Deriving the Ratio 1/X : 1/Y : 1/Z

The problem requires finding the ratio of the reciprocals of variables $X, Y,$ and $Z$, given two related ratios.

Step 1: Simplify the Given Ratios

First, simplify the individual ratios $X : Y$ and $Y : Z$ into their simplest whole number forms.

• For $X : Y = 1/2 : 1/5$, multiply both terms by the least common multiple (LCM) of the denominators (2 and 5), which is 10. $X : Y = \left(\frac{1}{2} \times 10\right) : \left(\frac{1}{5} \times 10\right) = 5 : 2$
• For $Y : Z = 1/3 : 1/4$, multiply both terms by the LCM of the denominators (3 and 4), which is 12. $Y : Z = \left(\frac{1}{3} \times 12\right) : \left(\frac{1}{4} \times 12\right) = 4 : 3$

Step 2: Combine Ratios to Find X : Y : Z

To find the combined ratio $X : Y : Z$, ensure the value representing $Y$ is the same in both simplified ratios ($X : Y = 5 : 2$ and $Y : Z = 4 : 3$).

• The values for $Y$ are 2 and 4. The LCM of 2 and 4 is 4.
• Adjust the ratio $X : Y = 5 : 2$ by multiplying by 2: $X : Y = (5 \times 2) : (2 \times 2) = 10 : 4$
• The ratio $Y : Z$ is already $4 : 3$.
• Now, combine the adjusted ratios: $X : Y : Z = 10 : 4 : 3$

Step 3: Calculate the Reciprocal Ratio $1/X : 1/Y : 1/Z$

The ratio $1/X : 1/Y : 1/Z$ is the reciprocal of the ratio $X : Y : Z$.

• Substitute the values from $X : Y : Z = 10 : 4 : 3$: $1/X : 1/Y : 1/Z = \frac{1}{10} : \frac{1}{4} : \frac{1}{3}$
• To simplify this ratio, multiply each term by the LCM of the denominators (10, 4, and 3). The LCM(10, 4, 3) is 60. $1/X : 1/Y : 1/Z = \left(\frac{1}{10} \times 60\right) : \left(\frac{1}{4} \times 60\right) : \left(\frac{1}{3} \times 60\right)$
• Calculate the final values: $1/X : 1/Y : 1/Z = 6 : 15 : 20$

The required ratio $1/X : 1/Y : 1/Z$ is $6 : 15 : 20$.