Decimals

Which proportion is shaded (in decimals)?

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Decimals and fractions (easy)

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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)

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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

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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)

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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:

240− 322562241600280160240÷32=73287560÷32=1280÷32=86000001..160÷32=532×825632×7224


Dividing decimals (medium)

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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:

240− 322562241600280160240÷32=73287560÷32=1280÷32=86000001..160÷32=532×825632×7224


Dividing decimals (difficult)

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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:

240− 322562241600280160240÷32=73287560÷32=1280÷32=86000001..160÷32=532×825632×7224


Converting repeating decimals to fractions

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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:
$$9x=35$$
$$x=35/9$$



   
   

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