) ÷ 2)} = ¹⁰⁸/_3,307

• The numerator and denominator have no common prime factors.
• Their prime factorization:
361 = 19²
• 217 = 7 × 31
• GCF (19²; 7 × 31) = 1

• The numerator and denominator have no common prime factors.
• Their prime factorization:
221 = 13 × 17
• 419 is a prime number
• GCF (13 × 17; 419) = 1

• The numerator and denominator have no common prime factors.
• Their prime factorization:
256 = 2⁸
• 479 is a prime number
• GCF (2⁸; 479) = 1

• 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.

• To perform the operation of calculating the fractions we have to:
• 1) find their common denominator
• 2) then calculate the expanding number of each fraction
• 3) then make 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.

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).

• 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.

• The prime factorizations of the numerator and denominator:
• 895,170,362,612,494,231 = 2⁷ × 97 × 101 × 34,693 × 20,575,991
• 11,227,644,273,190,443,504 = 2¹¹ × 31 × 1.7684671549254E+14

Multiply all the common prime factors: if there are repeating prime factors we only take them once, and only the ones having the lowest exponent (the lowest powers).

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

We could have simplified the fraction without calculating the GCF. Just factor the numerator and denominator and cross out the common prime factors.

• 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.

An improper fraction: the value of the numerator is larger than or equal to the value of the denominator.

Simply divide the numerator by the denominator, without a remainder, as shown below:

• 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.

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.