Greg’s house is 4/5 mile from the library.So far,Greg has walked 5/6 of the distance.How many miles has he walked?

Question

It’s my homework, and no one can help me

Answer 1

TL;DR: To find out how much a portion of an amount is, we multiply the amount by the portion. Since they are both fractions, you need to remember to multiply the top numbers (numerators) together and then the bottom numbers together (denominators).

In depth:

When we say 4/5 mile, it's just the short way of saying:

1 mile divided up into 5 pieces, and using 4 of those pieces.

We can shorten this phrase by using some symbols you'd probably recognize:

1 mile ÷ 5 pieces × 4 pieces

But remember, since we're using division and multiplication, it doesn't actually matter in what order these operations are done as long as you don't start with division:

4 pieces ÷ 5 pieces × 1 mile

So fractions are really just tiny division problems! That's what the division symbol, ÷, represents (it's called an obelus). There's two dots separated by a horizontal line – the top dot goes with the number to the left of the ÷ and the bottom dot goes with the number to the right of the ÷. Sometimes, instead of using the dots to represent the numbers on either side, it's easier to just put the numbers in directly:

4 pieces

––––––– × 1 mile

5 pieces

The top and bottom of the fraction are both an amount of "pieces", and since they're the same, we don't really need to know what they are:

4

– × 1 mile

5

And multiplying anything by 1 just gives you the same value, so we can hide that:

4

– mile

5

And of course, this format takes up a bunch of space, so we use a slanted line instead of a horizontal line to make it smaller:

4/5 mile

So using the same thought process but in reverse, we can do this with two fractions now:

4/5 mile divided up into 6 pieces, and using 5 of those pieces

4/5 mile ÷ 6 pieces × 5 pieces

5 pieces ÷ 6 pieces × 4/5 mile

5 pieces

––––––– × 4/5 mile

6 pieces

5

– × 4/5 mile

6

5/6 × 4/5 mile

Eventually, we see that we can skip to directly the line above after enough practice. You'll use this shortcut a lot! At this point, we have to remember that it doesn't matter the order we do the multiplication and division. Let's go back to using ÷ instead of /:

5 ÷ 6 × 4 ÷ 5 mile

Let's also rearrange everything to put similar operations next to each other:

5 × 4 ÷ 6 ÷ 5 mile

The multiplication we can do pretty easily:

20 ÷ 6 ÷ 5 mile

but the repeated division gets annoying! So instead of saying "Take 20 and split it up into 6 pieces, and then split each of those pieces into 5 smaller pieces", we can just say "Well, if each of the 6 pieces ends up being split into 5 smaller pieces, then we're really ending up with 30 pieces since 6 × 5 = 30." So we can write:

20 ÷ (6 × 5) mile

And order of operations (PEMDAS) says we should start with parenthesis, so:

20 ÷ 30 mile

20/30 mile

And there you have it! While that is a ton of work, the pattern of thinking that we use is the same no matter what fractions we're given: when multiplying fractions, you can just multiply the top numbers together to give you the top number of the new fraction. The same goes for the bottom numbers. So going back to the problem set up:

5/6 × 4/5 mile

5 4

– × – mile

6 5

5 × 4

–––– mile

6 × 5

20

–– mile

30

Would you look at that! We could have been done like 20 steps ago. But it's important to know not only how we can solve a problem, but also why we're able to solve it the way we do.

I'll leave it to you to simplify the fraction!