Ratios & Proportions
Ratio (ray-she-yo) A ratio is the comparison of two numbers by division. A classroom has 16 boys and 12 girls. Also written as16 boys, 16:12 or 16 to girls Generally, ratios are in lowest terms: 16 = 16/4 = /4 3
Ratio, continued Ratios can compare two unlike things: ◦ Joe earned $40 in five hours ◦ The ratio is 40 dollars or 8 dollars 5 hours 1 hour ◦ When the denominator is one, this is called a unit rate.
Ratio, continued Let’s look at a classroom: Ratios can be part-to-part ◦ 16 boys 15 girls Ratios can be part-to-whole ◦ 16 boys 31 students
Now, on to proportions! What is a proportion? A proportion is an equation that equates two ratios So we have a proportion :
Tell whether the ratios are proportional A. Since the cross products are equal, the ratios are proportional. 60 = ? Using Cross Products to Identify Proportions 60 = 60 Find cross products
A mixture of fuel for a certain small engine should be 4 parts gasoline to 1 part oil. If you combine 5 quarts of oil with 15 quarts of gasoline, will the mixture be correct? 4 parts gasoline 1 part oil = ? 15 quarts gasoline 5 quarts oil 4 5 = = ≠ 15 The ratios are not equal. The mixture will not be correct. Set up ratios. Find the cross products. Using Cross Products to Identify Proportions
Tell whether the ratios are proportional. Try This: Example 1A Since the cross products are equal, the ratios are proportional = 20 Find cross products A. = ?
A mixture for a certain brand of tea should be 3 parts tea to 1 part sugar. If you combine 4 tablespoons of sugar with 12 tablespoons of tea, will the mixture be correct? Try This: Example 1B 3 parts tea 1 part sugar = ? 12 tablespoons tea 4 tablespoons sugar 3 4 = = = 12 The ratios are equal. The mixture will be correct. Set up ratios. Find the cross products.
RATIOS & PROPORTIONS Are the following proportions? FALSE – Not a proportion TRUE – this is a proportion
Solving Proportions When you do not know one of the four numbers in a proportion, set the cross products equal to each other and solve.
Solve the proportion. 6p = 12 5 p = 10 6p = 60 Find the cross products. Solve p 12 = ; the proportion checks =
Solve the proportion = 2g 21 = g 42 = 2g Find the cross products. Solve g = ; the proportion checks =
RATIOS & PROPORTIONS Find the missing numbers to make the following proportions. 9 10
Proportions Proportion is a statement that says two ratios are equal. ◦ In an election, Damon got three votes for each two votes that Shannon got. Damon got 72 votes. How many votes did Shannon get? ◦ Damon 3 = 72 ◦ Shannon 2 n n = 48, so Shannon got 48 votes.
Proportions, continued Tires cost two for $75. How much will four tires cost? ◦ # of tires 2 = 4 cost 75 n n = 150, so four tires cost $150
Proportion, continued Three cans of soup costs $5. How much will 12 cans cost? # of cans 3 = 12 cost 5 n n = 20, so 12 cans cost $20
Now you know enough about properties, let’s solve the Mysterious problems! If your car gets 30 miles/gallon, how many gallons of gas do you need to commute to school everyday? 5 miles to school 5 miles to home Let x be the number gallons we need for a day: Can you solve it from here? x = Gal
So you use up 1/3 gallon a day. How many gallons would you use for a week? 5 miles to school 5 miles to home Let t be the number of gallons we need for a week: Gal What property is this?
So you use up 5/3 gallons a week (which is about 1.67 gallons). Consider if the price of gas is 3.69 dollars/gal, how much would it cost for a week? Let s be the sum of cost for a week: 5 miles to school 5 miles to home 3.69(1.67) = 1s s = 6.16 dollars
So what do you think? 10 miles You pay about 6 bucks a week just to get to school! What about weekends? If you travel twice as much on weekends, say drive 10 miles to the Mall and 10 miles back, how many gallons do you need now? How much would it cost totally? How much would it cost for a month? 5 miles Think proportionally!... It’s all about proportions!
Exit Ticket Tell whether each pair of ratios is proportional = ? = ? Solve each proportion yes no n = 30 n = n 12 = n =