Ratio A ration is a comparison between two

Ratio A ration is a comparison between two numbers. We generally separate the two numbers in the ratio with a colon(: ). Suppose we want to write the ratio of 8 and 12. We can write this as 8: 12 or as a fraction 8/12, and we say the ratio is eight to twelve.

Ratio Examples: Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange. 1. What is the ratio of books to marbles? Expressed as a fraction, with the numerator equal to the first quantity and the denominator equal to the second, the answer is 7/4. Two other ways of writing the ratio are 7 to 4 and 7: 4

Ratio Jeannine has a bag with 3 videocassettes, 4 marbles, 7 books, and 1 orange. What is the ratio of videocassettes to the total number of items in the bag? There are 3 videocassettes, and 3 + 4 + 7 + 1 = 15 items in total. The answer can be expressed as 3/15, 3 to 15, or 3: 15

Comparing Ratios To compare ratios, write them as fractions. The ratios are equal if they are equal when written as fractions. Example: Are the ratios 3 to 4 and 6: 8 equal? The ratios are equal if 3/4 = 6/8. These are equal if their cross multiplications are equal; that is, if 3 x 8=4 x 6 Since both of these products equal 24, the answer is yes, the ratios are equal.

Comparing Ratios REMEMBER TO BE CAREFUL! ORDER MATTERS! A ratio of 1: 7 is not the same as a ratio of 7: 1 Examples: Are the ratios 7: 1 and 4: 81 equal? NO! If we cross multiply we get 7 x 81 = 567 and 1 x 4 = 4. Since the products are not equal, the ratios are not equal.

Comparing Ratios Are 7: 14 and 36: 72 equal? If you cross multiply these numbers you receive 7 x 72 = 504 and 14 x 36 = 504. Since both products are the equal, the ratios are also equal

Proportion A proportion is a comparison between two RATIOS. If the cross products are equal then the proportions are equal. A proportion is an equation with a ratio on each side. It is a statement that two ratios are equal. 3/4 = 6/8 is an example of a proportion.

Proportions • When one of four numbers in a proportion is unknown, cross multiplying can be used to find the unknown number. This is called solving the proportion. Question marks or letters are often used in place of the unknown number. • Example: solve for n: 1/2 = n/4 • Using cross multiplication we see that 2 x n = 1 x 4, so 2 x n = 4. Dividing both sides by 2, n = 2 so that n = 2.

Rate • A rate is a ratio that expresses how long it takes to do something, such as traveling a certain distance. To walk 3 kilometers in one hour is to walk at a rate of 3 km/h. The fraction expressing a rate has units of distance in the numerator and units of time in the denominator. • Problems involving rates typically involve setting two ratios equal to each other and solving for an unknown quantity, that is, solving a proportion.

Rate Juan runs 4 km in 30 minutes. At that rate, how far can he run in 45 minutes? Give the unknown quantity the name n. In this case, n is the number of km Juan could run in 45 minutes at the given rate. We know that running 4 km in 30 minutes is the same as running n km in 45 minutes, that is the rates are the same.

Rates • So we have the proportion 4 km/30 min = n km/45 min, or 4/30 = n/45. • Finding the cross product and setting them equal, we get 30 x n = 4 x 45, or 30 n = 180. • We now divided both sides by 30, we find that n = 180/30 = 6 and the answer is 6 km.

Problems 1. Ratio: Out of the 248 seniors, 112 are boys. What is the ratio of boys to the total number of seniors? 2. Madison’s flower garden measures 8 feet long by 6 feet wide. What is the ratio of length to width?

Problems Solve the following proportions: 3. 4.

Problems 5. For the first four home football games, the concession stand sold a total of 600 hotdogs. If that ratio stays constant, how many hotdogs will they sell for all 10 home games?

Ratio and Proportions 6. If 2 inches represent 80 miles on a scale drawing, how long would a line segment be that represents 320 miles?

Ratio and Proportion 7. A drink contains 20% cranberry juice to the rest is apple juice. What is the ratio of cranberry juice to apple juice? A. 1: 4 B. 1: 20 C. 4: 1 D. 20: 1

Ratio and Proportions Triangle ABC is similar to triangle XYZ as shown. 8. What is the length of side BC? A. 3. 0 B. 3. 6 C. 6. 0 D. 6. 6

9. Shondra used 6 ounces of chocolate chips to make two dozen cookies. At this rate, how many ounces of chocolate chips would she need to make seven dozen cookies? A. 72 B. 21 C. 36 D. 18

ANSWERS: : : 1. Ratio: Out of the 248 seniors, 112 are boys. What is the ratio of boys to the total number of seniors? Ans. 112/248 2. Madison’s flower garden measures 8 feet long by 6 feet wide. What is the ratio of length to width? Ans. 8 to 6, 8/6 or 8: 6 3. 4. Ans. 15 Ans. 4

5. For the first four home football games, the concession stand sold a total of 600 hotdogs. If that ratio stays constant, how many hotdogs will they sell for all 10 home games? Answer – 1500 hotdogs 6. If 2 inches represent 80 miles on a scale drawing, how long would a line segment be that represents 320 miles? Answer – 8 inches

7. A drink contains 20% cranberry juice to the rest is apple juice. What is the ratio of cranberry juice to apple juice? A. 1: 4 B. 1: 20 C. 4: 1 D. 20: 1 Answer - A Triangle ABC is similar to triangle XYZ as shown. 8. What is the length of side BC? A. 3. 0 B. 3. 6 C. 6. 0 D. 6. 6 Answer - B

9. Shondra used 6 ounces of chocolate chips to make two dozen cookies. At this rate, how many ounces of chocolate chips would she need to make seven dozen cookies? A. 72 B. 21 C. 36 D. 18 Answer - B