Multiplying fractions word problem: bike

you can ride your bike 1/5 of a mile per minute if it takes you 3 and 1/3 minutes to get to your friends house how many miles away does your friend live and this here's pictures of these guys on bicycles it's pretty clear they're not riding to work or some of these guys aren't even riding a bicycle but let's focus on the question so you can ride your bike 1/5 of a mile per minute and you're going to do this for three and a third minutes times three and a third so we really have to figure out how do we multiply 1/5 times 3 and 1/3 so there's a couple of ways to think about it you could literally view a 3 and a third as this is the same thing as 1/5 times 3 times 3 plus 1/3 that's exactly what 3 and 1/3 is and then we can just apply the distributive property this would be 1/5 times 3 1/5 times 3 I'm going to keep the colors the same times 3 plus 1/5 plus 1/5 times 1/3 x times 1/3 and this is going to be equal to well we could rewrite 1/5 times 3 as 1/5 times 3 over 1 that's what 3 really is if we wrote it as a fraction and then of course we're going to have plus 1/5 1/5 times 1/3 times 1/3 and let's just think about what each of these evaluate to here you multiply the numerators and you multiply the denominator so this is going to be equal to 1 times 3 over 5 times 1 and this business right over here is going to be and remember order of operations we want to do our multiplication first so this is going to be 1 times 1 over 5 times 3 over 5 times 3 and so that's going to be equal to 3 over 5 plus plus 1 over 15 and now we have different denominators here but lucky for us 3 over 5 if we multiply the numerator and the denominator by 3 we're going to get it later of 15 and so that's equal to 9 over 15 plus 1 over 15 which equals 10 over 15 and if you divide the numerator and the denominator both by 5 you're going to get 2 over 3 so your friend lives 2/3 miles away from your house well that's kind of interesting and this tote was kind of a long way to do it let's think about if there's a simpler way to do it so this is the same thing as 1/5 times and I'm just going to write 3 and 1/3 as a mixed number so it's 1/5 times 3 and 1/3 can be rewritten as 9/3 I'm sorry I'm going to rewrite 3 and 1/3 is an improper fraction so this is the same thing as 9 thirds that's 3 plus 1/3 which is the same thing as 1/5 why switch colors arbitrarily which is the same thing same thing I'm still in the same color as 1/5 times 1/5 times 9 thirds plus 1/3 is 10 thirds and now we can just multiply the numerator and multiply the denominator the numerator or multiply the numerators so this is 1 times 10 1 times 10 I'm trying to stay good with the color-coding over 5 times 3 over 5 times 3 which is exactly equal to what we just got 1 times 10 is equal to 10 5 times 3 is 15 10 15 so we already established is the same thing as 2/3 so your friend lives 2/3 of a mile away from you