which fraction do I take the reciprocal of?

[e1e1e1e1]

I have a question about the 1st step on a problem on Khan Academy:



Since they took the reciprocal of 1/5, (which is 5/1) the result was 2/3
Had they taken the reciprocal of 2/15 instead, the result would have been 3/2, which is obviously a very different result.
So, clearly it matters which fraction they take the reciprical of. But they don't explain how they choose which fraction to take the reciprocal of!
So... how does one know which fraction to take the reciprocal of?

 

[HallsofIvy]

Mostly people know which fraction to invert because they learned, when they learned fraction arithmetic, that you invert the divisor and multiply. that is \(\displaystyle \frac{a}{b}\) divided by \(\displaystyle \frac{c}{d}\) is the same as \(\displaystyle \frac{a}{b}\) multiplied by \(\displaystyle \frac{d}{c}\). You invert the fraction you are dividing by.

 

[e1e1e1e1]

You invert the fraction you are dividing by.

Thanks. This much I understand.

However, when stated as a ratio...

either as:
2/15 : 1/5

or as:
1/5 : 2/15

are these not both equivalent statements?

And if they are equivalent, how does one determine which fraction will be the divisor when finding the value of the ratio?

For some reason they chose 1/5 as the fraction to be used as the divisor, and I don't understand how they made that choice.

 

[HallsofIvy]

Thanks. This much I understand.

However, when stated as a ratio...

either as:
2/15 : 1/5

or as:
1/5 : 2/15

are these not both equivalent statements?

No, they are not. "a:b" means "\(\displaystyle \frac{a}{b}\)".
2/15: 1/5 means \(\displaystyle \frac{\frac{2}{15}}{\frac{1}{5}}= \frac{2}{15}\times \frac{5}{1}= \frac{2}{3}\)
while 1/5: 2/15 mean \(\displaystyle \frac{\frac{1}{5}}{\frac{2}{15}}= \frac{1}{5}\times \frac{15}{2}= \frac{3}{2}\)

You may be thinking of a "proportion", a statement that two ratios are equal.
If a:b :: c:d then, as fractions, \(\displaystyle \frac{a}{b}= \frac{c}{d}\). If the two fractions are equal, then their reciprocals are also equal, \(\displaystyle \frac{b}{a}= \frac{d}{c}\) so we have the proportion b:a :: d: c. The individual ratios are different than they were before but these new ratios are still equal.

And if they are equivalent, how does one determine which fraction will be the divisor when finding the value of the ratio?

For some reason they chose 1/5 as the fraction to be used as the divisor, and I don't understand how they made that choice.

 

[e1e1e1e1]

Thanks HallsofIvy and Denis.

So, if I'm understanding this correctly, I now understand that the order of the items in the ratio absolutely matters, because the first item in the ratio is like the numerator of a fraction, and the second item in the ratio is like the denominator of a fraction.

And so therefore, for example, the ratio of "two to three" is NOT equivalent to the ratio of "three to two", just as the fractions 2/3 and 3/2 are obviously different. And so therefore also a ratio of "sugar to flour" would not be equivalent to a ratio of "flour to sugar". Correct?

I think I get it now. But I'm going to post an image below which might help explain my confusion, because they seem to keep changing the order of the items in the ratio willy-nilly.

First they describe the ratio as being one of "sugar to flour", then they show a chart which reverses that order to "Flour Sugar", then, in the first hint they order the items again as sugar to flour (2/15:1/5).

Are they deliberately trying to create ambiguity and confusion?

 

[HallsofIvy]

There is nothing ambiguous if you understand what I just said: "the ratio of sugar to flour" is the amount of sugar divided by the amount of flour. That has nothing to do with how the actual values are given.
In the first row of the table given the amount of sugar is 2/15 and the amount of flour is 1/5. The "ratio of sugar to flour" is \(\displaystyle \frac{\frac{2}{15}}{\frac{1}{5}}= \frac{2}{15}\times \frac{5}{1}= \frac{2}{3}\). If you look at the third and fourth rows you will see that the amount of sugar is 10/3 and flour is 5, in the third row and 20/3 and 10 in the fourth row. The ratios are \(\displaystyle \frac{\frac{10}{3}}{5}= \frac{10}{3}\\times \frac{1}{5}= \frac{2}{3}\) and \(\displaystyle \frac{\frac{20}{3}}{10}=\frac{2}{3}\) also.

The second row has the amount of sugar "?" and the amount of flour 1. To have the same ratio, you must have \(\displaystyle \frac{?}{1}= \frac{2}{3}\).

 

[Quaid]

Are they deliberately trying to create ambiguity and confusion?

No -- I think it's a case of unintentional sloppy wording. Khan Academy's statement that the table itself represents the ratio "sugar to flour" is not accurate. The table is simply a listing of associated pairs of numbers. It's up to you to put a pair of associated numbers in the correct order for representing the "sugar to flour" ratio.

2/15 to 1/5

2/15 : 1/5

(2/15)/(1/5)


So, although I understand your point, going forward try not to expect tabular data (or any given information, really) to be always be presented conveniently or in correspondence to some particular task.

Ciao

 

[Deleted member 4993]

No --... Khan's statement that the table itself represents the ratio "sugar to flour" is not accurate. ...

Ciao

hey ... hey ..hey - you are taking my name in vain!!

 

[Quaid]

you are taking my name in vain

My bad. Fixed it.

 

[e1e1e1e1]

Well, they may very well have reversed the order in the table deliberatly to try to teach a lesson about paying attention ONLY to the wording, but I really don't know.

At any rate, in the future, I'll just use the order in the wording as the actual/literal order for the ratio.