# Fractions

## Free Fraction Calculator (by Radu Turcan)

Operator: + - * /

=

Solution:

#### Definition of a fraction

A number written as $\frac{a}{b}$, where $a$ is an integer and $b$ is a non-zero integer, is called a fraction.
Number $a$ is called a numerator, and $b$ is called a denominator. A fraction represents a part of a whole or any number of equal parts.
The denominator shows how many equal parts make up a whole, and the numerator shows how many of these parts we have.

#### Examples of fractions

Example 1: Becky, Merry and John want to share a chocolate bar evenly.
What part of the bar will each of them take?
What part of the bar will Becky and Merry have together?

The children need to divide the bar into three pieces. So everyone will take $\frac{1}{3}$ of the chocolate bar.
Two girls together will have two pieces, hence, mathematically speaking, they will have $\frac{2}{3}$ of the bar.

Example 2: What part of the soldiers are yellow soldiers?

Example 3: What part of the apples is missing?

#### Fraction Rules

$\frac{A}{B} +\frac{C}{B} = \frac{A + C}{B}$

Subtraction:(same denominators)
$\frac{A}{B} -\frac{C}{B} = \frac{A - C}{B}$

$\frac{A}{B} +\frac{C}{D} = \frac{A\cdot D}{B\cdot D} +\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D + B\cdot C}{B\cdot D}$

Subtraction:(different denominators)
$\frac{A}{B} -\frac{C}{D} = \frac{A\cdot D}{B\cdot D} -\frac{B\cdot C}{B\cdot D} = \frac{A\cdot D - B\cdot C}{B\cdot D}$

Multiplication:
$\frac{A}{B}\times\frac{C}{D} = \frac{A\cdot C}{B\cdot D}$

Division:
$\frac{A}{B}\div\frac{C}{D} = \frac{A}{B}\times\frac{D}{C}= \frac{A\cdot D}{B\cdot C}$

#### Properties of fractions

Property I: All hatched parts of the circles represent one half $\frac{1}{2}, \frac{2}{4}$ and $\frac{3}{6}$, hence $\frac{1}{2} = \frac{2}{4} = \frac{3}{6}$

We get $\frac{2}{4}$ when we multiply the numerator and the denominator of the fraction $\frac{1}{2}$ by $2$.

We obtain $\frac{3}{6}$ by multiplying the numerator and the denominator of $\frac{1}{2}$ by $3$.

Let $a$ be an integer and $b$ and $c$ be non-zero integers.
Then:

$\frac{a}{b}=\frac{a\cdot c}{b\cdot c}$ and $\frac{a}{b}=\frac{a:c}{b:c}$

 a b
=
 a.c b.c
and
 a b
=
 a:c b:c

Property II: If two fractions have equal denominators, the fraction with the greater numerator is greater.
If $a$, $b$ and $c$ are integers and $c$ does not equal $0$:

$\frac{a}{c}>\frac{b}{c}$, if $a>b$

 a c
>
 b c
, if a > b

Property III: If two fractions have equal numerators, the fraction with the smaller denominator is greater.
If $a$, $b$ and $c$ are integers, and both $b$ and $c$ are non-zero,

$\frac{a}{b}>\frac{a}{c}$, if $b< c$

 a b
>
 a c
, if b < c

#### Fraction Test

1. A tennis player won $6$ out of first $12$ sets. Then he won all of the remaining $6$ sets. What part of the sets did the player win?
$\frac{1}{3}$      $\frac{2}{3}$      $\frac{1}{2}$

2. A boy had $\$36$. After a couple of hours of shopping he had$\$8$ left. What part of his money did he spend?
$\frac{2}{9}$      $\frac{2}{7}$      $\frac{7}{9}$

3. There were $12$ girls in a class of $30$ students. Then $6$ boys joined the class. What part of the class are the girls?
$\frac{1}{2}$      $\frac{3}{5}$      $\frac{1}{3}$

4. If the fraction $\frac{n}{40}$ is between $\frac{1}{5}$ and $\frac{1}{4}$ then n is
$8$      $9$      $10$

5. $\frac{6}{24}$ is equal to:
$\frac{1}{4}$      $\frac{3}{4}$      $\frac{6}{12}$

6. Which of the fractions is twice greater than $\frac{3}{8}$?
$\frac{6}{16}$      $\frac{3}{16}$      $\frac{3}{4}$

7.* Which of the following fractions is the largest: $\frac{12}{13}, \frac{13}{14}, \frac{14}{15}$ or $\frac{15}{16}$?
$\frac{15}{16}$      $\frac{12}{13}$      $\frac{14}{15}$

8. Which of the following sequences has fractions arranged in a descending order?
1: $\frac{7}{11}, \frac{5}{8}, \frac{3}{5}, \frac{2}{3}$;
2: $\frac{4}{3}, \frac{7}{11}, \frac{5}{8}, \frac{3}{5}$;
3: $\frac{21}{11}, \frac{2}{3}, \frac{3}{5}, \frac{5}{8}$
$2$      $3$      $1$

9.* Which of the following sequences has  fractions arranged in an ascending order?
1: $\frac{13}{19}, \frac{13}{23}, \frac{17}{23}$;
2: $\frac{13}{23}, \frac{17}{23}, \frac{13}{19}$;
3: $\frac{13}{23}, \frac{13}{19}, \frac{17}{23}$;
$1$      $2$      $3$

10. Calculate $\frac{20+4\cdot3}{120}$:
$\frac{2}{5}$      $\frac{3}{5}$      $\frac{4}{15}$

11. Calculate $\frac{1+2+3+4+5}{1\cdot2\cdot3\cdot4\cdot5}$:
$5$      $1$      $\frac{1}{8}$

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