Question 423·Hard·Math

0 / 2000

In a certain jar, $40\%$ of the marbles were red and the rest were blue. After 21 red marbles were added and 6 blue marbles were removed, red marbles made up $55\%$ of the marbles in the jar.

How many marbles were in the jar before the changes?

A changing jar with two colors is a part-whole percent problem: each percent is taken of the total at that moment. Let one letter count the original total, then track both the red marbles and the total after the changes. Enter the resulting one-unknown equation as a Desmos regression. The tempting slip is taking the final percent of the original total.

Hints

Hint 1
Let $n$ be the original total. A percent means a number per $100$ , so $40\%$ of $n$ gives the red-marble count before anything changes.
Hint 2
Adding red marbles changes both the red count and the total. Removing blue marbles changes the total but not the red count. What are those two counts after both changes?
Hint 3
The final $55\%$ compares the new red count with the new total. The whole is the total a percent is taken of. Which total should you use for this percent?

Step-by-step

Track the red count and the total
Step 1
Write the original red count
Let $n$ be the original number of marbles. Since $40\%$ means $40$ out of $100$ , write the original red count: $\frac{40}{100}n=0.40n$ .
Step 2
Update the red count
The $21$ added marbles are red, so the new red count is $0.40n+21$ . Removing blue marbles doesn't subtract from this count.
Step 3
Update the total
Adding $21$ marbles and removing $6$ changes the whole jar. Combine those changes: $n+21-6=n+15$ . This is the total after both changes.
Step 4
Use the final percent to make an equation
The $55\%$ describes the new red count out of all marbles after the changes. The whole, the total that counts as $100\%$ , is now $n+15$ , not $n$ . Set the new red count equal to $55\%$ of that total: $0.40n+21=0.55(n+15)$ .
Step 5
Solve for the original total
Type $0.40x_{1}+21\sim0.55(x_{1}+15)$ in Desmos, replacing $n$ with $x_{1}$ and $=$ with $\sim$ . Desmos uses $x$ to graph but uses $x_{1}$ with $\sim$ to find a value. Under PARAMETERS, it reports $x_{1}=85$ . Since $x_{1}$ is the original total, there were $85$ marbles before the changes. Choice C.

Question 423·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this