Understanding the Ratio Problem

The question asks us to find the ratio of green tokens to pink tokens in a bag. We are given two separate ratios:

• The ratio of red to green tokens is $5 : 12$. This can be written as $\text{R} : \text{G} = 5 : 12$.
• The ratio of pink to red tokens is $7 : 15$. This can be written as $\text{P} : \text{R} = 7 : 15$.

To find the ratio of green to pink ($\text{G} : \text{P}$), we need to relate the three quantities: red, green, and pink. The common token mentioned in both given ratios is 'red'. We can use this to link the ratios together.

Combining the Ratios Using the Common Term (Red Tokens)

We have the ratios:

• $\text{R} : \text{G} = 5 : 12$
• $\text{P} : \text{R} = 7 : 15$

Let's rewrite the second ratio to have red first, similar to the first ratio:

• $\text{R} : \text{G} = 5 : 12$
• $\text{R} : \text{P} = 15 : 7$ (simply reversed the $\text{P} : \text{R}$ ratio)

Notice that the 'red' part of the ratio is 5 in the first case and 15 in the second. To combine these ratios, the value representing the common term (red tokens) must be the same in both ratios. We can find the least common multiple (LCM) of 5 and 15, which is 15.

Now, we adjust the first ratio ($\text{R} : \text{G} = 5 : 12$) so that the red part becomes 15. To do this, we multiply both parts of the ratio by $\frac{15}{5} = 3$.

New $\text{R} : \text{G} = (5 \times 3) : (12 \times 3) = 15 : 36$.

The second ratio ($\text{R} : \text{P} = 15 : 7$) already has the red part as 15, so we don't need to adjust it.

Now we have:

• $\text{R} : \text{G} = 15 : 36$
• $\text{R} : \text{P} = 15 : 7$

Since the value for red is the same (15) in both ratios, we can combine them to find the ratio of red to green to pink ($\text{R} : \text{G} : \text{P}$).

The combined ratio is $\text{R} : \text{G} : \text{P} = 15 : 36 : 7$.

Finding the Ratio of Green to Pink Tokens

From the combined ratio $\text{R} : \text{G} : \text{P} = 15 : 36 : 7$, we can directly find the ratio of green to pink tokens. The green part of the ratio is 36, and the pink part is 7.

Therefore, the ratio of green to pink tokens is $\text{G} : \text{P} = 36 : 7$.

Let's summarize the ratios:

So, the ratio $\text{G} : \text{P}$ is $36 : 7$.

Revision Table: Ratio Calculation Steps

Additional Information on Ratios and Proportions

A ratio is a comparison of two or more quantities. It shows how much of one quantity there is compared to another. Ratios can be written in different ways, like $a : b$, $\frac{a}{b}$, or "a to b".

When combining ratios that share a common element, it is essential to make the value representing the common element the same in all ratios. This is done by finding a common multiple (usually the LCM) and scaling the ratios accordingly.

For example, if $\text{A} : \text{B} = x : y$ and $\text{B} : \text{C} = p : q$, the common term is B. To find $\text{A} : \text{B} : \text{C}$, you would find the LCM of $y$ and $p$ and scale the ratios so the B values are equal. If LCM is $L$, you multiply the first ratio by $L/y$ and the second by $L/p$. Then the combined ratio will be $(x \times L/y) : L : (q \times L/p)$.

Understanding how to combine ratios is a key skill in solving problems involving multiple proportional relationships.