Earlier this week, I posted a ACT Math Challenge Question making itself look all tough and scary in its cloak of probability and irrationality. Here’s a refresher if you missed it, then scroll down for the answer!
ACT Challenge Question #5
A box contains 50 cards. is written on the first card, on the second, on the third and so on through , with no numbers repeated. A card is drawn at random from the box. What is the probability that the number on the card is an irrational number?
Answer and Explanation
ANSWER: E. 43/50
First of all, we need to remember what rational and irrational numbers are. Rational numbers can be expressed as a fraction; irrational numbers cannot.
Now let’s think through the question. Do you really have to go through every number from 1 to 50? The ACT will never expect you to do that, so there must be a better way.
Let’s take a look at the pattern: , , , . As soon as we get to , we have something that can be simplified to an integer. = 2. is a perfect square, and guess what? The perfect squares are the only square roots that are rational numbers. Something like gets really messy because there are not two equal fractions that can be multiplied together to equal 5.
So we just need to count the perfect squares before 50. There are seven: 1, 4, 9, 16, 25, 36, 49. So that means out of our 50 numbers, 7 can be reduced to integers (or fractions) and 43 are irrational. So our answer is E.
Bonus hint: If you come across a problem like this on the ACT, and don’t know how to solve it, make sure you eliminate some answer choices! The moment you find an irrational number as you count through the series, you can eliminate answer choice A, for example. And this happens really quickly: is an irrational number. You can also take a logical guess if you just work through the first 11 numbers. You should find that out of the first 11, 8 numbers are irrational. This definitely eliminates B, and thinking logically, you could probably take a guess that the overall probability is going to be high.