# POW Answer – 0.999999 Repeating Equals One?

The following answers the Problem of the Week from May 7, 2012:

One of the simplest ways to prove that 0.999999 = 1 is by doing the following:

• 0.333333 = 1 / 3
• 3 × (0.333333) = 3 × (1 / 3)
• 0.999999 = 1

As shown above, since it is true 0.333333 is the way to represent 1 / 3 in decimal form, it follows mathematically that 0.999999 is an alternative way to represent 1 in decimal form. A more extensive discussion can be found here on Wikipedia.

# POW Answer – Differences In Languages

This is the answer to the Problem of the Week that was posted on April 30.

The desired answer is 0. The reason is because “`11`” in binary is actually equal to 3. Therefore 3 – 3 = 0. On the other hand, you could use the following table to see how an infinite amount of answers could be given since we don’t necessarily know what base was used for the first translation:

3 4 4 – 3 = 1
4 5 5 – 3 = 2
5 6 6 – 3 = 3
6 7 7 – 3 = 4
7 8 8 – 3 = 5
8 9 9 – 3 = 6
9 10 10 – 3 = 7
X X + 1 X + 1 – 3 = X – 2

# POW Answer – Party Balloons

This is the answer to last week’s Problem of the Week:

One thing that I like to do when solving these types of problems is determining what percentage of the work each participant will do. The problem stated that Tasheera takes 16 seconds for each balloon, whereas Omar takes 18 seconds. Knowing this, we can basically calculate the percentage of the work with the following table:

Name Time Comparative Speed Percentage of Work
Tasheera 16 18/16 (18 / 16) / (18 / 16 + 18 / 18) = 9/17
Omar 18 18/18 (18 / 18)(18 / 16 + 18 / 18) = 8/17

After doing this, we can multiply their percentage against the amount of balloons each will blow up:

Name Percentage of Work Balloons
Tasheera (18 / 16) / (18 / 16 + 18 / 18) = 9/17 100 × 9/17 ≈ 53
Omar (18 / 18)(18 / 16 + 18 / 18) = 8/17 100 × 8/17 ≈ 47

Finally we can use the amount of balloons and multiply it by the amount of time it takes per balloon (minus one increment of the rest time) to get the amount of seconds it would take to blow up 100 balloons:

Name Time w/ Rest Rest Time Balloons Total Time (secs)
Tasheera 16 5 53 53 × 16 – 5 = 843
Omar 18 3 47 47 × 18 – 3 = 843

Therefore, the answer is that it would take Tasheera and Omar 843 seconds (14 and 3 seconds) to inflate 100 balloons.

# POW – Party Balloons

A group of friends are preparing for a party. Tasheera and Omar are in charge of preparing the balloons. Tasheera takes eight seconds to inflate each balloon, three seconds to tie each, and a five second break in between each one. Omar takes ten seconds to inflate each balloon, five seconds to tie each, and a three second break in between each one. How long will it take them to finish 100 balloons? Will they finish at the same time? How many balloons will each one end up blowing?

The answer to this Problem of the Week can be found here.