Question : Let $x, y, z$ be fractions such that $x<y<z$. If $z$ is divided by $x$, the result is $\frac{5}{2}$, which exceeds $y$ by $\frac{7}{4}$. If $x+y+z=1 \frac{11}{12}$, then the ratio of $(z-x):(y-x)$ is:
Option 1: 6 : 5
Option 2: 9 : 5
Option 3: 5 : 6
Option 4: 5 : 9
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Correct Answer: 6 : 5
Solution :
Given that $z$ divided by $x$ is $\frac{5}{2}$.
$⇒z = \frac{5}{2}x$
Also, given that $\frac{5}{2}$ exceeds $y$ by $\frac{7}{4}$.
$⇒y = \frac{5}{2} - \frac{7}{4} = \frac{3}{4}$
Substituting $y = \frac{3}{4}$ and $z = \frac{5}{2}x$ into the equation $x + y + z = 1 \frac{11}{12}$.
$⇒x + \frac{3}{4} + \frac{5}{2}x = 1 \frac{11}{12}$
$⇒x + \frac{3}{4} + \frac{5}{2}x = \frac{23}{12}$
$⇒42x+9=23$
$⇒x = \frac{1}{3}$
Substituting $x = \frac{1}{3}$ into the equation $z = \frac{5}{2}x$,
$⇒z = \frac{5}{6}$
Therefore, the ratio of $(z - x)$ to $(y - x)$ = $\frac{\frac{5}{6} - \frac{1}{3}}{\frac{3}{4} - \frac{1}{3}} = \frac{6}{5}$
Hence, the correct answer is 6 : 5.
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