What is a fraction?
1- Definition
A fraction is a number, which is obtained by dividing an integer into equal parts. For example, when we say a quarter of the cake, we are dividing the cake into four parts and consider one of them.
A fraction is described in a scientific discipline by numbers that are written one on another and that are separated by a horizontal straight line denominated fractional line.
The fraction consists of two terms: the numerator and the denominator. The numerator is the number that is on the fractional line and the denominator is the number that is under the fractional line.
The numerator is the number of parts that is considered the unit or total.
The denominator is the number of equal parts in which the unit or total has been divided.
2- Reading fractions
All fractions are given a specific name, can be read as such, according to the numerator and denominator they have.
The number in the numerator reads the same, but not the denominator. When the denominator goes from 2 to 10, it has a specific name (if it is 2 is “means”, if 3 is “thirds”, if 4 is “quarter”, if 5 is “fifth”, if 6 is “sixth,” if 7 is “seventh,” if 8 is “eighth,” if 9 is “ninth”, if 10 is “tenth”), however, when it is greater than 10.
3- The meanings of the fractions in the different contexts of use
3.1 The fraction as an expression that links the part with the whole
In this case, it is used to indicate “the fracture” or “division into parts”, answering the question what part is it? of the integer in question or as considered parts of a collection of equal objects. It is agreed that the denominator of the fraction indicates the number of parts in which said integer is divided and the numerator the parts considered.
Issue:
From a basket of 36 flowers, 1/3 are pink roses; 1/4 are margaritas and rest are chocolates. How many flowers of each class are there?
To calculate the fraction of a number n, in this case, flowers, you can divide the number n by the denominator of the fraction and then multiply it by the numerator, or multiply the numerator of the fraction by n and divide the result by the denominator.
So in our problem:
– 1/3 of 36 are pink = 36: 3 = 12 x 1 = 12
Therefore of the 36 flowers that are in the basket: 12 are roses
Therefore of the 36 flowers that are in the basket: 9 are margaritas.
– If the rest of the flowers of the basket are chocolates, we must subtract the total of flowers, the sum of the other two.
roses + daisies = 12 + 9 = 21
36 – 21 = 15
Then we have that there are 15 chocolates.
Answer: Of the 36 flowers in the basket, 12 are roses, 9 are daisies and 15 are Chocolates.
3.2- The fraction as an equitable distribution
Answering the question, how much does each one have?
Similarly, if I am to distribute 3 chocolate bars between 4 children, each will receive 3/4 bar.
To make it clear, we will see another example:
– A group of 4 friends gets together to eat. They have 3 pizzas, which they will distribute in equal parts. What fraction of pizza corresponds to each?
Since the 3: 4 division is not accurate, we must do the following:
We will divide each pizza into 4 equal parts that is in quarters.
Then split the 12 pieces between the 4 friends
3.3- The fraction as reason including
– Two different sets, for example, the ratio or relation between a number of books in the class and the
number of students. Thus, 13 books for 26 students can be expressed as 13/26
reading “13 to 26” or what is the same, “1 for every 2”.
– A set and a subset of it, for example, the ratio of the 21 students in total and the male students (11) of a class can be expressed as 11/21 or “11 to 21”. A special case is a probability defined as the number of favorable cases on the number of possible cases of a given event. For example, in the roll of a die, the probability or odds ratio of a 2 “is one to 6” which is indicated as 1/6.
To divide two or more fractions, multiply “in the cross”.
This is the numerator (top number) of the first fraction by the denominator (number down) of the second fraction, so we get the numerator. To obtain the denominator, we have to multiply the denominator (number of down) of the first fraction by the numerator (number of above) of the second fraction.
In this post, we will learn how to do a division of fractions.
Fraction division method 1: Multiply in cross
This method consists of multiplying the numerator of the first fraction by the denominator of the second fraction and writing the result into the numerator of the resulting fraction.
On the other hand, we multiply the denominator of the first fraction by the numerator of the second and the result we write it in the denominator of the resulting fraction.
Finally, the fraction is simplified.
For example, to divide fractions 3/4 between 6/10.
We calculate the numerator by the denominator of the second. Now, we have the numerator of the final fraction 3 × 10 = 30
On the other hand, we multiply the denominator of the first (4) by the numerator of the second (6). In this way, we have the denominator of the final fraction 4 × 6 = 24
The last step is to simplify the fraction.
