[Arithmetic 6] Mixed numbers. Decimals. Percentages

\[\boxed{\text {Is this true: }\dfrac{9}{4} = 2\dfrac{1}{4} = 2.25 = 225\%  \text { ? }}\]

I. MIXED NUMBERS

Example: 7 divided by 4 is 1 with remainder 3.

Then, we write: $\boxed{\dfrac{7}{4} = 1 + \dfrac{3}{4} = 1\dfrac{3}{4}}$. The number $1\dfrac{3}{4}$ is a mixed number, where 1 is the integer part and $\dfrac{3}{4}$ is the fraction part of $\dfrac{7}{4}$.

❄ Covert mixed numbers to fractions
    $5\dfrac{1}{7} = \dfrac{{5.7 + 1}}{7} = \dfrac{{36}}{7}$

    $ - 4\dfrac{3}{5}$ = $\bf-$$\dfrac{{4.5 + 3}}{5}$ = $\bf-$$\dfrac{{23}}{5}$

❄ Covert fractions to mixed numbers
    $\dfrac{7}{3} = 2\dfrac{1}{3}$ (7 divided by 3 is 2 with remainder 1)

    $\dfrac{ - 7}{4}$ = $\bf-$$\dfrac{7}{4}$ = $\bf-$$ 1 \dfrac{3}{4}$

II. DECIMAL NUMBERS

- A decimal fraction is a fraction where the denominator is a power of 10 (such as: $\dfrac{3}{{10}}$, $\dfrac{{ - 152}}{{100}}$, $\dfrac{{73}}{{1000}}$, etc).

- The decimal fractions can be written as decimal numbers.

- A decimal number includes two parts:

      + Integer part is written on the left of the decimal separator,

      + Decimal part is written on the right of the decimal separator.

Example: $\dfrac{3}{{10}} = \color{black}{0.3}$; $\dfrac{-152}{{100}} = \color{black}{-1.52}$; $\dfrac{73}{{1000}} = \color{black}{0.073}$; etc are fractions.

- The number of digits in the decimal part is exactly equal to the number of 0 digits in the denominator of the decimal fraction.

❄ Covert decimal fractions to decimal numbers:

    $\dfrac{{27}}{{100}} = 0.27$

    $\dfrac{{ - 13}}{{1000}} =  - 0.013$

    $\dfrac{{261}}{{100000}} = 0.00261$

❄ Covert decimal numbers to decimal fractions:

    $1.21 = \dfrac{{121}}{{100}}$

    $0.07 = \dfrac{{7}}{{100}}$

    $-2.013 = \dfrac{{-2013}}{{1000}}$

III. PERCENTAGES
Fractions with denominator 100 are written as percentages with the symbol %.
Example: $\dfrac{3}{{100}} = 3\% $
               $\dfrac{107}{{100}} = 107\% $
               $\dfrac{7}{{25}} = \dfrac{{28}}{{100}} = 28\% $
               $\dfrac{{26}}{{65}} = \dfrac{2}{5} = \dfrac{{40}}{{100}} = 40\% $
               $3.7 = \dfrac{{37}}{{10}} = \dfrac{{370}}{{100}} = 370\% $
               $0.34 = \dfrac{{34}}{{100}} = 34\% $
❄ Covert percentages to decimal numbers
    $7\%  = 0.07$
    $45\%  = 0.45$
    $216\%  = 2.16$