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### Course: 5th grade (Eureka Math/EngageNY) > Unit 2

Lesson 7: Topic G: Partial quotients and multi-digit decimal division- Estimating decimal division
- Estimating with dividing decimals
- Dividing decimals
- Division strategies for decimal quotients
- Multi-digit division strategies for decimals
- Divide whole numbers with decimal quotients: 78÷12
- Divide whole numbers to get a decimal (2-digit divisors)
- Pattern when dividing by tenths and hundredths
- Divide whole numbers by 0.1 or 0.01

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# Division strategies for decimal quotients

There are various methods for division when the quotient is not a whole number. Different examples demonstrate how to use fractions, tenths, and reciprocals to multiply instead of divide so we can arrive at the correct answer.

## Want to join the conversation?

- how did you still get the anwser to 3 divieded by 2 i kept getting 5.6?(35 votes)
- 3/2 essentially means how many times will 2 go into three. Your answer multiplied times two should give you three. one and a half times 2 equals three. If your answer was 5.6 that would be 5.6 times two which is 11.2. So for example, try 11.2 divided by 2, the answer is 5.6 :P(47 votes)

- was it suposed to be 7/70 or 70/7?(19 votes)
- Sal's problem is 7/70 = 1/10 = 0.1 as a decimal.(14 votes)

- Why do we have to multiply with the reciprocal to divide?(15 votes)
- You don't - there are other methods of going about it. One such method is simply to divide the numerator of the first fraction by the numerator of the second fraction, and similarly for the denominators. In fact, this method helps us understand why we can simply invert and multiply in the first place!

Take the example (3/5)÷(4/7). Dividing across gives us (3÷4)/(5÷7), or (3/4)/(5/7). We can multiply by 4/4 (which is equal to 1) in order to produce a whole number for our numerator (3/(20/7), and we can multiply by 7/7 (which is also equal to 1) to give us a whole number for our denominator, producing a final result of (21/20). This is actually the same answer we get when multiplying by the reciprocal!

Here's what's going on here: after dividing 3 by 4, we then multiplied by 4 and then 7. Multiplication by 4 cancels out division by 4, so essentially all we did here is multiply by 7 (the denominator of the second fraction). Similarly, we divided 5 by 7, multiplied by 4 and then 7. The multiplication and division of 7 cancel each other out, so we essentially just multiplied by 4 - the numerator of the second fraction.

Now we can see that inverting and multiplying is actually just a shortcut to the perhaps more intuitive method of just dividing across.(17 votes)

- Wait.. How exactly does " 7 / 7 x 10 " equal " 7 / 7 /10 "?(12 votes)
- I've never done this myself but I think he applied the inverse property to (7 x 10) which makes it 10 / 7

So now you have 7 / 10 / 7 = 0.1

or 7 / 7 / 10 = 0.1

It seems that when it's solely division, as long as the 1st number remains the same, the order of the numbers that follow it does not matter.

3 / 5 / 1 / 2 / 6 = 0.05

3 / 1 / 2 / 6 / 5 = 0.05

I'm not sure what this property is called though. I hope someone can help us out!

Edit: looks like I'm wrong. The inverse property I was thinking of applies to fractions and I didn't apply it correctly.

It would have to be like this:

7 / 70

= 7 x 1 / 70

= 7 / 70 Which goes back to the original form lol.

= 1 / 10

= 0.1(7 votes)

- what dose 3.9875677498396545684 divdded by 68 equal(11 votes)
- .0586407022035 :) hope this helps!(9 votes)

- Why is the remainder the numerator in the mixed number?(8 votes)
- That's how many 'wholes' you have left. Take the mixed number 5 1/3. We have 5 wholes, and 1/3 of a whole extra, therefore the remainder will be 1 (because there is
*one*third left).(4 votes)

- how do you divide decimals with a whole number do you get a whole number or a decimal?(5 votes)
- if you divided a decimal by a whole number you would get another decimal.(7 votes)

- welp sadly i strongly dont uderstand(5 votes)
- what don't you understand?(3 votes)

- The reason why 3 divide 0.5 = 1.5 because it is the half.(3 votes)
- No, 3 divided by 0.5 is 6.

This is because 0.5 is 1/2, and dividing by 1/2 is the same as multiplying by 2.

Or you can think of it this way: 3 divided by 0.5 is the number of 0.5’s that make 3. Two 0.5’s make 1, so six 0.5’s make 3. The answer is 6.

Note that 3**times**0.5 would be 1.5.

Have a blessed, wonderful day!(6 votes)

- I have a question: Can you divide as if there is no decimal then add the decimal point? i tried it and i got the same answer. It could be shorter for solving the part of the video

"6:36". Please reply so i can add to my notes.(3 votes)- You can if your teacher is fine with you solving it that way. Mathematically, it works out to be the same answer as long as you make sure you put the decimal point in the right place.(4 votes)

