Problem Solving

The ratio of w to x is 5 to 4. The ratio of x to y is 8 to 1. If the ratio of y to z is 1 to 3, what is the ratio of w + x to z ?

6:1
8:1
10:1
24:1
30:1


I wrote out my ratios as so.. (please criticise my workings out!)


W X Y Z
5 : 4
8 : 1
1 : 3

I managed to get as far as going well W + X = 9 ... WX : Z = 9:3

I then completely lost track of where I was going with my workings out.. in fact, I can't see where I was going at all.

It's told me that answer A 6:1 is correct. I thought it was something to do with it being a multiple of 3, but that wouldn't make sense due to answer choices 24:1 and 30:1


Any help is most appreciated!!

The format hasn't allowed my workings to be posted properly! The numbers are spaced correctly under each letter.

W : X
5 : 4

X : Y
8 : 1

Y - Z
1 : 3

Step 1: write the ratios

W:X = 5:4 which is equivalent to 10:8
X:Y = 8:1
Y:Z = 1:3

Now write the combined ratios: ( if you nee help in how to do this please notify me)
W: X: Y :Z
10 :8 :1 :3

Now you can easily see if you have 3Z you will have 10W and 8X
So (W+X):Z is 18:3--->6:1
IMHO the answer is A

Hi macattack

Thanks for your help!

I'm assuming W:X 5:4 we double to make more manageable? And in terms of combining do you mind quickly debriefing?

Please excuse if none of this makes sense! But on reading again, I see that W:X and X:Y have 4 & 8, so is this the reason we've doubled W to make 10 so we can use X with a single value?

Thanks!

gemmox wrote:The ratio of w to x is 5 to 4. The ratio of x to y is 8 to 1. If the ratio of y to z is 1 to 3, what is the ratio of w + x to z ?

6:1
8:1
10:1
24:1
30:1

Like fractions, ratios can be MULTIPLIED:
w/x * x/y * y/z = w/z.
In the equation above, all of the values in red CANCEL OUT, leaving only w/z.
Plugging in the given values, we get:
5/4 * 8/1 * 1/3 = w/z.
10/3 = w/z.

To combine w/x = 5/4 and w/z = 10/3, the element COMMON to both ratios -- w -- must be represented by the SAME VALUE in each ratio:
w/x = 10/8 and w/z = 10/3.
Combining the ratios, we get:
w : x : z = 10:8:3.

Thus, if w=10, then x=8 and z=3.
Result:
(w+x)/z = (10+8)/3 = 18/3 = 6/1.

The correct answer is A.

I think Mitch explained it pretty well. Please let me know if you need further details and background on ratio rules and tricks.

GMATGuruNY wrote:

gemmox wrote:The ratio of w to x is 5 to 4. The ratio of x to y is 8 to 1. If the ratio of y to z is 1 to 3, what is the ratio of w + x to z ?

6:1
8:1
10:1
24:1
30:1

Like fractions, ratios can be MULTIPLIED:
w/x * x/y * y/z = w/z.
In the equation above, all of the values in red CANCEL OUT, leaving only w/z.
Plugging in the given values, we get:
5/4 * 8/1 * 1/3 = w/z.
10/3 = w/z.

To combine w/x = 5/4 and w/z = 10/3, the element COMMON to both ratios -- w -- must be represented by the SAME VALUE in each ratio:
w/x = 10/8 and w/z = 10/3.
Combining the ratios, we get:
w : x : z = 10:8:3.

Thus, if w=10, then x=8 and z=3.
Result:
(w+x)/z = (10+8)/3 = 18/3 = 6/1.

The correct answer is A.

Hi Mitch,

Thank you for your explanation, it's cleared up pretty much everything!! There is one thing, and I know it's right in front of me, but I'm failing to grasp it..

Where does the 10 come from.. I understand that the inner ratio's cancel themselves out, but unfortunately my brain power is failing me!

gemmox wrote:
Where does the 10 come from.. I understand that the inner ratio's cancel themselves out, but unfortunately my brain power is failing me!

w/x = 5/4 and w/z = 10/3.

To combine ratios with a COMMON ELEMENT, the common element must be represented by the SAME VALUE in each ratio.

Here, the element common to both ratios is w.
In the first ratio, w is represented by 5.
In the second ratio, w is represented by 10.
The LCM (lowest common multiple) of these two values is 10.

Thus, we want w to be represented by 10 in each ratio.
But we must take care not to CHANGE the value of either ratio: if we multiply the top of a ratio by a factor, we must multiply the bottom of the ratio by the SAME factor, so that the value of the ratio doesn't change:
w/x = 5/4 = (5*2)/(4*2) = 10/8.
w/z = 10/3.
Combining the ratios, we get:
w : x : z = 10:8:3.

Another example:
If we want to combine a:b = 2:3 and b:c = 5:4, we get:
a/b = 2/3 = (2*5)/(3*5) = 10/15.
b/c = 5/4 = (5*3)/(4*3) = 15/12.
Thus:
ac = 10:15:12.

I picked Values

w= 5
x = 4
y = 4/8
z = 12/8 = 3/2

(w+x)/z = 9*2/3 = 6:1

Not knowing much about ratio but still you can solve the problem . In the question there are variables (w , x , y,Z) and the answer is in the form of numbers (6:1 .. etc ) . So let us call this problem type variable to number substitution . variables in the question would mean flexibility in picking any number that I want subject to the condition that they fit the criterion given in the problem..(all this lecture to make you realise when can you substitute). Now.

let x= 80 so w must be 100 to fit the criterion of w/x = 5/4
similarly y must be 10 to fit the criterion x/y = 8/1.
and z= 30 to fit the criteria y/z =1/3

now (w+x )/z = (100+80)/30 = 180/30 = 6:1 .
Hence answer A .

while you prepare for a time based test such as the GMAT - try finding a method that works the fastest so that you can save time for tough questions .almost 30% of question that you would encounter in GMAT would be either be variable to number or variable to variable or number to variable or number to number .These I call it as flexible questions and if you substitute values in them you can crack them easily . The foolish test creators don't know this . Else they would never make such silly questions. I would love to change the way mathematics is taught in school.

Not knowing much about ratio but still you can solve the problem . In the question there are variables (w , x , y,Z) and the answer is in the form of numbers (6:1 .. etc ) . So let us call this problem type variable to number substitution . variables in the question would mean flexibility in picking any number that I want subject to the condition that they fit the criterion given in the problem..(all this lecture to make you realise when can you substitute). Now.

let x= 80 so w must be 100 to fit the criterion of w/x = 5/4
similarly y must be 10 to fit the criterion x/y = 8/1.
and z= 30 to fit the criteria y/z =1/3

now (w+x )/z = (100+80)/30 = 180/30 = 6:1 .
Hence answer A .

while you prepare for a time based test such as the GMAT - try finding a method that works the fastest so that you can save time for tough questions .almost 30% of question that you would encounter in GMAT would be either be variable to number or variable to variable or number to variable or number to number .These I call it as flexible questions and if you substitute values in them you can crack them easily . The foolish test creators don't know this . Else they would never make such silly questions. I would love to change the way mathematics is taught in school.