Fundamental Ideas
Similarity
Ratio is a concept that you have probably encountered in other math classes. It is a comparison of sizes.
Ratio
The
ratio of two numbers
a and
b is the fraction
, usually expressed in reduced form. An alternative form involves a colon. The colon form is most frequently used when comparing three or more numbers to each other. See Table
1 .
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Example 1: A classroom has 25 boys and 15 girls. What is the ratio of boys to girls?
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The ratio of boys to girls is 5 to 3, or 5/3, or 5 : 3.
Example 2: The ratio of two supplementary angles is 2 to 3. Find the measure of each angle.
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The angles have measures of 72° and 108°.
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2 x to 3 x reduces to 2 to 3.
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2 x + 3 x = 180° (The sum of supplementary angles is 180°.)
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5 x = 180°
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x = 36°
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Then, 2 x = 2(36°) and 3 x = 3(36°).
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So, 2 x = 72° and 3 x = 108°
The angles have measures of 72° and 108°.
Example 3: A triangle has angle measures of 40°, 50°, and 90°. In simplest form, what is the ratio of these angles to each other?
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40 : 50 : 90 = 4 : 5 : 9 (10 is a common divisor)
This means that:
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The ratio of the first to the second is 4 to 5.
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The ratio of the first to the third is 4 to 9.
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The ratio of the second to the third is 5 to 9.
Example 4: A 50-inch segment is divided into three parts whose lengths have the ratio 2 : 3 : 5. What is the length of the longest part?
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The longest part has a measure of 25 inches.
Proportion
A proportion is an equation stating that two ratios are equal.
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Means and extremes
The extremes are the terms in a proportion that are the farthest apart when the proportion is written in colon form ( a:b = c:d). In the foregoing, a and d are extremes. The means are the two terms closest to each other.
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In the preceding proportion, the values a and d are called extremes of the proportion; the values b and c are called the means of the proportion.
