Ratio and Proportion

Ratios

Writing Ratios as Fractions A ratio is the quotient of two quantities. The ratio of 1 to 3 can be written as 1 3 1 to 3 or or 1 : 3 fractional notation colon notation

Writing a Ratio as a Fraction The order of the quantities is important when writing ratios. To write a ratio as a fraction, write the first number of the ratio as the numerator of the fraction and the second number as the denominator. 1 3 3 1 The ratio of 1 to 3 is , not .

Simplifying Ratios To simplify a ratio, we just write the fraction in simplest form. Common factors can be divided out as well as common units. The ratio 4 to 6 in simplest form is the ratio 2 to 3 because .

Example 1 Example 2 A rectangle has length of 8.5 in and width 3.5 in. Find the ratio of length to width as a fraction reduced to the lowest terms. Slide 6

Rates

Writing Rates as Fractions A special type of ratio is a rate. Rates are used to compare different kinds of quantities. For example, a runner can run 3 miles in 30 minutes. Written as a rate, this is This means that the runner’s rate is 1 mile per 10 minutes.

Helpful Hint When comparing quantities with different units, write the units as part of the comparison. They do not divide out. Same units in ratios: Different Units in rates:

Finding Unit Rates hour A unit rate is a rate with a denominator of 1. A familiar example of a unit rate is 55 mph, read as “55 miles per hour.” This means 55 miles per 1 hour. 55 miles 1 hour denominator of 1

Rates are used extensively in sports, business, medicine, and science applications. One of the most common uses of rates is in consumer economics. When a unit rate is “money per item,” it is also called a unit price.

Martina bought 5 lb of russet potatoes for $4.99. Finding Unit Rates Example 1 Martina bought 5 lb of russet potatoes for $4.99. What was the rate in cents per pound? Example 2 One car travels 422 miles on 15 gallons of gasoline. Another car travels 354 miles on 13 gallons. Which car gets the better gas mileage? Round to the nearest tenth. Slide 12

Proportions

A proportion is a statement that two ratios or rates are equal. Writing Proportions A proportion is a statement that two ratios or rates are equal. Proportion A proportion states that two ratios are equal. If and are two ratios, then is a proportion. a b c d a b c d =

= 4 5 8 10 We read the proportion “4 is to 5 as 8 is to 10.” How to check, if this proportion true?

Using Cross Products to Determine Whether Proportions Are True or False b c d = b • c a • d cross product cross product If cross products are equal, the proportion is true. If cross products are not equal, the proportion is false.

= = For any true proportion, the cross products are equal. 4 5 8 10 if then 5 8 4 10 · = product of means product of extremes

Proportions and Problem Solving

Solving Problems by Writing Proportions Writing proportions is a powerful tool for solving problems in business, chemistry, biology, health sciences, and engineering. Given a specified ratio (or rate) of two quantities, a proportion can be used to determine an unknown quantity.

Recommended Dosage To control a fever, a doctor suggests that a child who weighs 28 kg be given 320 mg of a liquid pain reliever. If the dosage is proportional to the child’s weight, how much of the medication is recommended for a child who weighs 35 kg? Slide 21

1. Familiarize. 2. Translate. 3. Solve. Let t = the number of milligrams of the liquid pain reliever for a child who weigh 35 kg . 2. Translate. 3. Solve.

4. Check. The cross products are the same. 5. State the answer. The dosage for a child who weighs 35 kg is 400 mg.

Solve on your own. A basketball player completes 45 baskets out of every 100 attempts. How many attempts would she have to make to make 225 baskets?

Congruent and Similar Triangles

Congruent Triangles Two triangles are congruent when they have the same shape and the same size. Corresponding angles are equal, and corresponding sides are equal. equal angles a = 6 c = 11 d = 6 e = 11 b = 9 f = 9 equal angles equal angles

Similar Triangles Similar triangles are found in art, engineering, architecture, biology, and chemistry. Two triangles are similar when they have the same shape but not necessarily the same size.

In similar triangles, the measures of corresponding angles are equal and corresponding sides are in proportion. a = 3 d = 6 e = 10 b = 5 c = 8 f = 16 Side a corresponds to side d, side b corresponds to side e, and side c corresponds to side f. a d = 3 6 1 2 b e = 5 10 1 2 c f = 8 16 1 2

Solving Similar Triangles Given length of sides in similar triangles find the length of side x. 11in x in 6 in 9 in