Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a “ratio”? How do you write a ratio? ratio: a comparison of two numbers by division – There are three ways to compare two numbers: a to b a : b You can write a ratio to compare a part to a part or part to a whole. The first item being compared is always the first number and the second item being compared is the second number of the ratio. Example: A recipe calls for 4 cups of cereal and 2 cups of pretzels. Write this as a ratio. a b read as “to”

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a “rate”? What is a “unit rate”? rate: a ratio that compares two quantities measured in different units. Example: is read as “120 miles ‘per’ or ‘each’ 4 hours” unit rate: the rate in which the denominator is 1 (rate ‘per’ 1 unit of a given quantity) - Unit rate is used in everyday life to determine the best value. It can be found by simplifying the fraction to a denominator of “1” or simply dividing the numerator (top number) by the denominator (bottom number. Example: = or 120  4 = 30 mph The above unit rate is read as “30 miles ‘per’ hour” (30 m/h or 30 mph) 120 miles 4 hours 120 miles 4 hours  4 30 miles 1 hours

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is an Equivalent Ratio? Remember…… Write the ratios as fractions. Compare the two fractions to see Multiply both sides by 3 × 18 Simplify 2:3 and 10:18 Two ratios are equivalent if their cross products are equal. The cross products are not equal, so the ratios are not equivalent.

h What is a constant? What is a variable? Proportion Table? Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a constant? What is a variable? Proportion Table? constant: a term that has no variable Examples: 5, 104 h p h Proportional quantities have a constant ratio. This constant ratio is called the constant of variation. The constant of variation often represents a rate or a cost.

The constant of variation is 1. The cost is $1 per guest. Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is the cost per guest? The constant of variation, which often represents a rate or a cost, will tell you the answer. To find the constant of variation, divide "Cost" by "Guests". The constant of variation is 1. The cost is $1 per guest. Coordinate Plane Ratios Table

Start at the origin. The x-coordinate is -4 . Since the x-coordinate is negative, count four units to the left.

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a “proportion”? What is the “extremes of the proportion”? What is the “means of the proportion”? What are “cross products”? Why do “cross products” work? proportion: equal ratios (in other words, equal fractions) Example: extremes of the proportion: the first cross product of a proportion. In the above proportion, the “extremes of the proportion” are a and d. means of the proportion: the second cross product of a proportion. In the above proportion, the “extremes of the proportion” are b and c. cross products: the product of the means equals the product of the extremes (in the above example, ad = bc). Rule: ad = bc Example: Show that = . a b c d for b ≠ 0 and d ≠ 0 = 4 5 12 15 60 = 60 

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships POTD Find each unit rate. 1. You pay $4.50 for 2 gallons of orange juice. 2. You pay $21.60 for a 20-lb bag of dog food. 3. A bird flies at a speed of 384 in./s. What is its speed in units of ft/min? 4. A job takes 96 person-days. How many workers are needed to complete the job in 24 days? $2.25/gal $1.08/lb 1,920 ft/min 4

Does each pair of ratios form a proportion? Explain. 3. 4. Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships Lesson Quiz Solve each proportion. 1. 2. Does each pair of ratios form a proportion? Explain. 3. 4. 5. 100 nautical miles equals about 115 statute miles. About how far in nautical miles is 50 statute miles? Round to the nearest whole number. a 12 5 6 3 8 6 x = 10 = 16 2 9 0.4 1.8 21 50 14 25 = = Yes; the cross products are equal. No; the cross products are not equal. about 43 nautical miles

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships Solve. 1. Parallelogram ABCD ~ parallelogram EFGH. Find the value of x. 2. A girl who is 4 feet tall casts a shadow that is 6 feet long. The tree next to her casts a shadow that is 12 feet long. How tall is the tree? 3. The scale on a map is 3 in. : 100 mi. What is the actual distance between two towns that are 9 in. apart on the map? 12 8 ft 300 mi