30: 6 = 5
24: 6 = 4
Hence, the outcome of the division is 5/4
Fraction division method 2: Invert and multiply
Step 1: Change the second fraction, ie alter the numerator by the denominator and contrariwise.
Step 2. Simplify any numerator with any denominator.
For example, let’s split 12/5 between 6/4.
Step 1: Change the second fraction 6/4. This becomes 4/6
Step 2: We simplify numerators with denominators.
The numerators are:
12 = 2x2x3
4 = 2 × 2
The denominators are:
5 = 5
6 = 2 × 3
We can simplify both the numerator and the denominator by a 2 and a 3. Thus we have, 2x2x2 / 5
Step 3: Multiply online: 2x2x2 / 5 = 8/5
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Add fractions
There are two cases:
While of making the fractions we can find two cases: Fractions with same denominator and Fractions with different denominator
Fractions with the same denominator:
The sum of the fractions with the same denominator is very simple, just add all the numerators and divide by common denominator. However, if, the fractions have a different denominator. Then follow the below steps-
1. Multiply in a cross. The numerator of the first fraction is multiplied by the denominator of the second, and the denominator of the first by the numerator of the second. Both multiplications add up.
2. Multiply the denominators of the two fractions. The denominators of the two fractions are multiplied.
3. We resolve all operations.
We divide them by that number.
In this case, it is an improper fraction because the denominator (4) is smaller than the numerator (5).
Fraction with different denominator can also be divided through alternative ways, such as
1. between the two denominators, there is always least common multiple.
2. Calculate the numerator with the formula: old numerator x common denominator (the drawn with the common minimum multiple) and divided by the old denominator.
3. Once the denominator is equal, the fractions are added as in the first case (since the fractions have the same denominator).
1. We calculate the least common multiple (l.c. m.). The minimum common divisor (m.c.m) of 4 and 2 is 4.
2. Calculate the numerators. We work out the numerators with the formula. old numerator x common denominator (the drawn with the common minimum multiple) and divided by the old denominator.
Numerator of the first fraction: 3 x 4: 4 = 3
Numerator of the second fraction: 4 x 4: 2 = 8
3. Once the denominators are the same the operations are performed. The result of these operations is:
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Subtraction of Fractions
There are two cases:
In the subtraction of fractions we can find two different cases:
Fractions with the same denominator
with the different denominator
Fractions with the same denominator.
The subtraction fractions with the same denominator are very plain, just work out the numerators and leave the denominator.
2. Multiply the denominators of the two fractions. The denominators of the two fractions are multiplied.
We divide the denominator and the numerator by this number.
Another way to do it:
The subtraction of two or more fractions with different denominator is a bit less simple. Let’s go step by step:
1 The minimum common multiple of the two denominators
2 Calculate the numerator with the formula: old numerator x common denominator and divided by old denominator
3 Proceed as in the first case (since the fractions have the same denominator)
Example:
6/4 – 1/2
1 We calculate the least common multiple (L.c. m.). (4, 2) = 4.
2 We calculate the numerators.
Numerator of the first fraction: 6 x 4: 4 = 6
Numerator of the second fraction: 1 x 4: 2 = 2
3 Thus we have a fraction that is:
6/4 – 2/4
As the denominators are identical, we can subtract it as in case 1
Multiplication of fractions
The multiplication of fractions is very simple.
The multiplication of two or more fractions is done “in line”. That is, the numerator of the first fraction by the numerator of the second and the denominator of the first fraction by the denominator of the second.
Divide fractions
When dividing an integer, multiply by the reciprocal of its divisor. In the example of painting where you need 3 gallons of paint to apply a layer and you have 6 gallons of paint, you can find the total number of layers you can paint by dividing 6 between 3, 6 ÷ 3 = 2. You can also multiply 6 by the reciprocal of 3, which is, then the multiplication problem becomes.
If you have a sweet that needs to split in half, you can divide by 2, or you can multiply it to find the amount you want.
Similarly, with a mixed number, you can either divide between the whole number or you can multiply by the reciprocal. Suppose you have pizzas that you want to divide equally among 6 people.
Dividing a fraction of a whole number is the same as multiplying by its reciprocal, so you can always use the multiplication of fractions to solve problems of divisions.