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Adding Fractions

Adding fractions implies that the concept of unitizing

, (that is understanding what is the UNIT of the group of elements we are working with) is well understood.

The concept of partitioning goes along with it.

The unit is defined by the denominator (the bottom number).



The rules goes that you cannot add apples with oranges!

you cannot add fractions with different denominators!

( or, conversely, you can add fractions which have the same denominator only! )




adding fractions implies same units, that is, same denominator!


Do you see? the units (the denominators) which we used to divide our rectangle are different! How can we add the two fractions?

Instead, we must first find a common ground to work on. In Algebra, this is found through the Least Common Denominator.

So the following step lead us to "create" this common ground with a simple trick: to find the smallest common multiple of the two fractions and change the original partitioning where necessary.

1/3 expressed as 4/12, an equivalent fraction


adding fractions implies same units, that is, same denominator!


Ahh! Now we can proceed adding fractions! Easy!

the result is












One common mistake first algebra learners have is to mistakenly add two fractions which have no common unit (or denominator) like this

1/3 + 1/4 = 1/7 (wrong !)


adding numerator and denominator as in a normal addition, like this

1/5 + 2/3 = 3/8 (wrong !)

Instead, you must do like this:

1/2 + 5/2 = 6/2 (correct!)

So, mind the bottom numbers (the denominators)! Are they the same? If positive just add the top numbers together and leave the bottom numbers as you found them!


603. The examples which a beginner should choose for prac-
tice should be simple and should not contain very large num-
bers. The powers of the mind cannot be directed to two things
at once; if the complexity of the numbers used requires all the
student's attention, he cannot observe the principle of the rule
which he is following. DE MORGAN, A.

Study and Difficulties of Mathematics (Chi-
cago, 1902), chap. 8.

source: Memorabilia mathematica; or, The philomath's quotation-book - Moritz, Robert Édouard, 1868-1940


Related pages

Subtracting Fractions

Multiplying fractions

Dividing Fractions

*We transformed 1/3 into 4/12. these two fractions are called

equivalent fractions!






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