Not an easy question. The total number of bottles each person has is not known.


Part 1

Total capacity of Edwina's 5 small bottles = 5 × 400ml = 2000ml = 2l

Total capacity of Georgia's 16 big bottles = 16 × 0.6l = 9.6l

Difference in capacity here = 9.6l - 2l = 7.6l

If we only look at these bottles, then Georgia has 7.6l more capacity than Edwina.


But, question says the capacity of Edwina's bottles (all of them) is 0.8l more than Georgia's


So this tells us that :

In order to have this overall difference, the capacity of Edwina's big bottles must be greater than Georgia's small bottles by 0.8l more than 7.6l .

Difference here = 0.8l + 7.6l = 8.4l


Part 2
Important thing to note in the question : They have an equal number of bottles.

If we compare Edwina's 5 small bottles to Georgia's 16 big bottles, the difference is 11 bottles (16 - 5 = 11)

Since the number of big bottles Georgia has is 11 more than the number of small bottles Edwina has, in order to have the same total number of bottles,

Edwina's number of big bottles must be 11 more than Georgia's number of small bottles.

Part 3

We know the difference between the capacity here must be 8.4l.

But here the number of big bottles and small bottles they have is different. It is hard to compare.

So what we do first is to subtract that capacity of the 11 more big bottles that Edwina has. By doing this,

we are now comparing the difference between the capacity of small bottles Georgia has and the capacity of an equal number of big bottles belonging to Edwina.

11 × 0.6l = 6.6l
8.4l - 6.6l = 1.8l


Now we use grouping difference.

400ml = 0.4l

Difference between 1 big and 1 small bottle's capacity = 0.6l - 0.4l = 0.2l

This is 1 set.

How many big bottles would have a capacity that is 1.8l more than a the same number of small bottles?

1.8l ÷ 0.2l = 9

9 big bottles contain 1.8l more than 9 small bottles

Part 4

From part 3 we have found that :

Georgia' number of small bottles = 9
Edwina's number of big bottles = 9 + 11 = 20

So for (a), the answer is 20.



(b)

Total capacity of Edwina's bottles

= 20 big and 5 small
= 20 × 0.6l + 2l (already calculated in part 1)
= 12l + 2l
= 14l

Check :

Georgia's total number of bottles (9 small and 16 big)

= 9 + 16 = 25 (same as Edwina's total) ✔

Total capacity of Georgia's bottles

= 9 small and 16 big

= 9 × 0.4l + 9.6l (already calculated in part 1)

= 3.6l + 9.6l

= 13.2l


Difference between the total capacity of Edwina and Georgia


= 14l - 13.2l
= 0.8l ✔

A faster alternative method : 'what if'

We know that the capacity of Edwina's bottles is 0.8l more than Georgia's. They have the same total number of bottles.

What if we tried to make their total capacity the same, but also keeping the total number of bottles the same?

If we give Georgia 1 big bottle and Edwina small bottle :

Total capacity of Georgia's bottles increases by 0.6l.

But total capacity of Edwina's bottles increases by 0.4l only (= 400ml)

So the difference between their capacities will become smaller by : 0.6l - 0.4l = 0.2l

i.e Georgia's capacity is closing up the gap with Edwina's total capacity by 0.2l.

Since the capacity of Edwina's bottles is 0.8l more than Georgia's,

0.8l ÷ 0.2l = 4

We need to give Georgia 4 big bottles and Edwina 4 small bottles such that the gap closes to zero.


If we do this,

Now Georgia has 16 + 4 = 20 big bottles
Now Edwina has 5 + 4 = 9 small bottles


They now have the same total number of bottles and same total capacity.


But hold on. What does this mean?

If they have the same total number of bottles, and the same capacity, it means that :

The numbers of each type of bottle they have must be the same. If not, the total capacity will be different (which is the case in the original situation, where total number of bottles is the same but their bottles of each type are different in number)


So Georgia has 20 big bottles now, which means Edwina also must have 20 big bottles.

Since Edwina did not get any big bottles from the 'what if' scenario, this means she had 20 big bottles in the first place. (Answer for part a)

We can solve part b) from here since we know the number of small and big bottles she actually has.


Additional info we can find.

Edwina has 9 small bottles now, which means that Georgia also must have 9 small bottles.

Since Georgia did not get any small bottles from the 'what if' scenario, this means she had 9 small bottles in the first place.


Summary of info found :

Georgia : 9 small, 16 big, total 25 bottles
Edwina : 5 small, 20 big, goal 25 bottles.


With this method, we don't even need to calculate the total capacity of Edwina's small bottles and Georgia's big bottles.

Another way of thinking (but same steps as the 'what if' method) : grouping

They have the same total number of bottles, and a difference in capacity of 0.8l

Now, they must have some number of small and big bottles in common. There won't be any difference here as they will cancel out.

(For example,

If Edwina has 5 small, 18 big (total 23) and Georgia has 7 small, 16 big (total 23 also)

→Edwina : 5 small + 2 big + 16 big
→Georgia: 5 small + 2 small + 16 big

then they have 5 small and 16 big in common.
What causes the difference in capacity is the difference between the number of small bottles Georgia (2) has compared to an equal number of big bottles Edwina has (2) . These are not in common and don't cancel out.



1 large bottle has 0.6l - 0.4l = 0.2l more capacity than 1 small bottle.

0.8l ÷ 0.2l = 4

4 large will have 0.8l more capacity than 4 small.

So the difference separating the two of them is 4 small and 4 big.


Since Edwina has a higher capacity, Edwina must have more big bottles than Georgia.

She has 4 more big than Geogia and Georgia has 4 more small than Edwina.


Edwina has 16 + 4 = 20 big