Fractions, percentages and decimals

Fractions

A fraction is a number expressed as the division of two quantities, where the numerator is the number on top and the denominator is the number at the bottom.
If the numerator is smaller than the denominator, the fraction is proper. For example, 3/4 is a proper fraction.
If the numerator is larger than the denominator, the fraction is improper. For example, 4/3 is an improper fraction.
Reduction or simplification of fractions involves reducing the numerator and denominator to the lowest terms, by finding the highest common factor.
The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of 5/8 is 8/5.
When adding or subtracting fractions, common denominators are usually required. The least common denominator is the least common multiple of the two denominators.

Percentages

Percentages is a way of expressing a number as a fraction of 100. It is widely used for comparing quantities expressed in different units.
Calculation of percentages may involve: finding what certain per cent is of a number, increasing/decreasing a number by a certain per cent, and finding the original quantity after a certain per cent has been added or subtracted.
To convert a fraction to a percentage, you divide the numerator by the denominator, multiply by 100, and add the % sign. For instance, 1/4 becomes 25%.
To convert a decimal to a percentage, you multiply by 100 and add the % sign. For instance, 0.75 becomes 75%.

Decimals

Decimals are base-10 numbers and represent fractions, whereby, the point separates the whole number from the fractional part.
Decimals can be expressed as fractions with the denominator being a power of 10. For example, 0.8 equals 8/10 or can be simplified further to 4/5.
Recurring decimals are those that possess digits that repeat indefinitely. For instance, 2/3 = 0.666…
A terminating decimal is a number which can be written as a decimal number without a fractional component or with a fractional part that is a multiple of a power of 10. For example, the decimals 0.414, 0.45, and 0.3 are all terminating decimals.
To convert a fraction to a decimal, you simply divide the numerator by the denominator. For example, 3/4 becomes 0.75.
To convert a percentage to a decimal, you divide by 100. For example, 75% becomes 0.75.

Solving Problems involving Fractions, Percentages, and Decimals

Solving problems often involves conversion between fractions, percentages and decimals. Understanding the relation between these different forms is key.
Recognise the type of problem: is it a percentage increase or decrease, conversion, or comparison? Then decide the most practical method to use.
Remember to keep track of what the quantities represent, and always answer the question asked.