Question 51·Hard·Percentages
A jar contained some red beads and some blue beads, for a total of 300 beads. After 30% of the red beads and 10% of the blue beads were removed, the total number of beads in the jar decreased by 18%. How many red beads were originally in the jar?
For percentage mixture or population-change problems, define variables for each group, write an equation for the original total, and then express the final amounts using the given percentage increases or decreases. Convert any overall percent change into a multiplier (like becomes ) to get the new total, then write a second equation for the remaining amount. Solve the resulting system—often by substitution—rather than trying to reason it out mentally or by trial and error, which is slower and more error-prone on the SAT.
Hints
Start with variables and the total
Let be the number of red beads and be the number of blue beads. What equation can you write using the fact that there are 300 beads in total?
Interpret the 18% decrease
If the total number of beads decreased by , what percentage of the original beads remains? Use this to find the new total number of beads after the removal.
Write an equation for the remaining beads
After removing of the red beads, what fraction of is left? After removing of the blue beads, what fraction of is left? Add these to match the new total you found.
Solve the system
You should now have two equations: one from the original total and one from the remaining beads. Substitute into the second equation and solve for .
Desmos Guide
Set up the expressions in terms of red beads
In Desmos, let represent the number of red beads. Since the total is 300, the number of blue beads is . The expression for the remaining beads after removal is .
Graph remaining beads and the target total
Type y = 0.7x + 0.9(300 - x) as the first equation. Then type y = 0.82 * 300 as the second equation. These represent (1) the number of beads left in terms of , and (2) the final total number of beads.
Find the intersection
Use Desmos to find the point where the two graphs intersect. The -coordinate of this intersection is the number of red beads originally in the jar. Read this -value directly from Desmos.
Step-by-step Explanation
Define variables and use the total
Let be the number of red beads and be the number of blue beads originally in the jar.
We are told the total number of beads was 300, so we have the equation:
Translate the percentage decrease for the total
The total number of beads decreased by .
That means the new total is of the original total.
The original total is 300 beads, so the new total number of beads is:
So, after removing some beads, there are 246 beads left in the jar.
Express how many beads remain after removal
We removed of the red beads and of the blue beads.
- Remaining red beads: of , which is .
- Remaining blue beads: of , which is .
So the total remaining beads is:
But from Step 2, we know this must equal 246, so:
Solve the system of equations
We now have a system:
From the first equation, express in terms of :
Substitute this into the second equation:
Distribute :
Combine like terms :
Subtract 270 from both sides:
Divide both sides by :
So, there were originally 120 red beads in the jar.