#### Fractions, Percentages, and Decimals

**FRACTIONS**

- A fraction represents part of a whole quantity.
- A fraction is in two parts; a
*numerator*and a*denominator*

*Equivalent fractions*all represent the same amount. For example,^{4}/_{9},^{8}/_{18},^{40}/_{90}are equivalent fractions.- If two or more fractions have the same denominator, we say they have a
*common denominator*. - The
*lowest common denominator*of a set of fractions is the LCM of the denominators. - When a fraction is such that the numerator and denominator have no common factor, we say the fraction is in its
*lowest terms*or in its*simplest form*. - A
*mixed number*contains a whole number and a fractional number. E.g. 2^{3}/_{4}is a mixed number. - In a fraction, if the numerator is:
- less than the denominator, it is a
*proper*fraction. E.g.^{2}/_{5} - greater than the denominator, it is an
*improper*fraction. E.g.^{5}/_{2}

- less than the denominator, it is a
- To add or subtract fractions:
- change mixed numbers to improper fractions,
- find the lowest common denominator,
- express the fractions as equivalent fractions with the same denominators.

- To multiply fractions:
- change mixed numbers to improper fractions,
- multiply the numerators together to give the numerator of the product,
- multiply the denominators to give the denominator of the product.

- The
*reciprocal (also referred to as inverse)*of a fraction is the same fraction turned upside down. E.g.^{9}/_{4}is the reciprocal or inverse of^{4}/_{9}. - To divide by a fraction, multiply by its reciprocal.

**PERCENTAGES**

*Percentage*or*per cent*means hundredths. ∴ 34% is another way of writing^{34}/_{100}- The symbol ‘%’ is short for per cent.
- To change a percentage to a fraction, write it as a fraction of 100 and reduce it to its lowest terms, where possible.
- To change a fraction to a percentage, multiply the fraction by 100.
- To express a quantity as a percentage of another:
- make sure they are in the same units,
- express the first as a fraction of the second,
- multiply the fraction by 100.

**DECIMALS**

- Decimals are another way to write fractions in which the denominators are 10 and times 10. For example: tenths (
^{1}/_{10}), hundredths (^{1}/_{100}), etc. - They are written with a dot before them. This dot is called a
*decimal point*. - The decimal point separates the whole numbers from the decimal figures. For example:

- The decimal figures have their values reduced as they move away from the decimal point.
- Examples of decimals include: 23.45, 344.3, 10445.57712, etc.
- Decimals can be expressed as fractions with denominators 10, 100, 1000, 10000, etc. To achieve this:
- Write the number and draw a line under it,
- Write a 1 under the decimal point, and
- Replace each of the numbers to the right of the decimal point with zeros.
- For example:

- To change a fraction to a decimal:
- Count the number of zeros in the denominator.
- Use the number of zeros in the denominator to count backwards from the end of the numerator,
- At the digit where you stop, put a decimal point before the digit.
- For example:

__Examples__

__Examples__

- Express each of the fractions
^{3}/_{4},^{5}/_{8},^{7}/_{8},^{2}/_{3}with a denominator of 24. Hence arrange the fractions in ascending order (i.e., from lowest to highest).^{3}/_{4}=^{3 × 6}/_{4 × 6}=^{18}/_{24}^{5}/_{8}=^{5 × 3}/_{8 × 3}=^{15}/_{24}^{7}/_{8}=^{7 × 3}/_{8 × 3}=^{21}/_{24}^{2}/_{3}=^{2 × 8}/_{3 × 8}=^{16}/_{24}- The order is
^{15}/_{24},^{16}/_{24},^{18}/_{24},^{21}/_{24}, i.e.^{5}/_{8},^{2}/_{3},^{3}/_{4},^{7}/_{8}

- Express
^{42}/_{70}and^{26}/_{78}in their lowest terms.^{42}/_{70}=^{42 ÷ 7}/_{70 ÷ 7}=^{6}/_{10}=^{6÷2}/_{10÷2}=^{3}/_{5}**OR**by using prime factors:^{42}/_{70}=^{2 × 3 × 7}/_{2 × 5 × 7}=^{2÷2 × 3 × 7÷7}/_{2÷2 × 5 × 7÷7}=^{1 × 3 × 1}/_{1 × 5 × 1}=^{3}/_{5}

^{26}/_{78}=^{26 ÷ 2}/_{78 ÷ 2}=^{13}/_{39}=^{13 ÷ 13}/_{39 ÷ 13}=^{1}/_{3}

- Express 4
^{5}/_{8}as an improper fraction.- 4
^{5}/_{8}=^{4 × 8 + 5}/_{8}=^{32 + 5}/_{8}=^{37}/_{8}

- 4
- Express
^{19}/_{8}as a mixed number.- 19 ÷ 8 = 2, remainder 3.
- ∴
^{19}/_{8}= 2^{3}/_{8}

^{5}/_{6}+^{3}/_{8}- The LCM of 6 and 24 is 24.
- ∴
^{5}/_{6}+^{3}/_{8}=^{5 × 4}/_{6 × 4}+^{3 × 3}/_{8 × 3}=^{20}/_{24}+^{9}/_{24} - =
^{20 + 9}/_{24} - =
^{29}/_{24} - ∴
^{5}/_{6}+^{3}/_{8}= 1^{5}/_{24}

^{7}/_{10}–^{4}/_{15}- The LCM of 10 and 15 is 30.
- ∴
^{7}/_{10}–^{4}/_{15}=^{7 × 3}/_{10 × 3}–^{4 × 2}/_{15 × 2}=^{21}/_{30}–^{8}/_{30} - =
^{21 – 8}/_{30} - ∴
^{7}/_{10}–^{4}/_{15}=^{13}/_{30}

- 3
^{3}/_{5}+ 2^{2}/_{3}- =
^{18}/_{5}+^{8}/_{3} - =
^{54}/_{15}+^{40}/_{15} - =
^{94}/_{15} - ∴ 3
^{3}/_{5}+ 2^{2}/_{3}= 6^{1}/_{15}

- =
- Simplify
^{3}/_{8}of 2^{2}/_{9}× 1^{3}/_{5}.- =
^{3}/_{8}×^{20}/_{9}×^{8}/_{5} - =
^{4}/_{3} - ∴
^{3}/_{8}of 2^{2}/_{9}× 1^{3}/_{5}= 1^{1}/_{3}

- =
- Find the value of 2
^{1}/_{4}÷^{3}/_{7}.- = 2
^{1}/_{4}×^{7}/_{3} - =
^{9}/_{4}×^{7}/_{3} - =
^{21}/_{4} - ∴ 2
^{1}/_{4}÷^{3}/_{7}= 5^{1}/_{4}

- = 2
- Express 15% as a fraction in its lowest terms.
- 15% =
^{15}/_{100}=^{3 × 5}/_{20 × 5}=^{3}/_{20}

- 15% =
- Express
^{2}/_{5}as a percentage.^{2}/_{5}=^{2 × 20}/_{5 × 20}- =
^{40}/_{100} - = 40%

- A worker hammers a post 2.2 m long into the ground. 66 cm of the post is below the ground. What percentage of the post is above the ground?
- Fraction of post below the ground =
^{66 cm}/_{2.2 m}=^{66 cm}/_{220 cm}=^{66}/_{220} - Percentage of the post below the ground =
^{66}/_{220}× 100% =^{6}/_{20}× 100 = 30% - ∴ Percentage of post above the ground = 100% – 30% = 70%

- Fraction of post below the ground =