- 84/52 + 69/48 = ? Adding ordinary (common) fractions, online calculator, addition operation explained step by step. The answer, written in three ways. As a negative proper fraction (the numerator < the denominator). As a decimal number. As a percentage.

- 84/52 + 69/48 = ?

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: - 84/52 = - (22 × 3 × 7)/(22 × 13) = - ((22 × 3 × 7) ÷ 22 )/((22 × 13) ÷ 22 ) = - 21/13

The fraction: 69/48 = (3 × 23)/(24 × 3) = ((3 × 23) ÷ 3)/((24 × 3) ÷ 3) = 23/16


Internal link » Reduce (simplify) common (ordinary) fractions to the lowest terms (to their simplest form equivalent), online calculator


Rewrite the equivalent simplified operation:

- 84/52 + 69/48 =

- 21/13 + 23/16

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: - 21/13

- 21 ÷ 13 = - 1 and the remainder = - 8 ⇒ - 21 = - 1 × 13 - 8

- 21/13 = ( - 1 × 13 - 8)/13 = ( - 1 × 13)/13 - 8/13 = - 1 - 8/13


The fraction: 23/16

23 ÷ 16 = 1 and the remainder = 7 ⇒ 23 = 1 × 16 + 7

23/16 = (1 × 16 + 7)/16 = (1 × 16)/16 + 7/16 = 1 + 7/16



Rewrite the equivalent simplified operation:

- 21/13 + 23/16 =

- 1 - 8/13 + 1 + 7/16 =

- 8/13 + 7/16

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 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.


1) Find the common denominator
Calculate the LCM of the denominators:

The prime factorization of the denominators:

13 is a prime number

16 = 24

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 (13; 16) = 24 × 13 = 208


External link » Calculate LCM, the least common multiple of numbers, online calculator


2) Calculate the expanding number of each fraction:

Divide the LCM by the denominator of each fraction.

- 8/13 : 208 ÷ 13 = (24 × 13) ÷ 13 = 16

7/16 : 208 ÷ 16 = (24 × 13) ÷ 24 = 13


3) Make the fractions' denominators the same:

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.


- 8/13 + 7/16 =

- (16 × 8)/(16 × 13) + (13 × 7)/(13 × 16) =

- 128/208 + 91/208 =

( - 128 + 91)/208 =

- 37/208


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.


- 37/208 is already reduced to the lowest terms.

The numerator and denominator have no common prime factors.

Their prime factorization:

37 is a prime number

208 = 24 × 13

GCF (37; 24 × 13) = 1


Internal link » Reduce (simplify) common (ordinary) fractions to the lowest terms (to their simplest form equivalent), online calculator


Rewrite the fraction

As a decimal number:

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


- 37/208 =

- 37 ÷ 208 ≈

- 0.177884615385 ≈

- 0.18

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.


- 0.177884615385 =

- 0.177884615385 × 100/100 =

( - 0.177884615385 × 100)/100 =

- 17.788461538462/100 ≈

- 17.788461538462% ≈

- 17.79%


External link » Convert and write integer and decimal numbers, fractions and ratios as percentages, online calculator


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

As a negative proper fraction:
(the numerator < the denominator)
- 84/52 + 69/48 = - 37/208

As a decimal number:
- 84/52 + 69/48 ≈ - 0.18

As a percentage:
- 84/52 + 69/48 ≈ - 17.79%

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:

How to add the common ordinary fractions:
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Add common ordinary fractions, online calculator:

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All the operations with added fractions

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

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).

  • 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.

    • Calculate LCM, the least common multiple of numbers.

  • 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.

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

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