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Adding Fractions with Unlike Denominators

How do you add 6/5 + 1/4?

In general, when adding, you must add things that are alike.  For example, when adding 23 + 45 we add the 3 and the 5 together because they represent single unit values.  We add the 2 and the 4 together because they represent groups of ten.

Similarly when adding fractions we need to add together representations of the same thing.  That’s where the complication comes in with this problem.  The denominators are different, fifths versus fourths, so first we need to represent these values in a way that makes them alike.

VERY Briefly:

 6/5 + 1/4

 = 24/20 + 5/20

= 29/20 or 1 whole and 9/20

= 1 9/20 (or at least that's the best I can do with the formats offered in this blog!)
 

Using mathematical models only: (For a more detailed explanation – see the area model explanation below :-)

We find a common denominator.  Fifths and Fourths can both be represented by twentieths. 

Students often figure this out by finding a least common multiple for the denominators.  This can be done by skip counting or just thinking through your basic facts. 5 X 4 = 20 and 4 X 5 = 20. (Notice 5 X 1 = 5, 5 X 2 = 10, and 5 X 3 = 15, none of which produces a multiple of 4 as the product. The products 5, 10, and 15 are not multiples of 4 as well as 5.  They are not common multiples.  Twenty is therefore the least common multiple of both 4 and 5.) 

Next equivalent fractions are made using the least common multiple as a common denominator.

6/5 = 24/20. Students generally find the equivalent fraction by multiplying the numerator and denominator by the same value.  Since 5 X 4 brought us to 20 for the denominator, 6 X 4 brings us to 24 for the numerator.  Basically we are showing that when you cut up each 5th four times, the 6 fifths have now become four times the number of pieces. We have four times the number of pieces – but these pieces are smaller or ¼ their original size. 6x 4 pieces or 24 pieces… but each piece is only a small twentieth of one whole unit.

1/4 = 5/20.  Since 4 X 5 brought us to 20, 1 X 5 brings us to 5 for the numerator.  If you cut up each fourth 5 times, the 1 piece now becomes five times the value but the pieces are 1/5 the original size. 5 small twentieth pieces have replaced 1 large fourth piece.

24/20 + 5/20 can be added as each twentieth represents the same value.

24/20 + 5/20 = 29/20 

Now we can change this to a mixed number by remembering that 20/20 = 1 whole thing.

29/20 = 20/20 + 9/20 or 1 whole + 9/20 which can be represented as 1 9/20. 

 

Need a better understanding?  Read on!  (and please excuse misalignments as once again I'm limited by formats offered in this blog.)

An area model can help show this process and find the sum.  We’ll divide each area model into 5 equal parts vertically (columns going up and down.) This will show fifths.  We’ll divide each area model into 4 equal parts horizontally (rows going across.) This will show fourths.  I need more than one area model because 5 fifths is one whole thing and this problem needs a representation of 6 fifths which is more than one whole thing.  Notice 6 fifths below covers 6 columns where each column is 1 fifth of one whole thing.

                                                  6                                                                                                                                                                                                                                                      5  shown shaded below (Notice this value is more than 1 whole thing.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                     

 

 

 

 

 

Now let’s look at the rows.  Each area (large rectangle) has been broken into four equal rows to create our fourths.  Notice 1 fourth is just 1 row inside of just 1 area model (1 large rectangle.)

1                                                                                                                                                                                                                                                            4    shown shaded below (Notice this value is less than 1 whole thing.)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                     

 

 

 

 

 

 

If it helps to picture a real circumstance – think of each large rectangle as 1 whole pan of brownies.  We split each pan into fifths by cutting up and down each pan 5 times.  We then split each pan into fourths by cutting across each pan four times. If I want to give you 6 fifths of a pan of brownies, I would take out 6 of the columns for you.  If I want to give you 1 fourth of a pan of brownies, I would give you one row of a pan. 

 

NOTICE: By breaking the rectangles into BOTH fourths going across and fifths going up and down, NOW each whole area is broken into 20 equal parts (or 20 brownies per pan).  Now we can see why fifths and fourths can both be thought of as twentieths.   Each column is a fifth but it is also four twentieths.  Go ahead and count the boxes (or brownies) in each column.  There are four of them in each of the columns.  Below I have shaded in 6 fifths or 6 columns.  It is exactly equal to 24 twentieths (or 24 brownies.)

 

6                                        24  

   is shown below as     20      (Notice the amount shaded remains the same and is still more than 1 whole.)

 

1

5

9

13

17

2

6

10

14

18

3

7

11

15

19

4

8

12

16

20

 

21

 

 

 

 

22

 

 

 

 

23

 

 

 

 

24

 

 

 

 

                                    

 

 

 

  

 

 

 

Below we will just shade in 1 fourth.  Notice 1 fourth is exactly equal to 5 twentieths (or 5 brownies.)

1                                           5

4      is shown below as     20  (Notice the amount shaded remains the same and  is still less than 1 whole.)

 

1

2

3

4

5

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

                                    

 

 

 

  

  

 

If you are looking at this as brownies: 1 whole pan has 20 equal sized brownies, so each brownie is also 1 twentieth of a whole pan.

 

Now we can find the sum buy representing both values as twentieths.  Using a common denominator is also described as using LIKE terms. Fourths and Fifths could NOT be added because they represent different amounts. Each fourth was the same as 5 twentieths while each fifth was only 4 twentieths.  

 

Twentieths can be added because each twentieth represents the same amount or the same value. 

So now we can represent 6 fifths plus 1 fourth as:

24 twentieths plus 5 twentieths = 29 twentieths, or 29/20. (Just as 24 brownies plus 5 more brownies would be 29 brownies – and we already noticed that each brownie is a twentieth- 29 brownies is the same as 29 twentieths or 29/20.)

While 29/20 is already a correct sum, generally it is more understandable if we now break out any whole values to represent this amount as a mixed number:

20 twentieths makes one whole thing.  (If you don’t believe it, count up the small boxes inside of each large rectangle/brownie pan above.  There are 20 in each which is WHY we call them twentieths.  The name of any denominator tells us how many equal parts make one whole thing.)

29 twentieths can be broken up or thought of as 20 twentieths plus 9 more twentieths.

As 20 twentieths is 1 whole, this sum can also be represented as 1 whole and 9 more twentieths.

So this is how we can arrive at a final answer of 1 and 9 twentieths or 1 9/20.

 

Consider representing other fraction problems with area models or brownie pans.  Visualizing fractions really helps strengthen your understanding of what the problems mean.  This will help you apply your fraction knowledge to all kinds of problems.

 

Best of luck with your fractions work!