Howdy, folks! We’re back with another lesson in Math Etiquette, my series dedicated to building the kind of fundamental math understanding that will help you take on any math section, even the scary new SAT No-Calculator Math, with the confidence that comes from knowing you can work quickly and accurately. This time, we’re relearning how to do something you already know how to do: multiplication. Being able to multiply with ease, efficiency, and accuracy is one of the cornerstones of being able to do virtually every other type of math problem out there on the SAT and ACT—with enough time to get a top score.
Unfortunately, most students I know are very literal with their multiplication. They only think they know the answer if it’s written in EXACTLY the same way as on their flashcards from 2nd grade. Like, they will know that 6×3=18, but will stutter and stumble and spend 2 minutes on scratch-work when faced with 0.6×30 (also 18), 60×300 (18,000), or (God forbid!) 0.6×0.03 (0.018).
But with a little mental flexibility, you’ll see that you really DO know how to multiply ANYTHING—and quicker than you think! These Math Etiquette tips show you how.
1) Be smart about decimals and powers of 10
In other words, don’t get distracted by WHERE the digit is occurring—instead, locate the basic multiplication fact, and adjust according to a few rules:
A) Both are increased by powers of 10
This is really 6×3=18 in disguise. Don’t focus on the extra 0s. Just count them up and tack them onto your basic math fact answer of 18. There are three 0s, so it’s 18 with “000” at the end, or 18,000.
B) Both are decimals (or, decreased by powers of 10)
This is also 6×3=18 in disguise, but each is behind the decimal. Don’t get distracted by how scary you think the decimals are; just count how many digits are behind them instead. There are three digits hiding behind a decimal (“6” and “03”), so do your math fact of 6×3=18 and move the decimal over to the left three places: 0.018
C) One number is increased by a power of 10 while the other is a decimal.
These are my favorite! Please don’t take out your pencil and start doing long-form multiplication! Instead, they each counteract the other. In other words, you can move the decimal to the right on one number, so long as you move it to the left by the same number of spaces on the other. 0.6×30 becomes 6×3, which we know is 18. (I moved the decimal on the 0.6 to the right one space, while moving the decimal on the 30 to the LEFT one space, in essence, chopping off the 0.)
Here’s another: 6,000×0.03
Using our trick, I can move the decimal on the 0.03 twice to the right, so long as I “pay” for it with two 0’s from the 6,000. We now have 60×3, which is just 6×3=18 (math fact) with an extra “0” tacked on. So, 180.
2) How to quickly multiply a number ending in 5 by itself
This takes no scratch-work at all and only two steps:
STEP ONE: Write the last two digits as “25”: _ _ 25
STEP TWO: Multiply the tens-place digit (the “4” in “45”) by the next number up (next number up from “4” would be “5”). Write that in front. 4×5=20, so 45×45= 2025.
Here’s another: 75×75=_ _ 25; 7×8=56, so 5625. Easy!
3) How to quickly multiply numbers ending in 5 that are 10 apart
This also takes no scratch work at all and only two steps!
STEP ONE: Write the last two digits as “75”: _ _ 75
STEP TWO: Multiply the smaller tens-place digit (the “4” in “45”) by 1+ the larger tens-place digit (“5” in “55” is the larger, so 1 more than 5 is 6). Write that in front. 4×6=24, so 45×55= 2475.
Here’s another: 75×85=_ _ 75; 7×(8+1) = 7×9 = 63, so 6375. Piece of cake.
4) The ones digit tells all.
If you are working on a large calculation, and you’re just trying to find which multiple-choice answer it is, this one can save you tons of time. Basically, the one’s digit of your answer—no matter how gigantic the numbers you’re multiplying are!—is just the one’s digit when you multiply the one’s digits of the two original numbers.
It’s better to see this one than explain it.
Ex: 4,235,756×55,897= well, that’s obnoxious, but it ends in “2”. And if there’s only one answer choice that ends in 2, then I’m done and saved myself a TON of time!
How did I know? The ones place in “4,235,756” is 6 and the one’s place in “55,897” is 7. When I multiply 6×7, I get 42, which ends in 2. That's usually enough to answer the question—and if you approach it this way, you can ignore most of those digits. You’re welcome.
5) Difference of Squares
What—what?? You thought this was only for polynomials, did you? Ha!
You can use difference of squares (so long as you’ve memorized your perfect squares—remember those?) to multiply two numbers together that are an even number apart. This is how it works:
Ex: 17×23= ?
Notice that 17 and 23 are only 6 (an even number) apart, equally spaced around 20. In fact, you can replace 17 with (20−3) and 23 with (20+3).
Basically, you have (20−3) (20+3), which WE know is 202−32 = 400 − 9 = 391. Pretty cool, huh?
Here’s another: 21 × 29 = (25-4)(25+4) = 252 − 42 = 625 − 16 = 609.
(And since this ends in 9, and the 1 in 21 and the 9 in 29 multiply to 9, I know I’m right without having to check!)
When you’re able to manipulate how you multiply your numbers, it’s going to make every single step of harder or more complicated math problems go much more quickly, and you’re less likely to mess up along the way. Try these out while you’re doing your math homework for school as well as for your SAT or ACT prep. And happy multiplying!