What is the 'link' in a multiple ratio problem?

The link is the common variable or quantity that appears in both given ratios. It serves as the bridge that allows two separate ratios to be combined into one three-part ratio.

Why can't you combine $A:B = 2:3$ and $B:C = 4:5$ by just writing $2:3:5$?

Because the value of $B$ is different in both ratios (3 vs 4). You must scale both ratios so that $B$ has the same value (the LCM, which is 12) before they can be merged.

What is the most common error when scaling a ratio to match a link?

The most common error is only multiplying the link value and forgetting to multiply the other term in the ratio. This changes the actual proportion of the original ratio.

How do you find the total number of 'parts' in a combined ratio $A:B:C$?

You sum the values of all terms in the combined ratio ($A + B + C$). This total is used as the denominator when calculating fractions or sharing a total quantity.

When should you use the Least Common Multiple (LCM) in ratio problems?

Use the LCM when you have a shared variable with different values in two ratios. Setting the shared variable to the LCM in both ratios allows you to join them accurately.

What is the difference between a two-part ratio and a three-part ratio regarding the 'whole'?

In a two-part ratio $A:B$, the whole is $A+B$. In a three-part ratio $A:B:C$, the whole is $A+B+C$. The 'whole' changes because a new quantity has been introduced into the comparison.

If $X:Y = 1:2$ and $Y:Z = 1:2$, why is the combined ratio not $1:2:2$?

In the first ratio, $Y$ is 2. In the second, $Y$ is 1. To match them, the second ratio must be scaled by 2 to become $2:4$. The correct combined ratio is $1:2:4$.

How does combining ratios help in solving word problems with a 'total amount'?

It allows you to see how the total amount is distributed across all variables at once. You can find the value of 'one part' by dividing the total by the sum of all parts in the combined ratio.

Can you combine three ratios (e.g., $A:B$, $B:C$, and $C:D$)?

Yes, by using the same linking process sequentially. First combine $A:B$ and $B:C$ into $A:B:C$, then use $C$ as the link to join $C:D$ into a four-part ratio $A:B:C:D$.

What happens to the ratio $A:B$ if you multiply both $A$ and $B$ by 5?

The ratio remains equivalent. The relative proportion between $A$ and $B$ does not change, which is why scaling is a valid step in combining multiple ratios.

Library Podcasts

Courses

Referral & Rewards

Ratio, Proportion & Rates of Change

Multiple Ratios

Summary

Multiple ratios involve the process of combining two or more separate two-part ratios into a single, unified multi-part ratio. This is achieved by identifying a common element (the 'link') shared between the ratios and scaling them so that the shared element has the same numerical value in both expressions.

1. Definition & Core Concepts

Multiple Ratios occur when you have two separate comparisons, such as $A:B$ and $B:C$ , and need to determine the relationship between all three variables ( $A:B:C$ ).
The Link is the variable or quantity that appears in both ratios, serving as the bridge that allows them to be joined together.
A Combined Ratio represents the relative proportions of all involved quantities simultaneously, which is essential for solving problems involving a total sum shared among three or more parties.

Diagram showing two separate ratios linked by a common middle term (B) to form a single unified ratio (A:B:C).

2. Underlying Principles

The principle of Equivalence dictates that multiplying or dividing both parts of a ratio by the same non-zero number does not change the relationship between the quantities.
To merge ratios, the shared term must represent the same number of 'parts' in both ratios; this is conceptually similar to finding a Common Denominator when adding fractions.
The Least Common Multiple (LCM) of the link's values in the two separate ratios is the most efficient target value for scaling.

3. Methods & Techniques

4. Key Distinctions

5. Exam Strategy & Tips

6. Common Pitfalls & Misconceptions