The three numbers are in the ratio 1/2 : 2/3 : 3/4. The difference between greatest and smallest numbers is 36. Find the numbers

Step 1 (Convert Fractional Ratios to Integers): We are given the ratio of three numbers:

To convert this to an integer ratio, we multiply each term by the Least Common Multiple (LCM) of the denominators $\{2,3,4\}$, which is $12$:

• First term: $\frac{1}{2}\times 12=6$

• Second term: $\frac{2}{3}\times 12=8$

• Third term: $\frac{3}{4}\times 12=9$

So the integer ratio is $6:8:9$.

Step 2 (Formulate Equations using variables): Let the three numbers be $6k$, $8k$, and $9k$. The smallest number is $6k$ and the greatest is $9k$.
We are given that the difference between the greatest and smallest numbers is 36:

Step 3 (Compute the Numbers):

• First number: $6k=6(12)=72$

• Second number: $8k=8(12)=96$

• Third number: $9k=9(12)=108$

Step 4 (Conclusion): The three numbers are exactly 72, 96, 108.