Ratios

A ratio is a comparison between two or more like quantities in the same units. This suggests that a ratio can be simplified by dividing (or multiplying) its terms by the same number. Note: The ratio 1 : 2 is read as '1 to 2' or '1 is to 2'. Simplification of Ratios Example 1 Express the ratio 6 : 9 in its simplest form. Solution: Note: The ratio 2 : 3 is in its simplest form since the numbers 2 and 3 have no common factor. Scale Factor If the ratio is expressed in the form 1 : n, then n is called the scale factor. E.g.  10 : 100 = 1 : 10 So, 10 is the scale factor. Equivalent Ratios Clearly, 5 : 20 = 1 : 4 We say that 5 : 20 and 1 : 4 are equivalent. In general: Multiplying both terms of the ratio a : b by the same number, c, results in the equivalent ratio ac : bc. Finding the Ratio of Two Quantities We must express both quantities in the same unit of measurement to find the ratio of two quantities in its simplest form. Example 2 Find the ratio of 40 centimetres to 2 metres in its simplest form. Solution: Note: We can use ratios to compare more than two quantities conveniently.  Fractions are not usually suitable for this. Dividing a Quantity in a Given Ratio Consider a line segment AB of length 9 cm, which is divided into two line segments of length 3 cm and 6 cm, as shown in the diagram. That is, AP = 3 cm and BP = 6 cm. Conversely, if we know that the point P divides the line segment AB of length 9 cm in the ratio 1 : 2, we can determine the lengths of AP and BP as follows: So, the lengths of AP and BP are 3 cm and 6 cm, respectively. Check: 3 cm + 6 cm = 9 cm Example 3 Anne and Allen worked together on a project and received $250 for their completed work. Allen worked for 2 days and Anne worked for 3 days, and they agree to divide the money between them in the ratio 2 : 3.  How much should each receive? Solution: We picture the $250 divided into equal parts. Now, there are 5 parts and the smaller amount is 2 of them. So, the smaller amount (Allen's share) is $100, and so Anne's share is $250 – $100 = $150. Example 4 An urn contains red and black marbles in the ratio 2 : 3.  If there are 40 red marbles, find the total number of marbles in the urn. Solution: Ratio of red marbles to black marbles = 2 : 3 Number of red marbles (2 parts out of 5) = 40 Let x be the number of marbles in the urn. Hence the number of marbles in the urn is 100. Proportion The statement of equality of two ratios is called a proportion. E.g.  2 : 3 = 8 : 12 is a proportion. The proportion 2 : 3 = 8 : 12 is read as '2 is to 3' as '8 is to 12'. Example 5 If 2 : 3 = x : 6, find x. Solution: Example 6 If 6 : x = 18 : 27, find x. Solution: