# Decimals

Which proportion is shaded (in decimals)?

Decimals and fractions (easy)

## Fractions and decimals

Decimal numbers are another way to represent numbers that are smaller than the unit 1. Decimals are written to the right of the units place separated by a period called decimal point. The first digit to the right of the decimal point represents tenths, the second represents hundredths, the third represents thousandths, etc.

\$\$30/100=3/10=0.3\$\$
\$\$46/100=0.46\$\$

If you want to compare or do calculations with fractions and decimals, you need to convert all of them to one type (either decimals or fractions).

### Convert decimals to fractions:

1. Convert the decimal to fraction using tenths, hundredths, thousandths, etc. depending on the number of decimal places:

• one place=tenth, e.g. \$0.5=5/10; ; 3.3=33/10\$
• two places=hundredths, e.g. \$0.02=2/100 ;; 0.12=12/100 ;; 1.38=138/100\$
• three places=thousandth, e.g. \$0.002=2/1000; ; 0.304=304/1000\$

2. Simplify the fraction to the lowest common term:
\$\$5/10=1/2 ;; 2/100 = 1/50\$\$

### Convert fractions to decimals

1. Find a number you can multiply by the denominator (bottom of the fraction) to make it 10, or 100, or 1000, etc.

\$\$1/2={1×5}/{2×5}=5/bo10\$\$
\$\$1/4={1×25}/{4×25}=25/bo100\$\$

2. A fraction in this form can be converted to a decimal - just put the decimal point in the correct position (one space from the right hand side for every zero in the bottom number):
E.g. \$5/10=0.5;\$ ; \$25/100=0.25\$
For some numbers there is no way to multiply them to become tens, hundreds or thousands. For these numbers you can calculate an approximate decimal (finding a denominator close to 10, 100 or 1000):

\$\$1/3={1×333}/{3×333}=\$\$
\$\$=333/bo999≈333/bo1000=0.333\$\$

### Convert whole numbers to fractions

Just put 1 below the whole number:

\$\$8=8/1\$\$

Decimals and fractions (medium)

## Fractions and decimals

Decimal numbers are another way to represent numbers that are smaller than the unit 1. Decimals are written to the right of the units place separated by a period called decimal point. The first digit to the right of the decimal point represents tenths, the second represents hundredths, the third represents thousandths, etc.

\$\$30/100=3/10=0.3\$\$
\$\$46/100=0.46\$\$

If you want to compare or do calculations with fractions and decimals, you need to convert all of them to one type (either decimals or fractions).

### Convert decimals to fractions:

1. Convert the decimal to fraction using tenths, hundredths, thousandths, etc. depending on the number of decimal places:

• one place=tenth, e.g. \$0.5=5/10; ; 3.3=33/10\$
• two places=hundredths, e.g. \$0.02=2/100 ;; 0.12=12/100 ;; 1.38=138/100\$
• three places=thousandth, e.g. \$0.002=2/1000; ; 0.304=304/1000\$

2. Simplify the fraction to the lowest common term:
\$\$5/10=1/2 ;; 2/100 = 1/50\$\$

### Convert fractions to decimals

1. Find a number you can multiply by the denominator (bottom of the fraction) to make it 10, or 100, or 1000, etc.

\$\$1/2={1×5}/{2×5}=5/bo10\$\$
\$\$1/4={1×25}/{4×25}=25/bo100\$\$

2. A fraction in this form can be converted to a decimal - just put the decimal point in the correct position (one space from the right hand side for every zero in the bottom number):
E.g. \$5/10=0.5;\$ ; \$25/100=0.25\$
For some numbers there is no way to multiply them to become tens, hundreds or thousands. For these numbers you can calculate an approximate decimal (finding a denominator close to 10, 100 or 1000):

\$\$1/3={1×333}/{3×333}=\$\$
\$\$=333/bo999≈333/bo1000=0.333\$\$

### Convert whole numbers to fractions

Just put 1 below the whole number:

\$\$8=8/1\$\$

Compare decimals, fractions and whole numbers

## Fractions and decimals

Decimal numbers are another way to represent numbers that are smaller than the unit 1. Decimals are written to the right of the units place separated by a period called decimal point. The first digit to the right of the decimal point represents tenths, the second represents hundredths, the third represents thousandths, etc.

\$\$30/100=3/10=0.3\$\$
\$\$46/100=0.46\$\$

If you want to compare or do calculations with fractions and decimals, you need to convert all of them to one type (either decimals or fractions).

### Convert decimals to fractions:

1. Convert the decimal to fraction using tenths, hundredths, thousandths, etc. depending on the number of decimal places:

• one place=tenth, e.g. \$0.5=5/10; ; 3.3=33/10\$
• two places=hundredths, e.g. \$0.02=2/100 ;; 0.12=12/100 ;; 1.38=138/100\$
• three places=thousandth, e.g. \$0.002=2/1000; ; 0.304=304/1000\$

2. Simplify the fraction to the lowest common term:
\$\$5/10=1/2 ;; 2/100 = 1/50\$\$

### Convert fractions to decimals

1. Find a number you can multiply by the denominator (bottom of the fraction) to make it 10, or 100, or 1000, etc.

\$\$1/2={1×5}/{2×5}=5/bo10\$\$
\$\$1/4={1×25}/{4×25}=25/bo100\$\$

2. A fraction in this form can be converted to a decimal - just put the decimal point in the correct position (one space from the right hand side for every zero in the bottom number):
E.g. \$5/10=0.5;\$ ; \$25/100=0.25\$
For some numbers there is no way to multiply them to become tens, hundreds or thousands. For these numbers you can calculate an approximate decimal (finding a denominator close to 10, 100 or 1000):

