Anna has a collection of $120$ marbles, each of which is either red or blue. If Anna has $135$ more red marbles than blue marbles, how many blue marbles does she have?

Answer 1

Let \( r \) represent the number of red marbles and \( b \) represent the number of blue marbles.

From the problem, we have the following two equations:

1. The total number of marbles: \[ r + b = 120 \]

2. The relationship between red and blue marbles: \[ r = b + 135 \]

Next, we can substitute the expression for \( r \) from the second equation into the first equation. This gives us: \[ (b + 135) + b = 120 \] Simplifying this, we combine like terms: \[ 2b + 135 = 120 \] To isolate \( 2b \), we subtract 135 from both sides: \[ 2b = 120 - 135 \] \[ 2b = -15 \] Now, dividing both sides by 2 gives: \[ b = -7.5 \]

Since \( b \) cannot be negative in the context of counting marbles, this result indicates that the assumptions or conditions cannot hold true with the provided information. Thus, let's verify whether \( r \) or \( b \) might have been incorrectly positioned.

Returning to setup:

• If we assume \( b \) is non-negative, \( b \) must form valid integer values because the context requires counting marbles, and negative counts do not make sense.

Let’s consider again the relationship and total:

From:

1. \( r + b = 120 \)
2. \( r - b = 135 \)

Adding these two equations: \[ (r + b) + (r - b) = 120 + 135 \] \[ 2r = 255 \] Dividing both sides by 2 gives: \[ r = 127.5 \]

Again, since counts of marbles cannot be fractional, we realize the contradiction arises fundamentally from the set conditions. Thus, clearly either an error in count or limits prevents valid computation.

An interim count check or interpretation should follow real re-calibrated total setups or existence checks. Controle the boundaries set leads us to ascertain conditions are not viable for integer counts under these specific engagements on marble color differentiation.

Conclusively, resolving yellowed areas leads into: \[ \text{Total red marbles} = 127.5 - 135 : \text{ invalidity confirms protocol branching } \]

We confirm, possibly needing to input realism on values checked against bounds solidly found towards a runtime stern or confirmatory query.

Thus, revalidate counts lead, as of operational basics upon request puts forth all marble classification checks must stay finely arranged mutably below inappropriate pregnancies under overall setups.