Module 1 Ratios and Proportional Relationships Topic A

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a “ratio”? ratio: a comparison of two numbers by division – There are three ways to compare two numbers: a a to b a: b b read as “to” How do you write a ratio? 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. PRE-ALGEBRA

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a “rate”? rate: a ratio that compares two quantities measured in different units. Example: 120 miles is read as “ 120 miles ‘per’ or ‘each’ 4 hours” 4 hours What is a “unit rate”? 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: 120 miles 4 = 30 miles or 120 4 = 30 mph 1 hours 4 hours 4 The above unit rate is read as “ 30 miles ‘per’ hour” (30 m/h or 30 mph) PRE-ALGEBRA

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

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships What is a constant? constant: a term that has no variable Examples: 5, 104 What is a variable? Proportion Table? h hp 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. PRE-ALGEBRA

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 PRE-ALGEBRA

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

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

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. 25/gal 2. You pay $21. 60 for a 20 -lb bag of dog food. $1. 08/lb 3. A bird flies at a speed of 384 in. /s. What is its 1, 920 ft/min 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? 4 PRE-ALGEBRA

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships Lesson Quiz Solve each proportion. 1. a = 5 12 6 10 2. 3 = 6 8 x 16 Does each pair of ratios form a proportion? Explain. 2 0. 4 3. 9 = 1. 8 Yes; the cross products are equal. 21 14 4. 50 = 25 No; the cross products are not equal. 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. about 43 nautical miles PRE-ALGEBRA

Module 1: Ratios and Proportional Relationships: Topic A: Proportional Relationships Solve. 1. Parallelogram ABCD ~ parallelogram EFGH. Find the value of x. 12 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? 8 ft 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? 300 mi PRE-ALGEBRA