Deriving the Combined Ratio When Given Individual Ratios

In this article, we will explore how to find the combined ratio of three variables when given two separate ratios. Specifically, we will derive the highest-common ratio of A:B:C when given the ratios A:B 3:4 and B:C 5:6. This process involves expressing the variables in terms of common terms, equating them, and simplifying the combined ratio.

Given Ratios

We are given two ratios:

A:B 3:4 B:C 5:6

Step-by-Step Derivation

To find the combined ratio A:B:C, we need to express all variables in terms of a common variable and then equate them.

Step 1: Express Variables in Terms of Common Variables

For the first ratio A:B 3:4, let:

A 3x and B 4x

For the second ratio B:C 5:6, let:

B 5y and C 6y

Step 2: Equate B from Both Expressions

Since B is common in both expressions, we can set them equal:

4x 5y

Solving for y in terms of x:

y (4x) / 5

Step 3: Substitute y Back to Find C

Substitute y back into the expression for C:

C 6y 6(4x/5) (24x)/5

Step 4: Express All Variables in Terms of a Common Denominator

Now we have:

A 3x B 4x C (24x)/5

To eliminate the fraction, multiply all terms by 5:

A 3x * 5 15x B 4x * 5 2 C (24x)/5 * 5 24x

Thus, the combined ratio A:B:C is:

15x : 2 : 24x

Step 5: Simplify the Ratio

Since x is common in all terms, we can remove it:

A:B:C 15 : 20 : 24

This is the final simplified ratio.

Alternative Methods

Another method to find the combined ratio is to use the least common multiple (LCM) of the denominators of the given ratios. In this case, the LCM of 4 and 5 is 20.

Method 1: Use LCM

Let us express both ratios with a common base of 20:

A:B 15:20 (since 4 * 5 20 and 3 * 5 15) B:C 20:24 (since 5 * 4 20 and 6 * 4 24)

Thus, the combined ratio A:B:C is:

15:20:24

Method 2: Algebraic Approach

Using the algebraic approach:

A:B 3:4 and B:C 5:6

Let us take A 3k, B 4k, B 5y, and C 6y. Equate B:

4k 5y → y (4k)/5

Thus, C 6y 6(4k/5) (24k)/5

Substitute back to find the combined ratio:

A 3k, B 4k, C (24k)/5

Multiplying by 5 to eliminate the fraction:

A 15k, B 20k, C 24k

Hence, the combined ratio A:B:C is: