Math puzzles are among the most intimidating in the world of puzzles. Many people will happily dive into a crossword or tackle a word seek at a moment’s notice, but drop some numbers into a puzzle, and they hesitate.

But there’s no reason to fear!

Math puzzles are certainly a different form of puzzling, but like all puzzles, there’s always a way in, if you know how to look for it. Today, we’re going to solve two math puzzles together in the hopes of demystifying this style of puzzle.

Let’s take a look at our first math puzzle, “Count the Votes.”

A problem developed at a recent election where 5,219 votes were cast for four candidates. The victor exceeded his opponents by 22, 30, and 73 votes, yet not one of them knew how to figure out the exact number of votes received by each. Can you?

Okay, where do we begin?

Let’s start with what we know. We know the total number of votes, 5,219. That will be one side of our equation.

We also know that the winner beat his three opponents by 22 votes, 30 votes, and 73 votes, respectively. Which means that the number of votes the winner received is the key to solving this puzzle. Let’s call that number of votes “x.”

The winner beat one opponent by 22 votes (x – 22), another by 30 votes (x – 30), and the last by 73 votes (x – 73).

We can build our simple equation from that information:

x + (x – 22) + (x – 30) + (x – 73) = 5219

Still a little daunting, but we can simplify it, because it doesn’t matter in which order we add or subtract things. So let’s look at that formula without the parentheses:

x + x – 22 + x – 30 + x – 73 = 5219

Now let’s reorganize it, putting the addition parts together and the subtraction parts together:

x + x + x + x – 22 – 30 – 73 = 5219

Subtracting those three numbers separately is the same as subtracting their total, so let’s simplify again:

x + x + x + x – 125 = 5219

Adding four x’s together is the same as multiplying one x by 4, so let’s express that:

4x – 125 = 5219

Now we’re getting somewhere.

And subtracting 125 from 4x is the same as adding 125 to 5219, so let’s do that:

4x = 5344

Finally, we divide 5344 by 4 to give us the value of x:

x = 1336

Which means that our victor got 1336 votes, one opponent got 1314 (x – 22), another opponent got 1306 (x – 30), and the last got 1263 (x – 73), totalling 5129 votes.

Now, that wasn’t so bad, was it? Let’s try another that’s a little bit harder.

This one is called “The Mathematical Cop.”

“Top of the mornin’ to you, officer,” said Mr. McGuire. “Can you tell me what time it is?”

“I can do that same,” replied Officer Clancy, who was known on the force as the mathematical cop. “Just add one quarter of the time from midnight until now to half the time from now until midnight, and it will give you the correct time.”

Can you figure out the exact time when this puzzling conversation took place?

Okay, this one isn’t as obvious about providing us with information, but the info is there if you look.

Since everything relates to the time “now,” we’ll make “now” our x.

Then we take each part of Officer Clancy’s statement in turn. “Just add one quarter of the time from midnight until now.”

“The time from midnight until now” is the same as “now,” x, so one quarter of that time is x/4.

And we’re meant to add that to “half the time from now until midnight.”

That’s a little bit tougher. After all, “the time from midnight to now” was easy, but “the time from now until midnight” covers the rest of a 24-hour day. So, if x covers the time from midnight to now, then “1440 – x” covers the time from now until midnight.

(There are 1440 minutes in a day, 60 minutes times 24 hours, and it’s easier to do all this in minutes, rather than hours and minutes.)

So “half the time from now until midnight” is (1440 – x)/2.

Okay, so what does our equation look like?

x/4 + (1440 – x)/2 = x

That’s pretty daunting, but we know what our goal is: to combine all those x’s and get them on the same side of the equal sign. And like the equation we built for “Count the Votes,” we can simplify it with some careful applied math.

The first step is to get rid of those pesky fractions.

Let’s multiply everything by 2 in order to remove the “/2” below “(1440 – x),” which gives us:

2x/4 + (1440 – x) = 2x

We can use the same trick to remove the “/4” below 2x:

2x + 4(1440 – x) = 8x

Now we’re getting somewhere! Let’s get rid of that 2x on the left by subtracting 2x from both sides:

4(1440 – x) = 6x

Let’s go a step further by multiplying both 1440 and x by 4:

5760 – 4x = 6x

One more step, and we’ve got all of those x’s combined on one side of the equation, as we’d hoped:

5760 = 10x

Divide 5760 by 10 and we’ve got x:

576 = x

If you recall, x represented the time “now,” but it’s still in minutes. To get the actual time, divide 576 by 60 to get the number of hours. 540 minutes = 9 hours, so 576 is 9 hours, 36 minutes.

It’s 9:36 AM, Officer, though to be honest, if you tell everyone the time this way, I imagine people stop asking you the time after a while.

I realize these are only two examples, and math puzzles come in all shapes and sizes, but hopefully, they don’t seem quite so intimidating, now that you know how to pick them apart for the important information.

Good luck! And if you find any math puzzles you need help with, send them our way! They could end up the subject of a future blog post!

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