Write expressions for the numbers of red and blue marbles after adding 21 red and removing 6 blue. What is the new total number of marbles in terms of $n$?

After the changes, red marbles are $55\%$ of all marbles. How can you write an equation that says 'red marbles divided by total marbles equals 0.55' using your expressions from the previous hint?

Once you solve your equation for $n$, check which answer choice matches that value. You can also quickly plug that value back into the story to verify the red percentage is $55\%$.

In Desmos, type the expression (0.4x + 21)/(x + 15) and press Enter. This graphs the fraction of marbles that are red after the changes, as a function of the original total $x$.

On a new line, type y = 0.55 and press Enter. This is the line representing a red percentage of $55\%$.

Use Desmos to find the intersection point of the curve (0.4x + 21)/(x + 15) and the line y = 0.55 (tap or click where they cross, or use the intersection tool). The $x$-coordinate of that point is the original number of marbles; choose the answer option that equals this $x$-value.

Let $n$ be the number of marbles in the jar before any changes.

• Originally, $40\%$ of the marbles were red, so the number of red marbles was $0.40n$.
• The rest were blue, so blue marbles were $60\%$ of the total, or $0.60n$.

So initially:

• Red: $0.40n$
• Blue: $0.60n$
• Total: $n$

Now apply the changes described:

• 21 red marbles were added: new red count is $0.40n + 21$.
• 6 blue marbles were removed: new blue count is $0.60n - 6$.

The new total number of marbles is:

• New total $= (0.40n + 21) + (0.60n - 6) = n + 15$.

So after the changes:

• Red: $0.40n + 21$
• Total: $n + 15$

After the changes, red marbles made up $55\%$ of all the marbles.

This means:

This equation connects the original total $n$ to the new percentage of red marbles.

Start with the equation

Multiply both sides by $n + 15$:

Distribute on the right:

Move like terms to opposite sides:

Solve for $n$:

So there were $85$ marbles in the jar before the changes, which corresponds to choice C.