\$\$1/3={1×333}/{3×333}=\$\$
\$\$=333/bo999≈333/bo1000=0.333\$\$

### Convert whole numbers to fractions

Just put 1 below the whole number:

\$\$8=8/1\$\$

Dividing decimals (easy)

## Dividing decimals

At first multiply the dividend and the divisor by as many 10's as we need to make them whole numbers. This multiplying will not change the result:

\$\$ 0.3÷0.2=3÷2=1.5\$\$
\$\$ 0.9÷0.04=90÷4=22.5\$\$
\$\$ 0.001÷0.2=1÷200=0.005\$\$

We can also multiply the dividend and the divisor by the same number to change the divisor to 1, 10 or 100:

\$\$8÷0.2=(8×5)÷(0.2×5)=40÷1=40\$\$
\$\$4.5÷0.25=(4.5×4)÷(0.25×4)=18÷1=18\$\$

You can also convert decimals to fractions, especially when doing division:

\$\$0.7÷0.25={7/10}÷{25/100}=\$\$
\$\$={7/10}÷{1/4}=7/10×4=28/10=2.8\$\$
\$\$0.5×0.6=5/10×6/10=30/100=3/10=0.3\$\$

We can also use long division:

\$\$0.6÷3.2=?\$\$

At first, we multiply each number by 10 in order to turn these numbers into whole numbers:

\$\$0.6÷3.2=6÷32=?\$\$

Now we can do long division. However, we need to be sure to put the decimal point in the quotient right above the decimal point in the dividend:

Dividing decimals (medium)

## Dividing decimals

At first multiply the dividend and the divisor by as many 10's as we need to make them whole numbers. This multiplying will not change the result:

\$\$ 0.3÷0.2=3÷2=1.5\$\$
\$\$ 0.9÷0.04=90÷4=22.5\$\$
\$\$ 0.001÷0.2=1÷200=0.005\$\$

We can also multiply the dividend and the divisor by the same number to change the divisor to 1, 10 or 100:

\$\$8÷0.2=(8×5)÷(0.2×5)=40÷1=40\$\$
\$\$4.5÷0.25=(4.5×4)÷(0.25×4)=18÷1=18\$\$

You can also convert decimals to fractions, especially when doing division:

\$\$0.7÷0.25={7/10}÷{25/100}=\$\$
\$\$={7/10}÷{1/4}=7/10×4=28/10=2.8\$\$
\$\$0.5×0.6=5/10×6/10=30/100=3/10=0.3\$\$

We can also use long division:

\$\$0.6÷3.2=?\$\$

At first, we multiply each number by 10 in order to turn these numbers into whole numbers:

\$\$0.6÷3.2=6÷32=?\$\$

Now we can do long division. However, we need to be sure to put the decimal point in the quotient right above the decimal point in the dividend:

Dividing decimals (difficult)

## Dividing decimals

At first multiply the dividend and the divisor by as many 10's as we need to make them whole numbers. This multiplying will not change the result:

\$\$ 0.3÷0.2=3÷2=1.5\$\$
\$\$ 0.9÷0.04=90÷4=22.5\$\$
\$\$ 0.001÷0.2=1÷200=0.005\$\$

We can also multiply the dividend and the divisor by the same number to change the divisor to 1, 10 or 100:

\$\$8÷0.2=(8×5)÷(0.2×5)=40÷1=40\$\$
\$\$4.5÷0.25=(4.5×4)÷(0.25×4)=18÷1=18\$\$

You can also convert decimals to fractions, especially when doing division:

\$\$0.7÷0.25={7/10}÷{25/100}=\$\$
\$\$={7/10}÷{1/4}=7/10×4=28/10=2.8\$\$
\$\$0.5×0.6=5/10×6/10=30/100=3/10=0.3\$\$

We can also use long division:

\$\$0.6÷3.2=?\$\$

At first, we multiply each number by 10 in order to turn these numbers into whole numbers:

\$\$0.6÷3.2=6÷32=?\$\$

Now we can do long division. However, we need to be sure to put the decimal point in the quotient right above the decimal point in the dividend:

Converting repeating decimals to fractions

## Repeating decimals

A recurring/repeating decimal is a number whose digits are periodic (repeating its values at regular intervals) and keep repeating forever, e.g. \$1/3=0.333333 ...\$ The infinitely repeated digit sequence is called the repetend. It can be denoted by a horizontal line (a vinculum) or dots above it, e.g. \$0.2ov57=0.257575757 ...\$

Every repeating or terminating decimal is a rational number since it can be converted to a fraction.
To convert repeating decimal to fraction follow these steps:

Step 1:
Set the repeating decimal equal to fraction x:

\$\$3.888ov8=x\$\$

Step 2:
Move the repeating digit(s)/repetend to the left of the decimal point by multiplying the decimal by 10, 100, 1000 etc.
\$\$10x=38.888ov8\$\$

Step 3:
Subtract the number from both sides of the equation. This will help you to get rid off the decimal part:
\$\$10x−x=38.88ov8−3.88ov8\$\$

Step 4:
Solve the equation for x:
\$\$9x=35\$\$
\$\$x=35/9\$\$