Ratios and Proportions

There are many uses of ratios and proportions There are many uses of ratios and proportions. We use them in map reading, making scale drawings and models, solving problems.

The most recognizable use of ratios and proportions is drawing models and plans for construction. Scales must be used to approximate what the actual object will be like.

A ratio is a comparison of two quantities by division A ratio is a comparison of two quantities by division. In the rectangles below, the ratio of shaded area to unshaded area is 1:2, 2:4, 3:6, and 4:8. All the rectangles have equivalent shaded areas. Ratios that make the same comparison are equivalent ratios.

Using ratios The ratio of faculty members to students in one school is 1:15. There are 675 students. How many faculty members are there? faculty 1 students 15 1 x 15 675 15x = 675 x = 45 faculty =

A ratio of one number to another number is the quotient of the first number divided by the second. (As long as the second number ≠ 0)

A ratio can be written in a variety of ways. You can use ratios to compare quantities or describe rates. Proportions are used in many fields, including construction, photography, and medicine. a:b a/b a to b

Since ratios that make the same comparison are equivalent ratios, they all reduce to the same value. 2 3 1 10 15 5 = =

Two ratios that are equal Proportions Two ratios that are equal

A proportion is an equation that states that two ratios are equal, such as:

In simple proportions, all you need to do is examine the fractions In simple proportions, all you need to do is examine the fractions. If the fractions both reduce to the same value, the proportion is true. This is a true proportion, since both fractions reduce to 1/3. 5 2 15 6 =

In simple proportions, you can use this same approach when solving for a missing part of a proportion. Remember that both fractions must reduce to the same value. To determine the unknown value you must cross multiply. (3)(x) = (2)(9) 3x = 18 x = 6 Check your proportion (3)(x) = (2)(9) (3)(6) = (2)(9) 18 = 18 True!

So, ratios that are equivalent are said to be proportional So, ratios that are equivalent are said to be proportional. Cross Multiply makes solving or proving proportions much easier. In this example 3x = 18, x = 6. If you remember, this is like finding equivalent fractions when you are adding or subtracting fractions.

1) Are the following true proportions? 2 10 3 5 2 10 3 15 = =

2) Solve for x: 4 x 6 42 =

3) Solve for x: 25 5 x 2 =

Example A A forest ranger who is 150 cm tall casts a shadow 45 cm long. At the same time, a nearby tree casts a shadow 195 cm long. Write and solve a proportion to find the height of the tree. 45x = 29250 Since x is multiplied by 45, divide both sides by 45 to undo the multiplication. x = 650 The tree is 650 centimeters tall.

Example B A woman who is 5.5 feet tall casts a shadow 3.5 feet long. At the same time, a building casts a shadow 28 feet long. Write and solve a proportion to find the height of the building. 3.5x = 154 Since x is multiplied by 3.5, divide both sides by 3.5 to undo the multiplication. x = 44 The building is 44 feet tall.

Solve the following problems. 4) If 4 tickets to a show cost $9.00, find the cost of 14 tickets. 5) A house which is appraised for $10,000 pays $300 in taxes. What should the tax be on a house appraised at $15,000.

The two windows below are similar The two windows below are similar. Find the unknown width of the larger window.

These two buildings are similar. Find the height of the large building.