1. Ratios and Proportions
Skill 36

2. Objective
HSG-SRT.5: Students are responsible for writing ratios and using them to solve proportions.

3. Definitions
A ratio is a comparison of two quantities by division.
An extended ratio compares three or more numbers.
An equation that states that two ratios are equal is called a proportion.
The first and last numbers in a proportion are the extremes.
The middle numbers in a proportion are the means.

4. If 𝒂 𝒃 = 𝒄 𝒅 , then 𝒂𝒅=𝒃𝒄. (1) 𝒂 𝒃 = 𝒄 𝒅 ↔ 𝒃 𝒂 = 𝒅 𝒄
Properties
Cross Products Property
If 𝒂 𝒃 = 𝒄 𝒅 , then 𝒂𝒅=𝒃𝒄.
Properties of Proportions
(1) 𝒂 𝒃 = 𝒄 𝒅 ↔ 𝒃 𝒂 = 𝒅 𝒄
(2) 𝒂 𝒃 = 𝒄 𝒅 ↔ 𝒂 𝒄 = 𝒃 𝒅
(3) 𝒂 𝒃 = 𝒄 𝒅 ↔ 𝒂+𝒃 𝒃 = 𝒄+𝒅 𝒅

5. Example 1; Writing a Ratio
The bonsai bald cypress tree is a small version of a full-size tree. A Florida bald cypress tree, called the Senator, stands 118 feet tall. What is the ratio of the height of the 15 inch bonsai to the height of the Senator.
𝟏𝟏𝟖 𝒇𝒕. 𝟏 ∗ 𝟏𝟐 𝒊𝒏. 𝟏 𝒇𝒕.
=𝟏𝟒𝟏𝟔 𝒊𝒏.
𝑩𝒐𝒏𝒔𝒂𝒊 𝑺𝒆𝒏𝒂𝒕𝒐𝒓 = 𝟏𝟓 𝒊𝒏. 𝟏𝟒𝟏𝟔 𝒊𝒏.
= 𝟓 𝟒𝟕𝟐
𝒓𝒂𝒕𝒊𝒐 𝒊𝒔 𝟓 :𝟒𝟕𝟐

6. Example 2; Dividing a Quantity into a Given Ratio
Members of the school band are buying pots of tulips and pots of daffodils to sell at their fundraiser. They plan to buy 120 pots of flowers. The ratio tulips to daffodils will be two-thirds. How many pots of each type of flower should they buy?
𝒕𝒖𝒍𝒊𝒑𝒔 𝒅𝒂𝒇𝒇𝒐𝒅𝒊𝒍𝒔 = 𝟐 𝟑 = 𝟐𝒙 𝟑𝒙
𝒕𝒖𝒍𝒊𝒑𝒔+𝒅𝒂𝒇𝒇𝒐𝒅𝒊𝒍𝒔=𝟏𝟐𝟎
𝟐𝒙+𝟑𝒙=𝟏𝟐𝟎
𝟓𝒙=𝟏𝟐𝟎
𝒙=𝟐𝟒
𝒕𝒖𝒍𝒊𝒑𝒔=𝟐 𝟐𝟒
=𝟒𝟖
𝒅𝒂𝒇𝒇𝒐𝒅𝒊𝒍𝒔=𝟑 𝟐𝟒
=𝟕𝟐

7. Example 3; Using an Extended Ratio
a) The lengths of the sides of a triangle are in the extended ratio 3:5:6. The perimeter of the triangle is 98 in. What is the length of the longest side?
𝟑𝒙+𝟓𝒙+𝟔𝒙=𝟗𝟖
𝟏𝟒𝒙=𝟗𝟖 𝟓𝒙
𝒙=𝟕 𝟑𝒙
𝑳𝒐𝒏𝒈𝒆𝒔𝒕 𝒔𝒊𝒅𝒆=𝟔𝒙
𝟔𝒙=𝟔 𝟕 𝟔𝒙
=𝟒𝟐 𝒊𝒏.

8. Example 3; Using an Extended Ratio
b) The lengths of the sides of a triangle are in the extended ratio 4:7:9. The perimeter of the triangle is 60 cm. What is the length of the sides?
𝟒𝒙+𝟕𝒙+𝟗𝒙=𝟔𝟎 𝟕𝒙 𝟒𝒙
𝟐𝟎𝒙=𝟔𝟎
𝒙=𝟑 𝟗𝒙
𝑺𝒊𝒅𝒆=𝟒𝒙
=𝟒 𝟑 =𝟏𝟐 𝒄𝒎
𝑺𝒊𝒅𝒆=𝟕𝒙
=𝟕 𝟑 =𝟐𝟏 𝒄𝒎
𝑺𝒊𝒅𝒆=𝟗𝒙
=𝟗 𝟑 =𝟐𝟕 𝒄𝒎

9. Example 4; Solving a Proportion
What is the solution to each proportion?
a) 6 𝑥 = b) 𝑦+4 9 = 𝑦 3 c) 15 𝑚+1 = 3 𝑚
𝟗𝒚=𝟑 𝒚+𝟒
𝟏𝟓𝒎=𝟑 𝒎+𝟏
𝟓𝒙=𝟔 𝟒
𝟓𝒙=𝟐𝟒
𝟗𝒚=𝟑𝒚+𝟏𝟐
𝟏𝟓𝒎=𝟑𝒎+𝟑
𝒙=𝟒.𝟖
𝟔𝒚=𝟏𝟐
𝟏𝟐𝒎=𝟑
𝒚=𝟐
𝒎=𝟎.𝟐𝟓

10. Example 4; Solving a Proportion
a) Suppose 𝑥 6 = 𝑦 7 . Write complete the ratio to create an equivalent proportion.
𝑥 𝑦 =
𝟔 𝟕
Property of Proportions (2)
b) Suppose 𝑥 6 = 𝑦 7 . Write complete the ratio to create an equivalent proportion.
6 𝑥 =
𝟕 𝒚
Property of Proportions (1)
c) Suppose 𝑥 6 = 𝑦 7 . Write complete the ratio to create an equivalent proportion.
= 𝑦+7 7
𝒙+𝟔 𝟔
Property of Proportions (3)

11. #36: Ratios and Proportions
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