Whole-Number Math and Order of Operations

0:00 All right, welcome to this explainer. Look, if you're gearing up for the ASVAB, there's one specific rule that probably makes your stomach completely drop. No calculators allowed. If the thought of doing all that math by hand makes you, you know, a little anxious, just take a deep breath. You are absolutely not alone here. It's a totally normal feeling. But today, we're going to completely flip that anxiety on its head. Our goal for this explainer is to empower you with solid whole number fluency. By the time we wrap up, you'll be tackling these problems mentally and on paper with total confidence. So here's our game plan for today. We'll start with the no calculator rule itself. Then move into mastering the four operations. After that, we'll hit divisibility and the order of operations. Look at some estimation and fast patterns. Tackle real ASVAB examples and traps. And finally, we'll finish up with a practice section to check your skills. OK, let's dive into part one, the no calculator rule. Why whole number fluency is your secret weapon? Here's the reality of the ASVAB arithmetic reasoning and mathematics knowledge sections.

1:02 Almost every single question, whether it's fractions, percentages, ratios, or even algebra, eventually boils down to just a few basic whole number steps. And if those steps are slow or kind of shaky for you, well, you lose time. And you really risk making careless errors on problems you actually know how to solve. Imagine you're in the middle of a super stressful test. And you need to multiply 25 by 36. Now, a prepared test taker knows that four times 25 is 100, right? So they just mentally divide 36 by 4 to get 9. And then multiply that 9 by 100 to instantly snap to 900. Seeing this answer mentally, without scribbling out long multiplication, that saves you precious minutes. It literally gives you the extra time you need for the harder questions. And it keeps your brain fresh. Moving on to part two, mastering the four operations. Adding, subtracting, multiplying, and dividing by hand. Let's kick this off with a subtraction problem that trips up a lot of people. 2004 minus 1868. If you set this up the old school traditional way, working right to left, you're immediately forced to borrow across multiple zeros.

2:08 Guys, this is a massive trap. Borrowing across zeros is exactly where rushing produces careless errors. It's so common to get confused, forget to drop a 9, and just end up with a completely wrong answer. But there is a much better, way safer method. It's what we call the counting up method. Instead of borrowing, we just count up from the smaller number to the larger one in nice, easy chunks. Watch how this works. Starting at 1,868, let's just count up to the next friendly hundred, which is 1,900. Well, that's a difference of 32. Next, we count up from that 1,900 right to our target, 2,004. That's a difference of 104. Finally, you just add those two friendly chunks together. 32 plus 104 equals 136. We completely bypassed that nasty zero borrowing trap, and the math was practically stress-free. OK, next up is multiplication. Here we've got 36 multiplied by 24. Doing that entire thing at once in your head, pretty tough. But we can make it so much easier by breaking one of those factors down into tens and ones.

3:14 Let's take that 24 and break it down. We simply split the 24 into 20 and 4. First, we do 36 times 20, which is 720, then we do 36 times 4, which is 144. Just add those two partial products together. 720 plus 144 equals 864. Now, I do want to give a quick warning here. If you are doing this as long multiplication on scratch paper, you absolutely must watch out for the dreaded misalignment trap. When you multiply by the tens digit, you have to remember to shift your partial product one place to the left. If you forget to shift it, your addition at the very end is going to be completely off. Division. More specifically, the huge trap of ignoring the remainder. Let's say we divide 347 by 8. Well, 8 goes into 347 43 times, with 3 left over. So strictly mathematically speaking, it's 43 remainder 3. But on the ASVAB, context is everything. What if this is a word problem about 347 trainees riding in vans that seat 8? Well, this exact same division gives you three completely different answers depending on how the question is asked.

4:21 If they ask for completely full vans, the answer is 43. If they ask how many trainees are left over, it's three. But if they ask how many vans are needed, so nobody gets left behind, the answer is 44. Because those three left over trainees still need a ride, right? Always, always read the question carefully. All right, part three, divisibility and order of operations, working smarter, not harder. Here's a super handy overview of divisibility rules. These are basically cheat codes for simplifying numbers without having to grind through long division. Let's just run through them real quick. A number divides by two if it ends in an even digit. It divides by five if it ends in a zero or a five. And by 10 if it ends in zero. Pretty simple, right? Now for the fun ones. It divides by three if its digits add up to a multiple of three. Same exact logic for nine. It divides by nine if its digits add up to a multiple of nine. For four, you just look at the last two digits. If they form a multiple of four, the whole big number does, too. And for six, it just has to pass both the two and the three tests. Seriously, memorize this.