## Video transcript

- [Instructor] In this
video, we're gonna come up with some strategies for division when the quotient isn't a whole number, when it's going to be a decimal, so let's try to compute
three divided by two. Pause the video and see
if you can figure out what that is going to be,
and I'll give you a hint, it's not going to be a whole number. Alright, now let's try
to work this together. And like all things in mathematics, there's multiple
strategies that you can use to try to figure out what
three divided by two is. One strategy is, well,
let's just rewrite this as a fraction, so three divided by two, you can write that as, you could write that as three over two. Now, or you could say, this
is the same thing as 3/2. But how can we express this as a decimal? Well, you might recognize that 3/2 is the same thing as 2/2 plus 1/2, let me write that down. So this is the same thing
as two plus one over two, and I'm really doing every step here to hopefully make things clear, which is the same thing as two over two, so that's two over two, plus one over two, plus 1/2. I could break this up into
two over two plus 1/2. Now, two over two is just one, and so this is going to be equal to 1 1/2. Now, you might immediately say, "Hey, 1/2, I could write that as 5/10," and that would be exactly right. You could just, we don't
wanna spell out every step, we could say this is equal to one, and when we write it in decimal form, we express things as tenths
or hundredths or thousandths, so 1/2 is the same thing as 5/10, and if we wanna express that as a decimal, this would be equal to
one and five tenths. Now I did every step here,
as you get more practice, you say okay this is
the same thing as 3/2. 3/2, two goes into three one time, and there's a 1/2 left over, so writing this as a
mixed number, it's 1 1/2, and 1/2 written as a decimal
is 0.50, so this is 1.50. Now another way that we
could've thought about this is "Okay, I'm not getting a whole number, "when I divide three divided by two. "Maybe I'll get something
in terms of tenths, "so let me express each of
these in terms of tenths." So three is how many tenths? Well, three is 30 tenths, and we'd be dividing by two, we're gonna be dividing it by two, so 30 tenths divided by two, well that's going to
be equal to 15 tenths. This is equal to 15
tenths, which is equal to 10 tenths and five tenths, or 1.50. So both of these are equally
legitimate strategies for figuring out what
three divided by two is. I like the first one a little bit, it leverages what we know about fractions, but let's do another example. Let's do a few more examples, this is fun. Let's figure out what
34 divided by four is, and like before, pause this
video and try to figure it out and try to see if you can
use some of the strategies that we used in the last video. Alright, so as we just said, we can re-express this as a fraction, this is the same thing
as 34 divided by four, 34 divided by four, or 34/4. Now what is this going to be equal to? Well, four goes into 34 eight times, it's gonna go eight times, and you're gonna have two left over, so this is eight and 2/4. Eight, let me write it both out. Alright, eight and 2/4, let me do that two in
that blue just for fun. Eight and two over four, 8 2/4, so how do I do this again? I said four goes into 34 eight times, and then I have two left over, so I'm gonna have 2/4 left over. Another way, if you
wanna see all the steps, you can say, "Hey I can
rewrite this as 32 over four "plus two over four." The 32 over four is our eight, so 8 2/4. Well 2/4, that's the same thing as 1/2. That's the same thing as 1/2, and if we wanna express
that in terms of tenths, this is equal to eight and, 1/2 is the same thing as 5/10, 8 5/10, which if we wanna
express as a decimal is of course eighth and five tenths, or 8.50, and we are done. Let's do another one of these, and actually let's do one of them where we're dividing into a decimal, where a decimal itself is being divided. Let's say we wanted to calculate 8.4 divided by seven. Pause this video and see
if you can figure it out. When you look at this,
you might immediately say well I know 84 is divisible by seven. We know if you know
your seven times tables, we know that seven
times 12 is equal to 84, or that 84 divided seven is equal to 12, but this isn't 84, this is 8.4, so how do we think about it? Well one way, we can think
about it in terms of tenths, 8.4 is the same thing as 84 tenths, and so 84 tenths divided by seven, well 84 of anything divided by seven is going to be 12 of that thing, so it's gonna be 12 tenths, and 12 tenths we can
rewrite as one and 2/10, 1.20 and we are done, so this is equal to 1.2. Another way that we could
have thought about this is we could've said, you know what, 84 tenths is the same thing as 84 over 10. If fact, you would read this as 84/10, and now you wanna divided this by seven. So you wanna divide this by
seven, this is the same thing, when you divide by something, it's the same as multiplying
by the reciprocal, so it's 84 over ten times 1/7, which is equal to, we can
change, let me write it this way. This is equal to 84 over 10 times seven, over ten times seven, and
now we could simplify this. If we divide the numerator
and the denominator by seven, 84 divided by seven is 12,
seven divided by seven is one. 12 divided by ten, this is
gonna be equal to 12/10, which is 1.2, we could write
this as 1 2/10, or 1.2. Let's do one more example
that's kind of related. Let's say we wanna figure out
what seven divided by 70 is. Pause this video and try to figure it out. Well, we can rewrite this, as we've been doing, as seven over 70. Instead of writing it as 70, let me write that as seven times ten, and what's valuable about this is we can divide the numerator
and the denominator by seven. If we divide the numerator
by seven, we get a one, we divide the denominator
by seven, we get a one. Remember, we can do the same thing to the numerator and the denominator. If we multiply or divide
by the same value, we're not changing the value
of the actual fraction. And so you're left with 1/10, which if you express it as a decimal, you go to the tenths place, you say I have one of those
tenths, that's one tenth, so this is one tenth. Another way to think about it is this is the same thing as, and really this is what
we wrote over here, but you could write this
as seven divided by seven, divided by the blue seven,
and then you divide by 10. If you're dividing by seven times ten, actually let me write that down. If you're dividing by seven times ten, and this essentially
another way of writing what we have over here,
this is going to be equal to seven divided by seven divided by 10. Well, seven divided by seven is one, so you get one divided by
10, which is one tenth.