SOLUTION: Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1

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Question 208973: Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1
Answer by Theo(13342) About Me  (Show Source):
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Find the ratio a:c if a:b= 1:2, b:d = 4:5, d:c = 3:1
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x : y is the same as x+%2F+y is the same as x divided by y
see reference: http://www.icoachmath.com/sitemap/Ratio.html
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a:b means a%2Fb+=+1%2F2 which means that a+=+b%2F2 and b+=+2%2Aa
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b:d means b%2Fd+=+4%2F5 which means that b+=+4d%2F5 and d+=+5b%2F4
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d:c means d%2Fc+=+3%2F1 which means that d+=+3c and c+=+d%2F3
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now that you have these translations out of the way you can solve.
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a:c means a%2Fc
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to find a%2Fc you need to find relationships in terms of a and c and then solve.
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in order to get a relationship from a to c, you need to go through b and d because there is no direct relationship between a and c.
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start with the relationship between a and b.
a+=+b%2F2
get the relationship from b to d
b+=+4d%2F5
replace b with 4d%2F5 to get:
+a+=+%284d%2F5%29%2F2 which becomes:
a+=+4d%2F10
get the relationship from d to c
d+=+3c
replace d with 3c to get:
a+=+%284%2A3c%29%2F10 which becomes:
a+=+12c%2F10 which becomes:
a+=+6c%2F5
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a+=+6c%2F5 is the same as:
a%2Fc+=+6%2F5 which is the same as:
a%3Ac+=+6%3A5
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to confirm, let's try to find c directly and then try to find c indirectly.
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let a = 6
if a+=+6c+%2F+5, then c must equal to 5a%2F6 = %285%2A6%29%2F6 = 5
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we know that a = 6 and c = 5 solving directly.
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let's find c indirectly.
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we start off with a = 6
we know that b = 2%2Aa = 2%2A6 = 12
we know that d = %285%2Ab%29%2F4 = %285%2A12%29%2F4 = 15
we know that c = d%2F3 = 15%2F3 = 5
we have a+=+6 and c+=+5 solving indirectly.
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the formulas check out and the relationship you are looking for is:
a%3Ac+=+6%3A5
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is there an easier way to do this?
yes there is
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the shortcut method is as follows:
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you have a%3Ac = %28a%3Ab%29+%2A+%28b%3Ad%29+%2A+%28d%3Ac%29
you know those ratios are:
1%3A2 and 4%3A5 and 3%3A1
you know this is the same as:
1%2F2 and 4%2F5 and 3%2F1
so you have:
a%2Fc = %28a%2Fb%29+%2A+%28b%2Fd%29+%2A+%28d%2Fc%29 = %281%2F2%29+%2A+%284%2F5%29+%2A+%283%2F1%29 = %281%2A4%2A3%29%2F%282%2A5%2A1%29 = 12%2F10 = 6%2F5
your ratio for a%2Fc+=+6%2F5
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