In the figure above, if x, y and z are the lengths indicated, what is

z = 3/7 - 1/3 = 2/21

y = 1/3 - 1/4 = 1/12

x = 1/4 - 1/7 = 3/28

y < z (2/24 < 2/21)
z < x (6/63 < 6/56)

So, y < z < x

Answer E

x = \((\frac{1}{4}\) - \(\frac{1}{3})\) = \((\frac{7-4}{28})\) = \(\frac{3}{28}\)

y = \((\frac{1}{3}\) - \(\frac{1}{4})\) = \((\frac{4-3}{12})\) = \(\frac{1}{12}\)

z = \((\frac{3}{7}\) - \(\frac{1}{3})\) = \((\frac{9-7}{21})\) = \(\frac{2}{21}\)

Compare by cross multiplying. If first cross product is larger, first fraction is larger. If second cross product is larger, second fraction is larger.

x and z: \(\frac{3}{28}\) [?] \(\frac{2}{21}\) = 63 | 56 --> x, \(\frac{3}{28}\) is larger

z and y: \(\frac{2}{21}\) [?] \(\frac{1}{12}\) = 24 | 21 --> z, \(\frac{2}{21}\) is larger

y < z < x
Answer E

I'm landing on answer E:

Using 84 as a common denominator:

X = 21/84 - 12/84 = 9/ 84
Y = 28/84 - 21/84 = 7/84
Z = 36/84 - 28/84 = 8/84

Therefore, Y < Z < X, (E)

My first Kudos please if this answer was helpful!

My pleasure - it's a good answer (and not one I was energetic enough to try). Kudos are good things. Pay it forward, yes?

For questions like these where you are going to have to subtract different fractions from each other and compare the results you should, as several posters above have mentioned, always convert to a common denominator.

You can try converting to decimals but usually these questions will use non-terminating decimals to screw you up.

In this case the common denominator is 84, so convert all fractions to the same denominator.

z = 3/7 - 1/3 = 36/84 - 28/84 = 8/84
y = 1/3 - 1/4 = 28/84 - 21/84 = 7/84
x = 1/4 - 1/7 = 21/84 - 12/84 = 9/84

We can see that y < z < x, which is (E).

One other thing to note is it is often a good idea to avoid reducing these fractions when you are solving problems like this because not all fractions will reduce and you then partially obscure your point of comparison (which is the entire point of the problem).

Lets first solve for x:

x = 1/4 - 1/7 = 7/28 - 4/28 = 3/28

Next for y:

y = 1/3 - 1/4 = 4/12 - 3/12 = 1/12

Next for z:

z = 3/7 - 1/3 = 9/21 - 7/21 = 2/21

To compare x, y, and z, we can express them with the same numerator (which is much easier than using the same denominator). We will make the numerator 6. Also, keep in mind that if two positive fractions have the same numerator, the larger the denominator, the smaller the fraction.

x = 3/28 = 6/56

y = 1/12 = 6/72

z = 2/21 = 6/63

Thus, y < z < x.

Answer: E

Take a common denominator of 84.
The four points marked are at 12, 21, 28, 36
x=21-12=9
y=28-21=7
z=36-28=8
Ascending order is y,z,x

(Too easy to be 700 level)

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