Imagine, you live in Germany... ;-)
Fancy a sunny day on the beach? You can experience it in Italy, for example.
Or would you prefer a winter holiday with lots of snow? Then perhaps Austria is the right choice.
However, one thing is certain after a quick glance at the map: From Germany the distance to Austria is shorter than to Italy.
This is so easy to say because both holiday destinations are located in the same direction: South
If it comes to the question of whether Spain or Finland is the closer target, it becomes more difficult...
And what can be done with numbers? They can be sorted by their values.
In the case of Natural Numbers it is obvious which is the smaller and which is the larger number. But in case of fractions, sometimes it's not so easy to do this. Could you tell at a glance whether 3/7 (= distance from Germany to Spain) or 7/15 (= distance from Germany to Finland) is greater?
When do I know at a glance which fraction is greater?
It is simple if the two fractions are in "the same direction" (like Austria and Italy from the point of view of Germany):
Case 1: Same denominator but different numerator
Remember when we compared a fraction to a cake: The numerator represented the number of pieces, the denominator indicated in how many pieces the cake was divided.
The same denominator means that the pieces of cake are equal in size.
The more pieces the better - the fraction with the larger numerator is greater:
5/8 > 3/8
(5 pieces of cake is more than 3 pieces of cake)
Case 2: Same numerator but different denominators
The denominator indicates how many pieces of the cake were cut. A greater denominator means that the cake was divided into more pieces - of course the pieces are smaller then!
The same numerator for both fractions means that the number of pieces is equal.
With the same number of pieces of cake: I would prefer the larger ones:
3/8 > 3/12
(in both cases there are 3 pieces of cake: first, the cake was divided into 8 pieces. Second, the cake was divided into 12 pieces - in the first case the pieces are bigger.)
Case 3: Reciprocal
If a fraction is given together with its reciprocal, you can also easily decide which is the greater one.
But first you have to know:
In which case is a fraction greater than 1?
A fraction is greater than 1 if the numerator is greater than the denominator.
Why? For example, if you have more apples than friends and you divide them evenly among yourselves, then of course everyone gets more than one apple.
Remember that the fraction bar is just another spelling for "divided by". The numerator indicates the number of apples, the denominator indicates the number of children.
In which case is a fraction lower than 1?
A fraction is lower than 1 if the numerator is lower than the denominator.
Why? Same explanation: If you have fewer apples than friends and you divide them equally among yourselves, then everyone gets less than one apple.
When is a fraction equal to 1?
A fraction is equal to 1 if the numerator is equal to the denominator.
You can guess why... For example, if 4 apples are evenly divided among 4 friends, then everyone gets exactly one apple.
If a fraction is given together with its reciprocal, then always one of these two fractions is greater than 1 and the other is lower than 1 (at least if the numerator and denominator are different)!
Why? If the numerator is greater (or lower) than the denominator, then it is exactly the opposite in the case of the reciprocal.
Therefore, the fraction is greater than its reciprocal if it is greater than 1.
The fraction is smaller than its reciprocal if it is less than 1.
Example: 7/3 > 3/7 because 7/3 is greater than 1.
And what if the numerator and denominator are equal?
In this case, the fraction and its reciprocal are also the same. Both fractions have the same value: 1.
In which cases are two fractions of equal value?
Two fractions have exactly the same value if the numerators and denominators are the same multiple of each other.
Imagine you have a big piece of cake: 1/8 of a cake.
Now cut this piece right in the middle so that you get two pieces of cake of the same size.
How big are these pieces? If you divide each of the eight pieces of cake in this way, you would have twice as many: 16
So it is true:
1/8 = 2/16
Both the numerator and denominator have been multiplied by the number 2.
Another example: 9/3 = 36/12
Both numerators and denominators have been expanded with factor 4. Calculate: 9 divided by 3 and 36 divided by 12 both results in the same value 3.
But that's exactly what the next chapter is about....
Video: Which Fraction is Bigger?
Finally, a video about the question: Which Fraction is Bigger?:
Congratulations! Step 2 is done!
In the Step 2 you learned that a fraction can be represented as a number and vice versa. The set of all fractions corresponds to the set of Rational Numbers.
In some cases it is easy to decide which of two given fractions is the greater one: for example, if the numerator or denominator is equal, or if the two fractions are the reciprocal of each other.
To complete Step 2, please take a minute to answer the comprehension questions: