316 CHAPTER 6 Algebra, Graphs, and Functions Did You Know? Survival of the Fittest Symbols come and symbols go; the ones that find the greatest acceptance are the ones that survive. The Egyptians used pictorial symbols: a pair of legs walking forward for addition or backward for subtraction. Robert Recorde (1510 –1558) used two parallel lines, ,= to represent “equals” because “no 2 thynges can be moore equalle.” Some symbols evolved from abbreviations, such as the “ + ” sign, which comes from the Latin et meaning “and.” The evolution of others is less clear. The invention of the printing press in the fifteenth century led to a greater standardization of symbols already in use. Proportions A ratio is a quotient of two quantities. An example is the ratio of 2 to 5, which can be written 2:5 or 2 5 or 2/5. Ratios are used in proportions. Example 4 Sales Tax at a Craft Show Tallulah is selling her homemade jewelry. Determine the cost of a necklace before tax if the total cost of a necklace, including an 8% sales tax, is $34.56. Solution We are asked to determine the cost of a necklace before sales tax. Let x cost of the necklace before sales tax = Then x 0.08 8% of the cost of the necklace (the sales tax) = + = + = = = = x x x x x Cost of the necklace before tax tax on the necklace 34.56 0.08 34.56 1.08 34.56 1.08 1.08 34.56 1.08 32 Thus, the cost of the necklace before tax is $32. 7 Now try Exercise 37 Definition: Proportion A proportion is a statement of equality between two ratios. An example of a proportion is a b c d , = where b 0 ≠ and d 0. ≠ Consider the proportion x x 2 5 5 8 + = + We can solve this proportion by first multiplying both sides of the equation by the LCD, 40. + = + ⎛ + ⎝⎜ ⎞ ⎠⎟ = ⎛ + ⎝⎜ ⎞ ⎠⎟ + = + + = + + = = = x x x x x x x x x x x 2 5 5 8 40 2 5 40 5 8 8( 2) 5( 5) 8 16 5 25 3 16 25 3 9 3 8 5 Multiplication property A check will show that 3 is the solution. Proportions can often be solved more easily by using cross-multiplication. Hadynyah/iStock/Getty Images
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