Understanding Ratios
To find the combined ratio a:b:c from the given ratios a:b = 3:2 and b:c = 3:5, we need to express all components in terms of a common variable.
Step 1: Express a, b, and c in terms of a common variable
- Given a:b = 3:2, we can express:
- a = 3x
- b = 2x
- Given b:c = 3:5, we can express:
- b = 3y
- c = 5y
Step 2: Set the value of b equal
- Since b is represented in two different ways, we set them equal:
- 2x = 3y
- Solving for y in terms of x:
- y = (2/3)x
Step 3: Substitute y back to find c
- Now, substitute y into the expression for c:
- c = 5y = 5 * (2/3)x = (10/3)x
Step 4: Collect all variables
- Now we have:
- a = 3x
- b = 2x
- c = (10/3)x
Step 5: Find the combined ratio a:b:c
- To express a:b:c in the simplest integer form, we multiply through by 3 (the denominator of c):
- a = 3x * 3 = 9x
- b = 2x * 3 = 6x
- c = (10/3)x * 3 = 10x
Final Combined Ratio
- Therefore, the combined ratio a:b:c is:
- a:b:c = 9:6:10
This gives us the final result in the simplest form, representing the relationships among a, b, and c effectively.