1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 = ? Adding ordinary (common) fractions, online calculator, addition operation explained step by step. The answer, written in three ways. As a positive proper fraction (the numerator < the denominator). As a decimal number. As a percentage.

1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 = ?

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:


6 = 2 × 3


12 = 22 × 3


20 = 22 × 5


30 = 2 × 3 × 5


42 = 2 × 3 × 7


56 = 23 × 7


72 = 23 × 32


90 = 2 × 32 × 5


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 (6; 12; 20; 30; 42; 56; 72; 90) = 23 × 32 × 5 × 7 = 2,520



2) Calculate the expanding number of each fraction:

Divide the LCM by the denominator of each fraction.


1/6 : 2,520 ÷ 6 = (23 × 32 × 5 × 7) ÷ (2 × 3) = 420


1/12 : 2,520 ÷ 12 = (23 × 32 × 5 × 7) ÷ (22 × 3) = 210


1/20 : 2,520 ÷ 20 = (23 × 32 × 5 × 7) ÷ (22 × 5) = 126


1/30 : 2,520 ÷ 30 = (23 × 32 × 5 × 7) ÷ (2 × 3 × 5) = 84


1/42 : 2,520 ÷ 42 = (23 × 32 × 5 × 7) ÷ (2 × 3 × 7) = 60


1/56 : 2,520 ÷ 56 = (23 × 32 × 5 × 7) ÷ (23 × 7) = 45


1/72 : 2,520 ÷ 72 = (23 × 32 × 5 × 7) ÷ (23 × 32) = 35


1/90 : 2,520 ÷ 90 = (23 × 32 × 5 × 7) ÷ (2 × 32 × 5) = 28


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.


1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 =


(420 × 1)/(420 × 6) + (210 × 1)/(210 × 12) + (126 × 1)/(126 × 20) + (84 × 1)/(84 × 30) + (60 × 1)/(60 × 42) + (45 × 1)/(45 × 56) + (35 × 1)/(35 × 72) + (28 × 1)/(28 × 90) =


420/2,520 + 210/2,520 + 126/2,520 + 84/2,520 + 60/2,520 + 45/2,520 + 35/2,520 + 28/2,520 =


(420 + 210 + 126 + 84 + 60 + 45 + 35 + 28)/2,520 =


1,008/2,520


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.


Calculate the greatest common factor, GCF,
of the numerator and denominator of the fraction:

The prime factorizations of the numerator and denominator:


1,008 = 24 × 32 × 7


2,520 = 23 × 32 × 5 × 7


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 (1,008; 2,520) = GCF (24 × 32 × 7; 23 × 32 × 5 × 7) = 23 × 32 × 7


The fraction can be reduced (simplified):

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


1,008/2,520 =


(24 × 32 × 7)/(23 × 32 × 5 × 7) =


((24 × 32 × 7) ÷ (23 × 32 × 7))/((23 × 32 × 5 × 7) ÷ (23 × 32 × 7)) =


2/5 =


2/5



Rewrite the equivalent simplified operation:

1,008/2,520 =


2/5


Rewrite the fraction

As a decimal number:

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


2/5 =


2 ÷ 5 =


0.4

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.


0.4 =


0.4 × 100/100 =


(0.4 × 100)/100 =


40/100 =


40%



The final answer:
:: written in three ways ::

As a positive proper fraction:
(the numerator < the denominator)
1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 = 2/5

As a decimal number:
1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 = 0.4

As a percentage:
1/6 + 1/12 + 1/20 + 1/30 + 1/42 + 1/56 + 1/72 + 1/90 = 40%

How are the numbers being written on our website: comma ',' is used as a thousands separator; point '.' used as a decimal separator; numbers rounded off to max. 12 decimals (if the case). The set of the used symbols on our website: '/' the fraction bar; ÷ dividing; × multiplying; + plus (adding); - minus (subtracting); = equal; ≈ approximately equal.

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All the operations with added fractions

Add common ordinary fractions, online calculator:

How to: Adding ordinary (simple, common) fractions. Steps.

There are two cases regarding the denominators when we add ordinary fractions:

A. How to add ordinary fractions that have like denominators?

An example of adding ordinary fractions that have like denominators, with explanations

B. To add fractions with different denominators (unlike denominators), build up the fractions to the same denominator. How is it done?


Read the rest of this article, here > How to add ordinary (common) fractions

More on ordinary (common) fractions / theory:

(1) What is a fraction? Fractions types. How do they compare?


(2) Changing the form of fractions, by expanding or reducing (simplifying)


(3) How to reduce fractions (simplifying). The greatest common factor, GCF


(4) How to compare two fractions with unlike (different) numerators and denominators


(5) How to sort out fractions in ascending order


(6) Adding common (ordinary) fractions


(7) Subtracting common (ordinary) fractions


(8) Multiplying common (ordinary) fractions


(9) Fractions as rational numbers