If `(1)/(x) : (1)/(y) : (1)/(z) = 2 : 3 : 5` then determine x : y :z

Options

A. `6 : 15 : 10`
B. `3 : 15 : 10`
C. `15 : 3 :10`
D. `15 : 10 : 6`

Answer

To solve the problem, we start with the given ratio:

\[ \frac{1}{x} : \frac{1}{y} : \frac{1}{z} = 2 : 3 : 5 \]

This means we can express the fractions in terms of a common variable \( k \):

\[ \frac{1}{x} = 2k, \quad \frac{1}{y} = 3k, \quad \frac{1}{z} = 5k \]

Now, we can find \( x \), \( y \), and \( z \) by taking the reciprocal of each fraction:

1. For \( x \):
\[ x = \frac{1}{2k} \]

2. For \( y \):
\[ y = \frac{1}{3k} \]

3. For \( z \):
\[ z = \frac{1}{5k} \]

Next, we need to express the ratio \( x : y : z \):

\[ x : y : z = \frac{1}{2k} : \frac{1}{3k} : \frac{1}{5k} \]

To simplify this ratio, we can eliminate \( k \) from the denominators:

\[ x : y : z = \frac{1}{2} : \frac{1}{3} : \frac{1}{5} \]

Now, we can find a common denominator to express these fractions in a simpler form. The least common multiple of 2, 3, and 5 is 30. We can rewrite each term:

1. For \( \frac{1}{2} \):
\[ \frac{1}{2} = \frac{15}{30} \]

2. For \( \frac{1}{3} \):
\[ \frac{1}{3} = \frac{10}{30} \]

3. For \( \frac{1}{5} \):
\[ \frac{1}{5} = \frac{6}{30} \]

Now we can write the ratio:

\[ x : y : z = 15 : 10 : 6 \]

Finally, we can simplify this ratio by dividing each term by their greatest common divisor, which is 1 in this case. Thus, the final ratio is:

\[ x : y : z = 15 : 10 : 6 \]

### Summary of the solution:
\[ x : y : z = 15 : 10 : 6 \]