Fractions, decimals and recurring decimals

Per cent means out of 100, so divide by 100.

$25 \div 100 = 0.25$

Multiply by 100: $0.3 \times 100 = 30$, so $0.3 = 30\%$.

Per cent means out of 100, so $17\% = \frac{17}{100}$.

$\frac{31}{100}$ means 31 hundredths.

$\frac{31}{100} = 0.31$

Per cent means 'out of 100', so $23\% = \frac{23}{100}$.

The 9 is in the tenths column, so $0.9 = \frac{9}{10}$.

The 7 is in the tenths column, so $0.7 = \frac{7}{10}$.

The 3 is in the tenths column, so $0.3 = \frac{3}{10}$.

To change a decimal to a percentage, multiply by 100.

$0.4 \times 100 = 40\%$

$\frac{4}{5} = \frac{8}{10} = 0.8$

Per cent means out of 100, so $35\% = \frac{35}{100}$.

This simplifies to $\frac{7}{20}$ by dividing top and bottom by 5.

Percent means out of 100.

$\frac{3}{10} = \frac{30}{100} = 30\%$

Hundredths are the second decimal place.

$\frac{9}{100} = 0.09$

Per cent means out of 100, so $31\% = \frac{31}{100}$.

$\frac{3}{4} = 3 \div 4 = 0.75$

(One quarter is 0.25, so three quarters is $3 \times 0.25 = 0.75$.)

The 7 is in the hundredths column, so $0.37$ means 37 hundredths.

$0.37 = \frac{37}{100}$

$\frac{1}{4} = 1 \div 4 = 0.25$

$0.25 \times 100 = 25\%$

Per cent means out of 100.

$40\% = \frac{40}{100} = \frac{2}{5}$

Multiply by 100 to turn a decimal into a percentage.

$0.73 \times 100 = 73\%$

Multiply by 100 to turn a decimal into a percentage.

$0.23 \times 100 = 23\%$

$0.75$ means 75 hundredths: $\frac{75}{100}$.

Divide top and bottom by 25: $\frac{75}{100} = \frac{3}{4}$

The 7 is in the tenths column, so $0.7 = \frac{7}{10}$.

Per cent means out of 100, so $19\% = \frac{19}{100}$.

Multiply by 100: $0.6 \times 100 = 60$, so it is 60%.

Per cent means out of 100, so $3\% = \frac{3}{100}$

$\frac{9}{10}$ means 9 tenths.

Nine tenths is written $0.9$.

$\frac{3}{10} = \frac{30}{100}$

So it is $30\%$.

Write each as a decimal: $\frac{1}{2} = 0.5$, $0.55$, $45\% = 0.45$

Smallest to largest: $0.45, 0.5, 0.55$

So: 45%, $\frac{1}{2}$, 0.55

0.31 is 31 hundredths, so $0.31 = \frac{31}{100}$.

Multiply top and bottom by 2 to get a denominator of 100:

$\frac{4}{50} = \frac{8}{100} = 8\%$

Change each to a decimal:

$\frac{7}{12} = 0.583...$, $57\% = 0.57$, $\frac{6}{11} = 0.545...$

Smallest first: $\frac{6}{11}$ (0.545), 0.558, 0.56, 57% (0.57), $\frac{7}{12}$ (0.583)

The last digit, 1, is in the hundredths column.

So $0.21 = \frac{21}{100}$

Let $x = 0.4626262...$

$1000x = 462.626262...$

$10x = 4.626262...$

Subtract: $990x = 458$

$x = \frac{458}{990} = \frac{229}{495}$

Let $x = 0.0723723...$

Then $10x = 0.723723...$ and $10\,000x = 723.723723...$

Subtract: $9990x = 723$

$x = \frac{723}{9990} = \frac{241}{3330}$ (dividing top and bottom by 3)

Let $x = 0.1232323...$

$10x = 1.232323...$ and $1000x = 123.232323...$

$1000x - 10x = 122$, so $990x = 122$

$x = \dfrac{122}{990} = \dfrac{61}{495}$

Let $x = 0.1171717\ldots$

$1000x = 117.171717\ldots$ and $10x = 1.171717\ldots$

$990x = 116$

$x = \frac{116}{990} = \frac{58}{495}$

Let $x = 1.06222...$

$1000x = 1062.222...$

$100x = 106.222...$

Subtract: $900x = 956$

$x = \frac{956}{900} = \frac{239}{225} = 1\frac{14}{225}$

In $0.\dot{4}\dot{3}$ the two digits 4 and 3 both repeat.

$10x = 4.3434...$ and $x = 0.4343...$ have different digits after the point, so subtracting them does not get rid of the recurring part.

He should use $100x = 43.\dot{4}\dot{3}$, then $99x = 43$, so $x = \frac{43}{99}$.

Let $x = 0.7333\ldots$

$10x = 7.333\ldots$ and $100x = 73.333\ldots$

$100x - 10x = 90x = 66$

$x = \frac{66}{90} = \frac{11}{15}$

Let $x = 0.4181818\ldots$

$1000x = 418.181818\ldots$ and $10x = 4.181818\ldots$

$990x = 414$

$x = \frac{414}{990} = \frac{23}{55}$

Let $x = 0.2565656\ldots$

$10x = 2.565656\ldots$ and $1000x = 256.565656\ldots$

Subtract: $990x = 254$

$x = \frac{254}{990} = \frac{127}{495}$

Let $a = 0.151515\ldots$: $100a - a = 15$, so $a = \frac{15}{99} = \frac{5}{33}$

Let $b = 0.2272727\ldots$: $1000b - 10b = 227.27\ldots - 2.27\ldots = 225$, so $b = \frac{225}{990} = \frac{5}{22}$

$\frac{5}{33} + \frac{5}{22} = \frac{10}{66} + \frac{15}{66} = \frac{25}{66}$, so $m = 25$