A box contains only red and blue pens.
- Exactly 50% of the red pens and (one-third) of the blue pens are removed.
- After the removal, the box contains a total of 68 pens, with an equal number of pens of each color.
How many blue pens were in the box before any pens were removed?
A reverse-percent question gives the amount left but asks for the starting amount. First split an equal final total between the groups. Then find what fraction of the requested group remains and use a one-variable Desmos regression to work backward. The other group's removal rate doesn't determine the requested group's starting count.
Hints
- Hint 1
The final total contains equal numbers of red and blue pens. Split it into two equal groups to find the blue pens left after removal, not the blue pens there at the start.
- Hint 2
A complement is the share left after some is removed. If one-third of the blue pens are taken out, what fraction of the original blue pens stays in the box?
- Hint 3
The blue pens left are a fraction of the original blue count. Set that fraction of the unknown starting count equal to the blue count you found, then solve.
Step-by-step
Work backward from the blue pens left
Step 1Find the blue pens left
The pens left are split equally between red and blue, so each color has pens left. Half the final total isn't half the number of pens originally in the box.
- Step 2
Find the fraction of blue pens that remains
The complement is the fraction left after removal. One-third of the blue pens were removed, so the fraction left is . The remaining blue pens are two-thirds of the original blue pens.
- Step 3
Connect the original count to the remaining count
Let be the original number of blue pens. Two-thirds of that count is the blue pens left, so . Using instead would count the blue pens removed.
- Step 4
Solve for the original blue count
Type . Desmos uses for the unknown and to fit its value in a regression. Under PARAMETERS, it shows . The box originally contained 51 blue pens. Grid in 51.