February 2026 Problem of the Month

Mr. P has two vacuum cleaners which are pictured below.

When he went to the store to buy a new vacuum, the salesman told him, "This vacuum will cut your work in half," so Mr. P said, "I'll take two." (the math didn't work out the way he thought it would).
Both models are cordless and run on a battery.
The Dyson V-8 has two modes, but the Dyson V-11 has three modes (Eco, Med, and Boost), and that is the vacuum that is used in this problem.

When fully charged, the Dyson V-11 can clean for 67 minutes in Eco mode, or it can clean for 33 minutes in Med mode, or it can clean for 15 minutes in Boost mode.

Mr. P needs to use the Boost mode for three minutes to do some heavy cleaning and he needs to use the Med mode for 8 minutes to clean some moderate areas.
How much time will he have left in Eco mode to finish cleaning his house before the vacuum needs to be recharged?

Solution to the Problem:

Here is the solution:

Mr. P will have about 37 minutes of cleaning time left in Eco mode before the vacuum needs to be recharged.

First determine the battery usage rates for each mode:
Eco:   1/67 charge per minute
Med:   1/33 charge per minute
Boost:   1/15 charge per minute

Now we need to determine how much of the charge was used in each mode:
Boost for 3 minutes:   3 x 1/15 = 1/5 of the charge
Med for 8 minutes:   8 x 1/33 = 8/33 of the charge

So now determine how much of the battery has been used:
1/5 + 8/33 = 33/165 + 40/165 = 73/165

Now determine how much battery is remaining:
1 − 73/165 = 92/165

Now convert remaining battery to Eco time:
92/165 x 67 = 6164/165 ≈ 37.4 minutes.