Question

What is the smallest number of red rods needed for an all-red train to equal the length of an all-blue train, assuming a 2:3 length ratio between red and blue rods?

Answer 1

Let's denote the lengths of the red and blue rods as \( R \) and \( B \) respectively. According to the problem, there exists a length ratio between red and blue rods given by:

\[
\frac{R}{B} = \frac{2}{3}
\]

From this ratio, we can express the length of a blue rod in terms of the length of a red rod:

\[
B = \frac{3}{2} R
\]

Next, we want to determine the smallest number of red rods, \( r \), needed for an all-red train to equal the length of an all-blue train composed of \( b \) blue rods. The lengths of the all-red and all-blue trains can be expressed as:

- Length of the all-red train:

\[
\text{Length of all-red train} = r \cdot R
\]

- Length of the all-blue train:

\[
\text{Length of all-blue train} = b \cdot B = b \cdot \left(\frac{3}{2} R\right) = \frac{3}{2} b R
\]

We set these two lengths equal to each other:

\[
r \cdot R = \frac{3}{2} b R
\]

Since \( R \) is common to both sides and non-zero, we can divide both sides by \( R \):

\[
r = \frac{3}{2} b
\]

To find the smallest integer \( r \) that satisfies this equation, we need \( \frac{3}{2} b \) to be an integer. This will occur when \( b \) is an even integer. Let’s denote \( b \) as \( 2k \) for some integer \( k \):

\[
r = \frac{3}{2} (2k) = 3k
\]

The smallest integer \( k \) we can take is \( 1 \):

\[
r = 3 \cdot 1 = 3
\]

Thus, when \( k = 1 \), we have:

\[
b = 2 \cdot 1 = 2
\]

This tells us that for \( b = 2 \), the smallest corresponding value of \( r \) is \( 3 \).

Therefore, the smallest number of red rods needed for an all-red train to equal the length of an all-blue train is:

\[
\boxed{3}
\]