Well hello there! I’ve got another lesson in Math Etiquette in store for you. This series is dedicated to helping you memorize and practice the key math building-blocks that will help you move quickly and accurately through any math section—from the ACT’s Math section to the dreaded SAT No-Calculator Math! Past installments in this series have taught you what core math facts you need to simply memorize and how to work efficiently with fractions. Today’s post drills down into a key skill: multiplication.
“But Kristina,” you protest, “I already know how to multiply!” Well, that’s a great start, BUT: time is of the essence in most standardized tests’ math sections—so are you able to multiply quickly enough to get the high score you need? As it turns out, multiplication is one of the cornerstones of many other types of math problem on the SAT and ACT. So learning some shortcuts in this area NOW will pay out huge dividends for you LATER!
Something I’ve noticed among my private tutoring clients is that they often come to me with a very literal approach to multiplication. They only feel confident that they know the answer if it’s written in EXACTLY the same way as on their flashcards from 3rd grade. Like, they will know that 7×3=21, but will stutter and stumble and spend 2 minutes on scratch-work when faced with 0.7×30 (also 21), 70×300 (21,000), or (God forbid!) 0.7×0.03 (0.021).
But with a little mental flexibility, you’ll see that you really DO know how to multiply ANYTHING—and lightning-fast, at that! These Math Etiquette hacks show you how.
ARTICLE CONTENTS
1. Decimals and powers of 10 on SAT and ACT Math
2. Quickly multiply a number ending in 5 by itself
1) Decimals and powers of 10 on SAT and ACT Math
The trick here is to avoid getting caught up on 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
Ex: 70×300
This is really 7×3=21 in disguise. Don’t focus on the extra 0s. Just count them up and tack them onto your basic math fact answer of 21. There are three 0s, so it’s 21 with “000” at the end, or 21,000.
B) Both are decimals (or, decreased by powers of 10)
Ex: 0.7×0.03
This is also 7×3=21 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 (“7” and “03”), so do your core math fact of 7×3=21 and move the decimal over to the left three places: 0.021.
C) One number is increased by a power of 10 while the other is a decimal.
Ex: 0.7×30
I love this one! Please don’t get out your scratch paper and start doing long-form multiplication. Instead, know that the number of decimal places in each number counteract one another. 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. 0.7×30 becomes 7×3, which, of course, is 21. (I moved the decimal on the 0.7 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 instance of this: 7,000×0.03
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 7,000. We now have 70×3, which is just 7×3=21 (math fact) with an extra “0” tacked on. So, 210.
2) Quickly multiply a number ending in 5 by itself
Ex: 45×45
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-peasy!
3) How to multiply numbers ending in 5 that are 10 apart
Ex: 45×55
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. Sweet!
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 to pick, 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.
This one becomes a lot clearer with an example:
Ex: 4,235,756×55,897= well, that’s icky….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. Fill in the bubble and move on to grabbing more points. 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 multi-part math problems go much more quickly, and you’re less likely to make an error along the way. Try today’s tips out while you’re doing your math homework for school as well as for your SAT or ACT prep. Happy multiplying! And if you’d like the help of a true numbers wiz on this or any other element of SAT/ACT math, learn more about working with me one-on-one, or check out my complete guides to SAT and ACT Math.
