Get the ratios of x:y:z

x:y=4:3=28:21

y:z=7:4=21:12

x:y:z=28:21:12

To get the least value put x=1,y=1,z=1

x+y+z=28+21+12=61

The condition mentioned in the stem is 3x=4y=7z, so you need to find the LCM among 3,4,7 and that is 84, then you divide 84 by 3,4,7 to get values of x,y,z (28,21,12) and then x+y+z(61).

I am not sure though how to clarify your method, but if you add the fractions, you do get a 61 in the numerator.

thanks both of you...i like your methods but still not sure what's wrong with my method
indeed, i get 61/28....but how do i get from there to just plain 61
instructors please help!

If you put only x=1 it means you are assuming least value only for 1.y and z can be of maximum values.

Similarly for y=1 or z=1

To get the least value of x+y+z,we need to substitute x,y and z values as 1(ie) x=1,y=1 and z=1.

We need to find the ratios of x,y,z so that we can get all the values.

Hope this clarify

san2009 wrote:If x,y and z are positive integers and 3x=4y=7z, then the least possible value of x+y+z is

A)33
B)40
C)49
D)61
E)84

OA is D

The way I approached problem was to assume the smallest possible value for x. Since x must be a positive integer, then x =1, from that we get y=3/4 and z=3/7
when we add those up we don't get anywhere close to the answer choices
if you were to take y=1 and solve with that or z=1 and solve from there - you would get a higher value for x+y+z

Can an instructor please advise what am i missing here??

thanks in advance

Hi San,

The question stem tells us that x, y and z are positive integers. As you wrote yourself, if x=1 then "we get y=3/4 and z=3/7". This contradicts the question stem because y and z must be positive integers.

My approach is to focus on 3x=4y=7z. Since the variables are integers, 3x, 4y and 7z must also be integers. They all equal some mystery number. This mystery number is a multiple of 3, 4 and 7. The smallest values of x, y, z will result in the smallest possible mystery number. Thus this number is the least common multiple (LCM) of 3, 4, and 7. This is 84. Thus 3x=4y=7z=84. x=28, y=21, z=12. x+y+z=61

Hope that helps,
-Patrick

Thanks Patrick. Well explained!