Pls tell me how we find the ratio b/c ..... If k1/ k2 is 2/3 after 1 hr

To find the ratio of two quantities, such as \( b \) and \( c \), given that the ratio of \( k_1 \) to \( k_2 \) is \( \frac{2}{3} \) after one hour, we can follow a systematic approach. Let's break it down step by step.

Understanding the Ratios

Ratios are a way to compare two quantities by division. When we say \( \frac{k_1}{k_2} = \frac{2}{3} \), it means that for every 2 parts of \( k_1 \), there are 3 parts of \( k_2 \). This relationship can help us find other ratios, such as \( b \) to \( c \), if we know how they relate to \( k_1 \) and \( k_2 \).

Setting Up the Problem

Let's assume that \( b \) and \( c \) are directly proportional to \( k_1 \) and \( k_2 \), respectively. This means we can express \( b \) and \( c \) in terms of \( k_1 \) and \( k_2 \). For instance, we can write:

• \( b = m \cdot k_1 \)
• \( c = n \cdot k_2 \)

where \( m \) and \( n \) are constants that define how \( b \) and \( c \) scale with \( k_1 \) and \( k_2 \).

Finding the Ratio

Now, to find the ratio \( \frac{b}{c} \), we can substitute the expressions for \( b \) and \( c \):

\[ \frac{b}{c} = \frac{m \cdot k_1}{n \cdot k_2} \]

Substituting the known ratio of \( k_1 \) to \( k_2 \):

\[ \frac{b}{c} = \frac{m \cdot (2/3 \cdot k_2)}{n \cdot k_2} \]

Notice that \( k_2 \) cancels out:

\[ \frac{b}{c} = \frac{2m}{3n} \]

Example Calculation

Let’s say we assume \( m = 1 \) and \( n = 1 \) for simplicity. Then:

\[ \frac{b}{c} = \frac{2 \cdot 1}{3 \cdot 1} = \frac{2}{3} \]

This means that if \( b \) and \( c \) are directly proportional to \( k_1 \) and \( k_2 \), respectively, then the ratio \( \frac{b}{c} \) is also \( \frac{2}{3} \).

Final Thoughts

In summary, to find the ratio of \( b \) to \( c \) based on the ratio of \( k_1 \) to \( k_2 \), you can express both \( b \) and \( c \) in terms of \( k_1 \) and \( k_2 \) and then simplify. This method can be applied to various scenarios where you have proportional relationships. If you have specific values for \( m \) and \( n \), you can substitute those in to get the exact ratio.