5:23 It's a massive time saver on test day. Now, let's talk about the strict order of operations. When a math expression has several operations going on, you have to follow this exact sequence. Step one, grouping symbols like parentheses or brackets, and you work from the inside out. Step two, exponents. Step three, multiplication and division. And finally, step four, addition and subtraction. Now, notice how I placed a major emphasis on the left to right rule for steps three and four. Multiplication and division are on the exact same level. You just do them from left to right as they appear. And the exact same thing goes for addition and subtraction. Let me show you exactly why that left to right rule is absolutely critical. We call this the deadly PEMDAS trap. Look at this problem. Evaluate 18 divided by three times two. So many test takers mistakenly think the M in PEMDAS means multiplication always comes before division. But if you fall for that trap, you might multiply three by two first to get six, and then 18 divided by six is three. That is completely wrong.

6:26 No way. The correct way is to read it left to right. 18 divided by three is six. Then six times two is 12. 12 is the actual correct answer. Don't let that trap catch you. Moving into part four. Estimation and fast patterns. Mental shortcuts for quick calculations. Check out this crazy fast no calculator pattern for multiplying by 99. Let's say you need to solve 37 times 99. Please don't write out long multiplication for this. Since 99 is just one less than 100, all you do is multiply the number by 100 instead and then subtract the original number once. So 37 times 100 is 3700. Then just subtract 37. 3700 minus 37 is 3663. You can literally do that in your head in about five seconds. Here's another brilliant trick. This time for multiplying any two digit number by 11. To instantly multiply by 11, you just add the two digits together and pop that sum right in the middle. So for 43 times 11, four plus three is seven. Put the seven in the middle and you get 473.

7:30 Super easy. Now if the sum is 10 or more, you just carry the one. So for 78 times 11, seven plus eight is 15. Put the five in the middle and carry the one over to the seven, making it an eight. So 858. These patterns are absolute gold for saving time. Part five, ASVAB examples and traps, putting it all together. All right, here is a classic packaging problem right out of our source material. A range has 380 rounds to pack and they fit 32 rounds per box. Now the test might ask how many boxes are completely filled? How many boxes are needed in total? Or how many rounds go in the partial box? The basic setup here is just straightforward division. 380 divided by 32. So let's solve it step by step. Let's do a quick estimation first. 32 times 10 is 320, right? So 11 seems like a really solid guess. 32 times 11 is 352. Now we just subtract that from our total. 380 minus 352 equals 28. So our mathematical answer is 11 remainder 28. But returning to that remainder trap we talked about earlier, what does this actually mean in the real world?

8:38 Well, it means we have 11 completely full boxes. The 28 left over rounds mean we have a partial box, which means we actually need 12 boxes total to hold everything. See how absolutely crucial that context is? Okay, part six, check yourself. Practice questions from the source. It's time to test out these skills. Here's the first one. Question two, what is 3,005 minus 1,748? Is it A, 4,753, B, 2,743, C, 1,257, or D, 1,267? I actually want you to pause this explainer right now, grab a pencil and try solving this by hand using the methods we just discussed. All right, welcome back. So the correct answer is C, 1,257. If you got that, excellent job. If you didn't quite get there, no worries at all. Let's quickly recap that counting up method to show how it completely avoids those messy borrowing mistakes across all those zeros in 3,005. We just count up from 1,748 to the nearest friendly number, which is 1,800.

9:45 That gives us a jump of 52. Then we count from 1,800 all the way up to 3,005, which is a jump of 1,205. Add those two chunks together, 52 plus 1,205 equals 1,257. No carrying, no borrowing, and absolutely zero careless errors. All right, one last challenge for you. Question four. 150 recruits ride to the range in buses that seat 40. How many buses are needed so everyone has a seat? Is it A, three, B, 3.75, C, 30, or D, four? Go ahead and pause the explainer one last time and solve for the exact number of buses. Let's see how you did on this one. The correct answer is D, four buses. Here's why. When you divide 150 by 40, you get three, because three times 40 is 120. That leaves a remainder of 30. So mathematically, it's three remainder 30. But remember, context is literally everything on these tests. The question specifically asks how many buses are needed so everyone has a seat.

10:49 You obviously can't just leave 30 recruits standing on the curb. Those 30 left over recruits absolutely require a fourth bus. And just like that, you've tackled ASVAB math without a calculator. You've successfully bypassed major traps and you've used mental patterns to speed up your work. You are honestly so much more capable than you probably realize. So as we wrap up today, I wanna leave you with this thought. Now that you've conquered your fear of the no calculator rule, what other math fears are you ready to absolutely crush next? Keep practicing, you've got this, and I'll see you in the next explainer.