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You are here: Experts > Science > Mathematics > Advanced Math > Ratio
Expert: Sherry Wallin - 11/6/2009
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
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