Writing ratio as fraction

[Lighthouse50]

I have a ratio like this: apples:bananas = 3:4

Could someone please explain why bananas = 4/3 X apples

 

[pka]

I have a ratio like this: apples:bananas = 3:4. Could someone please explain why bananas = 4/3 X apples

We are told that there are three apples for every four bananas.
If there are sixteen bananas then there are twelve apples.
If there are twenty-one apples then there are twenty-eight bananas.
Can you write an answer for us now?

 

[Dr.Peterson]

I have a ratio like this: apples:bananas = 3:4

Could someone please explain why bananas = 4/3 X apples

The proportion can be written as an equality of fractions: [MATH]\frac{apples}{bananas} = \frac{3}{4}[/MATH]. Flipping both sides over (that is, taking reciprocals), [MATH]\frac{bananas}{apples} = \frac{4}{3}[/MATH]. Multiplying both sides by "apples", [MATH]bananas = \frac{4}{3}\cdot apples[/MATH].

 

[Steven G]

...Or 4 apples = 3 bananas. Divide both sides by 3 and get(4/3) apples = bananas

 

[Harry_the_cat]

...Or 4 apples = 3 bananas. Divide both sides by 3 and get(4/3) apples = bananas

To Jomo:
No no no!!
4 apples = 3 bananas ?? Never! This sort of "fruit salad" algebra causes so much confusion with beginner algebrists.
Perhaps you mean:
4 x number of bananas = 3 x number of apples

To the OP:
apples:bananas = 3:4 is a shorthand way of saying "number of apples":"number of bananas" =3:4
In other words, for every 3 apples you have 4 bananas.
Letting a be the number of apples and b be the number of bananas, you can write
a:b = 3:4
Ratios can also be written as fractions:
\(\displaystyle \frac{a}{b} =\frac{3}{4}\)

Reciprocating both sides gives:
\(\displaystyle \frac{b}{a} =\frac{4}{3}\)

Multiplying both sides by a gives:
\(\displaystyle b =\frac{4}{3} *a\)

That is:
number of bananas = \(\displaystyle \frac{4}{3}\) x number of apples.

 

[Steven G]

It all a matter of opinion and I respect your opinion especially since you seem to really believe in it. My opinion is that students should learn early on that they can treat variables and units (and apples is a unit) just like they treat number. After all a variable is just a way of representing an unknown number. I believe that algebra students should learn that for example that (6ft/sec)*sec = 6 ft since the unit of sec cancel out just like variables.
They need to know this to convert from one unit to the next which I think is basic and isn't even algebra. The thing is I tell my students that 3seconds = 3*seconds all the time. Unfortunately the posters on this forum do not have the benefit of hearing me say certain things all the time. I guess I have to think about that when I make posts!

 

[Harry_the_cat]

I can see what you are saying, but what about this:

3 feet = 36 inches Agree?

But
3 x number of feet = 36 x number of inches
is not the same thing.

 

[Dr.Peterson]

In my opinion, there is a danger in confusing variables with units. I use "3 seconds = 3 * 1 second"; that's quite different from "3 seconds = 3 * number of seconds", which is what a variable would mean. Yes, you can treat a unit like a quantity, but in a different sense than a variable. A variable represents a number; a unit does not - it states the meaning of the number it is used with. And, in particular, "(6 ft/sec) * sec = 6 ft" is meaningless; you never use a unit without a number attached. Rather, "(6 ft/sec) * 1 sec = 6 ft".

In our ratio, apples : bananas = 3 : 4, we really mean # apples : # bananas = 3 : 4. We are using "apples" as a variable. not a unit.

If I wrote "4 apples = 3 bananas", where they are units, I would probably mean, say, that 4 apples cost or weigh as much as 3 bananas. If I then had equal values of the two, the ratio of apples to banana would be 4:3, not 3:4. But the equation itself would say nothing about how many of each I actually had.

For an example of the trouble you can get into, see here.

EDIT: Harry's example is exactly what I'm talking about.

 

[amin123]

The proportion can be written as an equality of fractions: [MATH]\frac{apples}{bananas} = \frac{3}{4}[/MATH]. Flipping both sides over (that is, taking reciprocals), [MATH]\frac{bananas}{apples} = \frac{4}{3}[/MATH]. Multiplying both sides by "apples", [MATH]bananas = \frac{4}{3}\cdot apples[/MATH].

Good answer -- probably the easiest of the explanations to understand