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If the ratio of grapes to walnuts to oranges is 9:3:11, how many grapes are there if the total number of fruits is 161?

If the ratio of grapes to walnuts to oranges is 9:3:11, how many grapes are there if the total number of fruits is 161?

add the numbers together: 9+3+11 = 23

you want the ratio of total number of grapes to total number of fruits

this should be consistent no matter how many fruits you have, so:

9/23 = x/161

cross-multiply and solve for x (number of grapes in a total of 161 fruits)

23x = 9*161 ---> x = 63

This ratio tells you that for every 9 grapes, there are 3 walnuts and 11 oranges. The purpose of a ratio is to tell you how much of something you have with respect to other objects, in the simplest numbers possible.

So from this information, if I said I have 6 walnuts, I would know that there are 18 grapes and 22 oranges on hand. If I have double the amount of walnuts, I would have to have double the amount of everything else.

Onto the question at hand, this question is telling us that we have 161 fruits. As you can see, this is a trick question, because a walnut is not a fruit, so we can just ignore it for now. So what the ratio is telling us, is that for every 9 grapes, we have 11 oranges. It is also telling us that for every 9 grapes, or for every 11 oranges, we have 20 total fruits.

The ratio of grapes to fruits is therefore 9:20. If you want to think about it in fractional form, its 9/20. 9 twentieths of the total fruit are grapes. Now that we are thinking about this in terms of fractions, we can jus go ahead and solve the problem. If we have 161 fruits, and 9 out of 20 of those fruits are grapes, how many grapes are there?

The answer is 72.45 grapes. Im not very happy with this answer because it is not a whole number, and you would think the author of the question wouldn't want to have an answer where theres just a half eaten grape laying on your fruit tray. I believe this is a trick question a typo because if you were not to remove the walnuts from the ratio, making it 9 grapes for every 23 objects, the numbers suddenly work out with all whole numbers (63 grapes). You might want to ask your teacher if there was a mistake, because 63 is a much more appealing number and it doesnt make much sense to have decimal places when you are working with discrete objects like pieces of fruit.

The number of grapes as a fraction of the total number of fruits is 9/23, because there are 23 total fruits as expressed in the ratio statement.  So,   9/23  = x/161.  Cross-multiply.   9 * 161  = 23 x.  Divide both sides by 23.

x=63

If you notice the ratio 9:3:11 is the same as saying 9 out of every 23 (or 9/23) pieces of fruit are grapes,  3 out of every 23 (or 3/23) pieces of fruit are walnuts,  and 11 out of every 23 (or 11/23) pieces of fruit are oranges,

because 9 grapes + 3 walnuts + 11 oranges = 23 total pieces of fruit (and nuts)

So there is a ratio of each type of fruit to the total amount of fruit, not just the ratios to each other.

Your next question should be whether or not 161 is a multiple of 23 and...

23*7 = 161 is the same as 161/23 = 7

So,

(9/23)*(7*23) = 63 grapes

(3/23)*(7*23) = 21 walnuts

(11/23)*(7*23) = 77 oranges

Another way to write that would've been:

(9/23)*161 = 9*(161/23) = 63

(3/23)*161 = 3*(161/23) = 21

(11/23)*161 = 11*(161/23) =77

Since 63 + 21 + 77 = 161, the answer checks out

If author of a problem means "... total numbers fruits and nuts ..." then:
9 + 3 + 11 = 23 parts,
161 ÷ 23 = 7 fruits or nuts in 1 part.
9 × 7 = 63 drapes
optional (just for check): 3 × 7 = 21 walnuts, 11 × 7 = 77 oranges. 21+77+63=161

Notice that each such "minimal collection" of 9 grapes, 3 walnuts and 11 oranges (9g,3w,11o) contain 23 pieces of fruit.

Using your calculator if you prefer, notice that 161 is equal to 23*7 (23 times 7) by carrying out the division "161/23 = ?" and solving it.

So to get 161 fruit you want 7 such "minimal collections", where we'll have 7*(9g,3w,11o)

This will give us 7*9g,7*3w,7*11o which tells you how many grapes (7*9 or 63) and how many walnuts and how many oranges too if you're curious.

I prefer this method to the "set up a proportion and cross multiply" method which many think of as a bit faster, because students make mistakes very often with that method. Not surprisingly since that method is pure Memorized Mechanical Manipulation without showing us why is works, as opposed to this method where each algebraic step is rooted concretely in the underlying meaning, of how many collections and so forth.

A third method is even more meaning-based, and which is good for younger students since it helps them develop their "proportion sense" and it uses Tables..I'll add that if I have time later, or drop me a note if curious. Let me know if you have any questions and I hope this helped!