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#### 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.
• mixed number contains a whole number and a fractional number. E.g. 23/4 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
• 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
1. 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
2. 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
3. Express 45/8 as an improper fraction.
• 45/8 = 4 × 8 + 5/8 = 32 + 5/8 = 37/8
4. Express 19/8 as a mixed number.
• 19 ÷ 8 = 2, remainder 3.
• 19/8 = 23/8
5. 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 = 15/24
6. 7/104/15
• The LCM of 10 and 15 is 30.
• 7/104/15 = 7 × 3/10 × 34 × 2/15 × 2 = 21/308/30
•                      = 21 – 8/30
• 7/104/15 = 13/30
7. 33/5 + 22/3
•             = 18/5 + 8/3
•             = 54/15 + 40/15
•             = 94/15
• ∴ 33/5 + 22/3 = 61/15
8. Simplify 3/8 of 22/9 × 13/5.
•                               = 3/8 × 20/9 × 8/5
•                        = 4/3
• ∴ 3/8 of 22/9 × 13/5 = 11/3
9. Find the value of 21/4 ÷ 3/7.
•                   = 21/4 × 7/3
•                   = 9/4 × 7/3
•                   = 21/4
• ∴ 21/4 ÷ 3/7 = 51/4
10. Express 15% as a fraction in its lowest terms.
• 15% = 15/100 = 3 × 5/20 × 5 = 3/20
11. Express 2/5 as a percentage.
• 2/5 = 2 × 20/5 × 20
•      = 40/100
•      = 40%
12. 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%