Learn the Fractions
The fractions express the idea of dividing a whole into equal parts and taking some of them.
A fraction also represents the quotient of two numbers.
The classic example is that of a cheese that matches portions. In the picture, we have made 8 portions, 3 roses and 5 green.
If we take the 3 roses, represents 3 portions of eight in which we have divided the cheese, ie 3 / 8 of the cheese, and if we take the 5 green , represents 5 portions of eight in which we have divided the cheese, ie 5 / 8 of the cheese.
The parts we take (3-5) are called numerator (top) and the parts that divide the cheese ( 8 ) denominator (bottom).
To read a fraction, the numerator is read normally but, as we will see next, the denominator has a special way of reading.
| Denominator and Reading | Examples | ||
|---|---|---|---|
| 2 | halves | 5/2 | five halves |
| 3 | thirds | 2/3 | two thirds |
| 4 | fourths | 3/4 | three fourths |
| 5 | fifths | 4/5 | four fifths |
| 6 | sixths | 5/6 | five sixths |
| 7 | sevenths | 6/7 | six sevenths |
| 8 | eighths | 7/8 | seven eighths |
| 9 | nineths | 8/9 | eight nineths |
| 10 | tenths | 9/10 | nine tenths |
Classification Of Fractions
Fractions can be classified in different ways, in the following table shows the characteristics of the most important.
| Type | Features | Examples |
|---|---|---|
| Proper | The numerator is smaller than the denominator | 1 / 2, 7 / 9 |
| Improper | The numerator is greater than the denominator | 4 / 3, 5 / 2 |
| Whole | The numerator equals the denominator; represent an integer | 6 / 6 = 1 |
| Homogeneous | They have the same denominator | 2 / 5, 4 / 5 |
| Heterogeneous | They have different denominator | 3 / 7, 2 / 8 |
| Equivalents | When they have the same value. Two fractions are equivalent if their cross product are equal | 2/3 y 4/6 2x6 = 3x4 |
Equivalent fractions
If a fraction we multiply or divide the numerator and denominator by the same number, we get a fraction equivalent to the first, as both have the same value. Example:
| 1 | (1 x 4) | 4 | 3 | (3 : 3) | 1 | |||||||||
| — | = | ——— | = | — | = | 0,5 ; | — | = | ——— | = | — | = | 0,2 | |
| 2 | (2 x 4) | 8 | 15 | (15 : 3) | 5 |
Simplify or Reduce fractions
Simplify or Reduce a fraction is to find the smallest equivalent fraction; for this, the first thing we do is find the greater number that divides exactly (rest = 0) the numerator and the denominator (greater common divisor) and then, we divide numerator and denominator by the factors common to both and we get an equivalent fraction (of equal value).
Simplify 30/42
The divisors of 30 (numbers that exactly divide to 30) are:
The divisors of 42 (numbers that exactly divide to 342) are:
| 30 | 30/6 | 5 | ||
| — | = | —— | = | — |
| 42 | 42/6 | 7 |
When in a fraction, the numerator and the denominator have no common divisor (other than 1), it is said to be an irreducible fraction.
Addition and Subtraction of Fractions
If the fractions have the same denominator, add or subtract the numerators and put the same denominator.
| 3 | 2 | (3 + 2) | 5 | 5 | 2 | (5 - 2) | 3 | |||||||
| — | + | — | = | ——— | = | — | ; | — | - | — | = | ——— | = | — |
| 6 | 6 | 6 | 6 | 7 | 7 | 7 | 7 |
If the fractions have different denominator (heterogeneous), the first thing we have to do is equalize the denominators. To achieve this, we seek two fractions equivalent to those given by multiplying the numerator and denominator each by the denominator of the other. Once you get the same denominator, we proceed as above, add or subtract the numerators and put the common denominator.
| 2 | 3 | (2 x 7) | (3 x 5) | 14 | 15 | 29 | ||||||
| — | + | — | = | ——— | + | ——— | = | —— | + | —— | = | —— |
| 5 | 7 | (5 x 7) | (7 x 5) | 35 | 35 | 35 |
Multiplication of Fractions
The product of various fractions is equal to another fraction whose numerator is the product of the numerators and denominator is the product of the denominators.
| 3 | 4 | 2 | (3 x 4 x 2) | 24 | 2 | |||||
| — | x | — | x | — | = | ————— | = | —— | = | — |
| 4 | 5 | 3 | (4 x 5 x 3) | 60 | 5 |
Fraction of a Number
Calculate the fraction of a number is the same as multiplying the fraction by that number.
Calculate the 2 / 3 of 60 :
| 2 | 2 | (2 x 60) | 120 | |||||||||
| — | de | 60 | = | — | x | 60 | = | ——— | = | —— | = | 40 |
| 3 | 3 | 3 | 3 |
Division of Fractions
The quotient of two fractions is another fraction whose numerator is the product of the first numerator by the denominator of the second, and the denominator is the product of the denominator of the first by the numerator of the second.(Cross productss).
| 4 | 3 | (4 x 5) | 20 | |||
| — | : | — | = | ——— | = | —— |
| 9 | 5 | (9 x 3) | 27 |
In the games, the fraction line we will write with the slash " / ", by simplicity in the handling of the keyboard.