Notation of ratios

Question

Hi everyone,

I have a question about the notation of ratios.
I do understand the math behind it yet sometimes it seems to me that questions use the notation interchangeably.
Here are two examples of what I mean:

1. Example 
If a=2b, 1/2b=c and 4c=3d what is the ratio of d to a?

I thought it's 3/1 but the correct answer is 1/3.

2. Example 
Given that a is the average (arithmetic mean) of the first nine positive multiples of six and b is the median of the first twelve positive multiples of six, what is the ratio of a to b?

Following the first example I answered 13:10 yet here the correct answer is 10:13.

My question:
Can someone please explain the logic behind this and give a rule how to chose the right ratio.

Thanks already in advance!

Bunuel · Answer

\frac{1}{2}*b = c , thus  b = 2c ;
 a = 2b = 4c .
 4c = 3d , thus  d = \frac{4c}{3} .

\frac{d}{a}=\frac{\frac{4c}{3}}{4c}=\frac{4c}{3}*\frac{1}{4c}=\frac{1}{3} .

Answer: A. Discussed  HERE .

The first nine positive multiples of six are {6, 12, 18, 24,  30 , 36, 42, 48, 54}
The first twelve positive multiples of six are {6, 12, 18, 24, 30,  36 ,  42 , 48, 54, 60, 66, 72}

Both sets are evenly spaced, thus their median=mean:
a=30 and b=(36+42)/2=39 --> a/b=30/39=10/13.

Answer: B.

Discussed  HERE .

Hope it helps.

Bunuel · Answer

Please follow the rules when posting a question: https://gmatclub.com/forum/rules-for-po ... 33935.html

Bunuel · Answer

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 Terminating and Recurring Decimals 
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Check below for more:
 ALL YOU NEED FOR QUANT ! ! ! 
 Ultimate GMAT Quantitative Megathread

Hope it helps.

JaeTPU · Answer

Rephrase "what is the ratio of d to a" algebraically to "d/a=?"

To get a numerical answer for your ratio, you will need the variables to cancel out. Put d in terms of c and a in terms of c (a->b->c)
d=(4/3)c, a=2b=4c. ((4/3)c)/4c = (4/3)/4 = 1/3

Same thing. If you rephrase "what is the ratio of a to b" into algebra, you get "a/b=?"

The average of numbers that are evenly spaced apart is the median. The median of the first nine positive multiples of 6 is 30.
Median of the first 12 positive multiples of 6 is the mean of the 6th and 7th positive multiples of 6, so (36+42)/2= 39

30/39 = 10/13

ccooley · Answer

If you're unsure about which order to write a ratio in, think about it in terms of actual numbers.

When the problem says "what is the ratio of a to b," you would write the value of a first, then the value of b.

If it says "what is the ratio of d to a," you would write the value of d first, then the value of a.

So, for instance, look at your first problem. One set of numbers that works is a = 2, b = 1, c = 1/2, and d = 2/3.

The ratio of D to A is 2/3 to 2, because D could equal 2/3 and A could equal 2.

Then, simplify that ratio by multiplying by 3:

D to A = 2 to 6

Then, divide by 2:

D to A = 1 to 3

That's how you know it's 1 to 3 and not 3 to 1.

Notation of ratios

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