Ratios: a comparison of two numbers using division To see if a ratio is equivalent 1. Compare the fractions by finding the LCD 2. Simplify the ratios 3. Compare by using cross products

Ratio, Proportions, Percent Mr. Pontrella Grade 6

Ratios, Rates, Proportions and Percent Grade 6 Mr. Pontrella

Ratios We are comparing rectangles to triangles. We use ratios to make comparisons between two things. We are comparing rectangles to triangles. Ratios can be written 3 ways. 1. As a fraction 3 5 2. Using the word to 3 to 5 3. Using a colon 3:5   equivalent ratios Ratios that name the same comparisons To see if to ratios are equivalent 1. Change each to a decimal and compare the decimals. 2. Reduce both ratios and compare. 3. Use cross products.

Rate: is a ratio of 2 measurements with different units Rates Rate: is a ratio of 2 measurements with different units Here we are comparing days to inches Example: It rained 4 inches in 30 days The rate is 4 30 We can reduce to 2 15 Unit Rates A rate that has 1 unit as its second term (denominator) If a car travels 325 miles and uses 11 gallons of gas what is the mile per gallon? This is an example of a unit rate. How many miles per 1 gallon? Create a ratio Miles Gallons 325 11 Since every fraction is a division problem we divide 325/ 11 Our Unit Rate is 29. 54 miles per gallon

Proportions We can write proportions in 2 forms. a:b = c:d An equation that shows that two ratios are equal We can write proportions in 2 forms. a:b = c:d If 2 ratios are equal then their cross product will be equal. a * d = b * c

Using Proportions to Solve problems A car travels 125 miles in 5 hours. How many miles will the car travel in 8 hours? Solve using proportions. 125 = m 5 8 Proportion Set an equation using cross products 125 * 8 = 5 * m Simplify 1000 = 5m Solve by inverse operation (The opposite of multiplication is division ) 1000 /5 = m 200 = m In 8 hours a car can travel 200 miles

Scale Drawings and Proportions On a map 1.5 inches is equal to 5 miles. If the distance in real life is 22 mile how big will it be on the map? Proportions can help us with this problem. We know 1 ratio is 1.5 in: 5 m. We know 1 part of the second ratio is 22 m. Proportion 1.5 in = X m 5 m 22 m Notice we lined up m to m and in to in Cross Products 1.5 x 22 = 5 x X Simplify 33 = 5X 22 miles is equal to 6.6 inches on the map Inverse Operation 33/5 = X 6.6 = X

Numbers and Their Pieces Percents Another form of writing a piece of a number is by using percents. Percent means "out of 100.” With percents we are using a comparison of decimals and fractions to 100 pieces. 5 out of 100 = 5% = 0.05 = 1 10 out of 100 =10% = 0.1 = 1 20 10

Converting Percents and Decimals A decimal can be written as a percent, by moving the decimal point two places to the right like this: Follow the same procedure for decimals larger than 1 2.35 = 2.35 = 235% A percent can always be written as a decimal by moving the decimal point two places to the left like this: 68% = 68. = 0.68 Follow the same procedure for percents larger than 100 345% = 345. = 3.45

Converting Percents and Fractions To convert the fraction 3 to a percent 5 To convert 138% to a fraction Change the fraction to a decimal Place the percent over 100 0.6 5 3 138 Then simplify Then move the decimal 2 places right 100 38 19 138 = 1 = = 1 0.6 60% 100 50 100

The Percent Proportion Dinner cost $75 and you wish to leave a 20% tip. How much will the tip be? We can use the percent proportion to solve. P is the percentage ( a value that is a number for the percent P = R B B is the base or the original amount R is the rate(the percent number over 100) In this problem the Base is $75, the Rate is 20 over 100 and we are solving for the Percentage ( how much money is equal to 20%) Cross products 75 x 20 = P x 100 P = 20 $75 100 Simplify 1500 = 100P Inverse operation 1500/ 100 = P $15 = P The tip will be $15

3 Types or Percent We have seen 1 type of percent problem. Let’s look at 2 others. If we left a 20% tip which was $25, how much was the bill? We know R is 20% and P is $25. We need to find B. Proportion 25 = 20 B 100 Cross products 25 x 100 = 20 x B Simplify 2500 = 20B The dinner bill was $125 Inverse Operation 2500/ 20 = B $125 = B

If Dinner cost $125 and we left a $35 tip what percent of the bill was the tip? P is $35, B is $125. We are trying to find R. Proportion 35 = R 125 100 Cross Products 35 x 100 = 125 x R Simplify 3500 = 125 R The tip was 28% of the bill Inverse Operation 3500/ 125 = R 28% = R