Proportional Relationships 04:45 minutes

Video Transcript

Transcript Proportional Relationships

My dear Aunt Sally, who sometimes gets a little confused, is reading a postcard she just got from her niece, Flo. Flo is writing to see if her aunt would like to come watch her run her first 5K race. Apparently, 5 kilometers is only about 3.1 miles, so Aunt Sally decides to surprise her niece and run WITH her in the race. To figure out how long it will take Aunt Sally to run the 5K, we're going to need to use proportional relationships. Let’s look at the facts. Flo figures it will take her nine minutes to run one mile. We can write this information as a ratio. Remember, the numbers in a ratio tell us how much there is of one quantity when compared to another. We could use a variety of different formats, but let's write our ratio as a fraction, comparing time to distance. Flo runs at a constant rate of 9 minutes for every 1 mile. Since the rate will never change, we can say that the relationship between time and distance is proportional. Just like equivalent fractions, a proportion is a statement that relates two quantities. So, how long will it take Flo to run the race? We know that she wants to run 3.1 miles, so we can plug that in for distance. We want to find the time it will take, so we will let 't' equal time. We can solve for 't' using cross-multiplication. First, we multiply the numerator on the left side of the equal sign by the denominator on the right side of the equal sign. Then, we set that equal to the product of the numerator on the right side and the denominator on the left side. 9 times 3.1 is 27.9. So this means it will take Flo 27.9 minutes to complete the race. Let's just round that up to 28 minutes to make things easier. But what about Aunt Sally? Unlike her niece Flo, Aunt Sally is a little out of practice. So it takes her 13 minutes to run 1 mile. Let’s write this constant rate as a ratio. If Flo can run the race in 28 minutes, where will Aunt Sally be when her niece finishes? Just like we did before, we can write a proportion and then solve for the unknown quantity by using cross-multiplication. This time 'd' is equal to the distance in miles Aunt Sally can run in 28 minutes, while running at her constant rate of 13 minutes per mile. Just like before, cross multiply and set the two products equal to each other, giving us 13d is equal to 28. Next, use the opposite operation of division to isolate the 'd'. Dividing both sides by 13, 'd' is equal to 2.15 miles. Which means Aunt Sally can go 2.15 miles in 28 minutes. I guess Flo will be waiting for her at the finish line! But how long will Sally need to finish the whole race? To find out, again we can set up a proportion using the constant rate 13 minutes for every mile. This time, ‘t’ is equal to the time it will take Sally to run 3.1 miles. Cross-multiply like we did before, giving us 13 times 3.1 is equal to 't'. So it looks like Aunt Sally will finish the race in just over 40 minutes, if she runs at a constant rate of 13 minutes per mile. That's not bad for somebody who's a little out of practice! So, to review... A proportional relationship describes a relationship between two or more numbers, like the relationship between time and distance. When solving problems involving proportional relationships, you first have to set up your proportion, then substitute in the known values, cross-multiply and simplify. It looks like Aunt Sally has finished the race, or actually, the race has finished HER.