6 + 1 x 0 + 2 / 2 = ?

It’s 7. It’s not 1, or 0, or 5, or 3.5, or any of the other answers people have imagined. This has been going round and round on Facebook, with over 300,000 responses last time I looked. Sadly, most of the answers are wrong.

There’s a thing called order of operations. Arithmetic isn’t a matter of opinion or voting. There’s a memory aid — Please Excuse My Dear Aunt Sally — to help one remember the right sequence: parentheses, exponents, multiplication and division, addition and subtraction.

Let’s apply that to 6 + 1 x 0 + 2 / 2:

  1. No parentheses. Move on.
  2. No exponents. Move on.
  3. Multiplication and division: 1 x 0 = 0 and 2 / 2 = 1. Now, our exotic puzzle becomes 6 + 0 + 1.
  4. Addition and subtraction: 6 + 0 + 1 = 7.

Smart and Stupid Calculators

My iPhone’s calculator app comes up with 7 because it uses the proper order of operations.

The calculator app on Windows 7 comes up with 7 because it also uses the proper order of operations.

My TI calculator from 20+ years ago gets 7. It, too, uses the proper order of operations. (Yep, I still have an old calculator, and it still works.)

However, I’ve seen cheap calculators that use the wrong order of operations. They act like everything is from left to right, no matter what. You enter 6 + 1, and they immediately show 7. Multiply by 0 and you get 0. Add 2 and you get 2. Divide by 2 and you get 1. I imagine this is why 1 is a popular answer, when people assume all arithmetic is strictly left to right, or they’ve been misled by a stupid calculator.

Counting Cookies

Here’s an illustration of why multiplication and division come before addition and subtraction.

Let’s say I’m counting my cookie intake over the last month. On 5 occasions, say I had 3 chocolate chip cookies: 5 times, 3 cookies, or 5 x 3. On 3 occasions, I had 2 oatmeal raisin cookies: 3 times, 2 cookies, or 3 x 2.

5 x 3 + 3 x 2 = ?

Do you think I had 21 cookies, or 36 (or some other number)? The correct answer, certainly, is that I had 21 cookies: 5 x 3 = 15 chocolate chip cookies, plus 3 x 2 = 6  oatmeal raisin cookies, for a total of 21 cookies.

If you do the arithmetic in the wrong order, like strictly left to right, you’d think I had 36 cookies. 5 times, 3 cookies = 15 so far. 15 cookies + the 3 times I had oatmeal raisin = 18 cookies so far. (Makes no sense now, right?) 18 cookies, times 2 for the two times I had oatmeal raisin = 36. Wrong!

If you think 5 x 3 + 3 x 2 = 36, try laying out pennies. Take 5 sets of 3, and then 3 sets of 2, and then count how many pennies you’ve laid out. You’ll get 21, not 36.

Or say the problem in a different order, oatmeal raisin before chocolate chip. 3 times, I had 2 oatmeal raisin cookies. 5 times, I had 3 chocolate chip cookies.

3 x 2 + 5 x 3 = ?

If you follow the proper order of operations, you’ll find that I had 21 cookies, same as before. If you pretend arithmetic always runs left to right, you get a different answer this time: 3 x 2 = 6, 6 + 5 = 11, 11 x 3 = 33.

If strict left-to-right arithmetic was correct, then I had either 36 or 33 cookies, depending on which cookies you count first. Does that convince you?

Vending Machines Help Make the Point

You go to a vending machine. The item you want costs 75 cents. You put in 2 quarters, 2 dimes, and a nickel. Does that add up to 75 cents? Only if you follow the correct order of operations.

2 x 25 + 2 x 10 + 1 x 5 = ?

The correct order of operations has us do the multiplications before the additions: 2 x 25 = 50, and 2 x 10 = 20, and 1 x 5 = 5. Then we add up 50 + 20 + 5 and get 75.

If you think arithmetic is only left to right, you’d get 2 x 25 = 50, plus 2 = 52, times 10 = 520, plus 1 = 521, times 5 = 2,605.

Which is it? Do you think 2 quarters, 2 dimes, and a nickel add up to 75 cents, or $26.05?

Now switch the order. If you take those same coins, but you put in the 2 dimes, the nickel, and then the 2 quarters, is it still 75 cents?

2 x 10 + 1 x 5 + 2 x 25 = ?

Of course it’s still 75 cents. The correct order of operations says so: 2 x 10 = 20, and 1 x 5 = 5, and 2 x 25 = 50. Then 20 + 5 + 50 = 75 cents.

If you use left-to-right order instead of the correct order: 2 x 10 = 20, plus 1 = 21, times 5 = 105, plus 2 = 107, times 25 = 2,675. You’d think you had put $26.75 into the machine this time, instead of $26.05 when you started with the quarters.

If you believe your two quarters, two dimes, and one nickel add up to 75 cents in any order, you just made a case for using the correct order of operations.

And that’s my $0.02.

Jim

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2 Comments

Filed under Math

2 responses to “6 + 1 x 0 + 2 / 2 = ?

  1. olaceyj

    There are now over 400,000 responses. I think the reason your cheap 4-function calculator gives the wrong result is that it only considers two values and one operator at a time. It cannot store the entire expression. So to use a 4-function calculator you have to know the Order of Operations and find the product 1×0 and the quotient 2÷2 before proceeding with the 6-0+1, which yields 7, of course.

  2. olaceyj

    I should have added….and this is why students of pre-Algebra, who are learning the Order of Ops, can use a 4-function calculator on their math tests but not a TI-83 or better.

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