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. 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
- 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
- 42/70 = 42 ÷ 7/70 ÷ 7 = 6/10 = 6÷2/10÷2 = 3/5 OR by using prime factors:
- Express 45/8 as an improper fraction.
- 45/8 = 4 × 8 + 5/8 = 32 + 5/8 = 37/8
- Express 19/8 as a mixed number.
- 19 ÷ 8 = 2, remainder 3.
- ∴ 19/8 = 23/8
- 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 = 15/24
- 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
- 33/5 + 22/3
- = 18/5 + 8/3
- = 54/15 + 40/15
- = 94/15
- ∴ 33/5 + 22/3 = 61/15
- 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
- Find the value of 21/4 ÷ 3/7.
- = 21/4 × 7/3
- = 9/4 × 7/3
- = 21/4
- ∴ 21/4 ÷ 3/7 = 51/4
- Express 15% as a fraction in its lowest terms.
- 15% = 15/100 = 3 × 5/20 × 5 = 3/20
- 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%