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You are here: Experts > Science > Mathematics > Advanced Math > Ratio
Expert: Sherry Wallin - 11/7/2009
Question
QUESTION: I have a ratio problem I solved it with 2 ways one I know to be mathematically correct but the other is more on logic I guess I want to know if the other one is mathematically correct or if there is a way to correct it, the problem is :
The ratio of apples to oranges is 5/8 when we take 15 apples the ratio becomes 1/4 what is the total number of fruits after taking the apples:
the first method:
x/y = 5/8 y = 8x/5
(x-15)/y = 1/4 y=4x-60
8x/5 = 4x-60 >>>>>> 8x=20x-300 >>>>12x=300
x=25
y = 4x-60 = 100-60 = 40
(x+y) = 65
x+y-15= 65-15 =50
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the second:
x/y= 5/8
for every 13 fruits there is 5 apples and 7 oranges.
a= total number of fruit before a-15 = after
so the total number of fruits = 13 * the number of groups we are making I mean we are separating each 13 of the total number and making it into 5apple:8orange and since we dont have half oranges or anything like that
a= 13 * numbers of groups we make 13 of the fruits (g)
a/13=g
in each group there are 5 apples ,so the number of apples is 5*g
x(1)=5a/13
y=8a/13
after we take 15 apples
the ratio becomes 1x :4y
so for every 5 fruits we have there will be 1 x and 4 y
(a-15)= the number of groups we have * 5fruits(4oranges and 1 apple)
g=(a-15)/5
in each group we have 1 apples so x(2)=g*1=g
=(a-15)/5
in each group we have 4 oranges y(2)=g*4= 4/5(a-15)
but since I didnt change the amount of oranges
y(1)=y(2)
8a/13=4/5(a-15)
10/13*a - a =-15
-3/13 a =-15
a=65 a-15=50
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I know its alot of words but is seemed to work so I need to find a way to turn these sentences into expressions can you help me?
ANSWER: Hamad~
About your 2nd method: first you can't have 13 fruits if you start with 5 apples and 7 oranges since 5 + 7 = 12. Secondly most of your logic seems reasonable and you did arrive at a 'correct' answer but some of your moves are not 'correct'. How about solving the problem this way instead? This is assuming you are looking for a 2nd method that arrives at the correct answer with all the correct moves:
let x = # of apples and y = # oranges you begin with. So you have an initial ratio of 5x/8y which is a ratio of 5/8. Now you know you removed 15 apples so your new ratio is [5x-15]/8y = 1/4. Solve for y, since it does NOT change. A side note: If two fractions are equal then their reciprocals are equal so we also have 8y/[5x-15] = 4. So y = 4(5x-15)/8 = [5x-15]/2. Put this value of y into the original ratio 5x/8y and solve for x: 5x/8y = 5x/8[5x-15]/2 = 5/8. 40x = 100x - 300 -> 60x = 300 -> x = 5. Now put x = 5 back into 8y = 4[5x-15] getting y = 4[5*5-15] = 40. So you originally had 25 apples and 40 oranges or 25+40 = 65 fruit and then when you removed the 15 apples this left you with 50 fruit which is what you go tin your first answer.
Math Prof
---------- FOLLOW-UP ----------
QUESTION: I meant we start with 5 apples and 8 oranges... , what other moves aren't correct?
ANSWER: If you are starting with 5 apples and 8 oranges, then why would you begin letting a = total number of fruit before and then let a - 15 be the total number of fruit after? Clearly you didn't start with 13 fruit and then take away 15 apples. I'm sorry if I am not giving you the answer you are seeking. But starting out incorrectly and the proceeding incorrectly makes every thing that follows not correct, if if you do somehow come out with a right answer.
Math Prof
---------- FOLLOW-UP ----------
QUESTION: well because I learned this method for good math teacher via youtube
salman khan
http://www.youtube.com/watch?v=ItA_hhRtUuw
but the problem that he explained it in words and I am trying to turn it into pure math
Answer
Hi Hamad~
I just viewed the video and he is a good math teacher. You want to do what he did with pure math and it is necessary to use some language in order to clarify what the variables represent. I hope you see that my original explanation is very similar to what he did and I used a lot less words too :)
Let me see if I can do what he did with fewer words and more math symbols.
You begin with 13 fruit in a ratio of 5 apples for every 8 oranges. You remove 15 apples which leaves you with a new ratio of 1 apple for every 4 oranges. He used x/13 to represent the number of groups of 5 apples to 8 oranges and so you ended up with 5x/13 apples and 8x/13 oranges. When you remove 15 apples from 5x/13 apples you have 5x/13 - 15 apples. Again as he says and I also said above, the one constant is that the number of oranges remains the same. If you end up with a ratio of 1 apple for every 4 oranges this is the same as saying that you will have 5 fruits in a group and (x-15)/5 groups of fruit when you finish. Since the ratio of oranges after removing the 15 apples is for every 1 apple there are 4 oranges now that you know you have (x-15)/5 apples this means you end up with 4(x-15)/5 oranges. #oranges before = #oranges after thus 8x/13 = 4(x-15)/5. I would multiply both sides by 5/4 as he did getting 10x/13 = x-15. But here is where I would do different. I would multiply both sides by 13 getting 10x = 13x - 195 then subtract -13x from bot sides getting:
-3x = -195 and then divide both side by -3 getting x = 65.
In order for the reader or student to understand where these ratios are coming from it is imperative that one explain them or else they seem to be coming out of thin air and the student will not be able to apply this to another similar problem. I hope my explanation has continued to help you.
Math Prof
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