An ice-cream seller sold 10 durian ice-cream and some red bean ice-cream. 1/5 of the red bean ice-cream were sold in cones and the remaining 24 red bean ice-cream were sold in cups. What percentage of the total ice-cream sold were red bean ice-cream sold in cones?


(1)  25%

(2)  20%

(3)  15%

(4)  10%

 Aug 30, 2021

First thing to answer: What are we solving for?
From the last part of the prompt, we can assume we are solving for the proportion of red bean ice cream cones sold, versus the total number of ice cream cones.


Second thing: What are the knowns, and the unknowns?

We know how many durian ice-cream sales were completed, and how many red bean ice-cream cups were sold.
The only unknown we have here is the number of red bean ice-cream cones sold. However, this has the side effect of leaving the total number of ice-cream sales completed unknown.


How might we calculate the total number of red-bean ice cream cones sold?
We're given a relation between the total number of red bean ice cream sales completed, and the number of red bean ice cream cones. We're also given a relationship between this and the total number of red bean ice-cream cups sold, which we're also given an exact number for. Source:
"The remaining 24 red bean ice-cream were sold in cups."


Our ratio here is: 1/5 of red bean ice-cream sales in cones, and the remainder (24) in cups.
The remainder after subtracting 1/5 from a whole is 4/5, so we can say that 4/5 of the red-bean ice-cream sales were in cups.
Reducing this to the smallest factor and multiplying to get the value of the whole, we first divide 24 by 4 (the numerator), then multiply by 5 (the denominator).

This leaves us with a total of 30 red-bean ice cream sales.


Now, we multiply the total number of red-bean ice cream sales (30) by the proportion of those sales that are in cones (1/5) to get the number of red-bean ice cream cones sold. (6)


We must return now to the total number of ice cream cones sold, which we can assume to be limited to the information provided. We add the durian ice cream purchases (10) to the number of red-bean ice cream purchases (30) to get the total ice cream purchases (40).


Now, we have our proportion: 6 red bean ice-cream cones :: 40 ice-cream sales.

This may be represented by the fraction, 6/40.

We can also reduce this fraction to 3/20, simply because they are equal quantities. This makes it easier to convert to a percent, as 20 is a factor of 100.


If we want to get a percentage from a fraction, we can use a conversion ratio. In this case, our conversion ratio is \(\frac{100\%}{1}\).

This might look odd, but the term "percent" can be traced back to a definition meaning "out of 100" (from the latin "per centum" [that's latin right?]), so 100% would be equal to 100/100, or 1.
In any case, we can then do some multiplication. \(\frac{3}{20} \times \frac{100\%}{1} = \frac{15\%}{1} = 15\%\)


To expand the multiplication, I first divided the 100% value by the denominator, in order ot figure out the value I multiply the numerator by. 100/20 = 5, so I multiplied the 3 on the top left by 5%.

This gives me 15%, which I believe to be the answer.

On a side note, it would be appreciated if you were to post how far you got into the problem before getting stuck next time, so that it might be possible to avoid a massive wall of text. I think that doing so would help both of us :)

 Aug 30, 2021

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