Did you solve it? Are you smarter than an English major? | Mathematics

Earlier today, I presented you with these puzzles for English majors, that is, people who studied English in college. Here they are again with solutions and commentary from Ben Orlin, whose book Mathematics for English majors is out in September.

For each question below, which option is larger? No calculators allowed!

1. Square vs cube

[Note: squares are the numbers 1², 2², 3², … and cubes are 1³, 2³, 3³, …]

[Update: There was a typo in the setting of this question. Sorry everyone. Now it is fixed]

Solution the blocks

It is tempting to choose the latter. Cubing grows a number faster than squaring it. Moreover, the individual numbers in that series are larger. Oh, that’s exactly the problem. Because squares grow slowly, the first sum includes 10 squares. But because cubes grow so fast, the second only includes one cube: 5³ = 125. So, the first sum is much larger.

2. Mr. Pentacles

Solution They are the same

Each is 17 x 32/100.

3. Breaking offense

Solution 3997/4001

Despite its many merits, the language of fractions has a fundamental problem: It is difficult to tell which of two is greater! In this case, each of the fractions is a tiny bit smaller than 1. The first is 4/4001 shy, and the second is 5/5001 shy. Set aside the original fractions and focus on these “missing piece” fractions. Rather than giving them a common denominator, go for a common numerator: the first is 20/20,005, while the second is 20/20,004. Each offers 20 slices of an extremely thinly sliced ​​pizza. The first pizza, with its one extra slice, has slightly thinner slices, so the first missing fraction (4/4001) is just a bit smaller than the others. So, the first of the original fractions (3997/4001) is just a bit bigger. Indeed, if you convert them to decimals, you will see that the fractions are identical for the first seven places, finally differing at the seventh: 0.99900025 versus 0.99900020.

4. Carrot shoots

• the square root of 6

• the cube root of 15

Solution cube root of 15

These numbers are close to each other (difference by less than 0.02). They are also, to use a crude epithet for irrational, nasty. We need an operation that can simultaneously increase their difference, and clean them up into a more beautiful form, without changing their relative size. The first number begs to be squared. The second is begging to be diced. So why not both? Raise the numbers to the sixth power! The first number becomes 6³which is 216. The second turns 15²which is 225. So, the second number is bigger.

5. Tick tock

• the number of seconds in a year,

• the number of hours in a millennium

Solution the number of seconds in a year

Instead of wading through the tiresome multiplication (60 x 60 x 24 x 365 vs. 24 x 365 x 1000), let’s just focus on how the numbers differ. The second duration is 1000 times longer. The first mark 60 x 60 = 3600 times more often. The first number is therefore 3.6 times larger.

6. Power shower

Solution 5⁴⁵

Comparing a long string of 2 x 2 x 2s to a long string of 5 x 5 x 5s is annoying. So let’s convert those 5s, as best we can, into 2s, rewriting each as 2 x 2 x 1.25. By rearranging those factors, the second number becomes 2⁹⁰ x (1.25)⁴⁵. Since the first number can be rewritten as 2⁹⁰ x 2¹⁰we have now simplified our equation to:

Whichever is greater, 2¹⁰ or (1.25)⁴⁵?

In other words, which is better: doubling ten times, or growing forty-five times by 25%?

If you have any experience with compound growth, you would suspect that the latter is better. Indeed, it is. Four consecutive increases of 25% are better than a single doubling. Now, we compare more than 11 of these quartets with just 10 doublings. So, 5⁴⁵ is bigger! By a factor, it seems, of about 22.

Thanks to Ben Orlin for suggesting today’s puzzles. Mathematics for English majors is out on September 26th.