Find Each Product By Factoring The Tens: Complete Guide

Find Each Product by Factoring the Tens

Ever stare at a multiplication problem and feel your brain melt before you even get to the answer? You’re not alone. Most of us learned the standard algorithm in elementary school, but there’s a shortcut that feels like a magic trick once you get the hang of it: factoring the tens.

It’s the kind of trick that lets you turn a “9 × 8” into “(10 – 1) × 8” in a split second, then pull the answer out of thin air. If you’ve never tried it, or you think it’s only for math‑whizzes, keep reading. The short version is: you already know the pieces, you just need to see them differently.

And yeah — that's actually more nuanced than it sounds And that's really what it comes down to..

What Is Factoring the Tens?

When we talk about “factoring the tens” we’re not diving into high‑school algebra. It’s a simple mental‑math strategy that rewrites a multiplication problem so that one of the numbers is expressed as a multiple of ten plus or minus a small adjustment.

As an example, instead of multiplying 7 × 6 directly, you could think of 7 as 10 – 3 and then do:

(10 – 3) × 6 = 60 – 18 = 42

The idea is the same for any two‑digit number that sits near a round ten—​12, 18, 27, 34, 59, you name it. You “factor” the ten out of one factor, multiply, then adjust.

In practice it’s just a way of breaking a product into a big, easy part (the tens) and a small, easy part (the ones). The mental load drops dramatically because you’re mostly adding or subtracting, not memorizing a whole table.

Why It Matters / Why People Care

First, speed. Teachers love it because it lets kids solve 8 × 9 in under five seconds. For adults, it’s the difference between fumbling with a calculator and answering a client’s quick‑fire question in a meeting.

Second, confidence. When you can see the structure behind a problem, you stop fearing it. You realize multiplication isn’t a mysterious wall of numbers; it’s a set of patterns you can manipulate.

Third, it builds a foundation for more advanced tricks—​like mental division, estimating large products, or even mental algebra. If you can factor the tens now, you’ll find it easier to handle “(100 + 25) × 48” later on.

And let’s be real: anyone who can pull off this trick feels a little smarter. That boost in self‑esteem is worth the extra few minutes of practice.

How It Works (or How to Do It)

Below is the step‑by‑step process you can apply to any multiplication where at least one factor is within ten of a round ten.

1. Identify the “near‑ten” number

Look at the two numbers you’re multiplying. If one of them is 6‑12 away from a multiple of ten, you’ve got a candidate That's the part that actually makes a difference. Practical, not theoretical..

• 9 × 7 → 9 is 1 away from 10
• 14 × 6 → 14 is 4 away from 10 (but we’ll use 20 later)
• 27 × 5 → 27 is 3 away from 30 (a multiple of ten)

2. Rewrite that number as (10 ± k) or (20 ± k) etc.

Pick the nearest round ten (or twenty, thirty—​anything ending in 0) Simple, but easy to overlook..

• 9 → (10 – 1)
• 14 → (10 + 4) or (20 – 6) – choose whichever makes the next step easier
• 27 → (30 – 3)

3. Multiply the round ten by the other factor

This is the “big part.”

• (10 – 1) × 7 → 10 × 7 = 70
• (10 + 4) × 6 → 10 × 6 = 60
• (30 – 3) × 5 → 30 × 5 = 150

4. Adjust using the small “k”

Now add or subtract the product of k and the other factor.

• Subtract: 70 – (1 × 7) = 70 – 7 = 63
• Add: 60 + (4 × 6) = 60 + 24 = 84
• Subtract: 150 – (3 × 5) = 150 – 15 = 135

5. Put it together – that’s your answer

You’ve just found the product without ever writing a full multiplication table.

Quick reference table

6. When both numbers are near a ten

Sometimes both factors sit close to a round ten, like 12 × 13. You can pick either one, but you can also combine both adjustments:

12 = 10 + 2
13 = 10 + 3

(10 + 2) × (10 + 3) = 100 + 30 + 20 + 6 = 156

Notice we used the distributive property: a + b times c + d expands to ac + ad + bc + bd. The mental math is still easy because each piece is a round ten multiplied by a small number Small thing, real impact..

Common Mistakes / What Most People Get Wrong

1. Choosing the wrong round ten – If you pick 10 for 19 × 6, you’ll end up with (10 + 9) × 6 = 60 + 54 = 114, which is correct but slower than using 20: (20 – 1) × 6 = 120 – 6 = 114. The key is to minimize the size of k.

2. Forgetting the sign – It’s easy to add when you should subtract, especially with numbers like 8 × 9. (10 – 2) × 9 = 90 – 18 = 72, not 90 + 18.

3. Mixing up the other factor – The adjustment always uses the other factor, not the one you rewrote. A common slip: (10 – 1) × 7 → “subtract 10” instead of “subtract 1 × 7”.

4. Applying it to numbers far from a ten – Trying to force the method on 23 × 47 wastes time. In those cases, other strategies (like splitting into 20 + 3 and 40 + 7) work better The details matter here..

5. Skipping the mental check – After you get a result, a quick estimate (round both numbers, multiply, see if you’re in the ballpark) catches most slip‑ups.

Practical Tips / What Actually Works

• Practice with the 9‑times table first. Everything from 9 × 2 up to 9 × 12 fits the (10 – 1) pattern. You’ll build speed instantly.

• Keep a “nearest ten” cheat sheet in your head. 5 → 10, 12 → 10, 18 → 20, 27 → 30, 34 → 30, 46 → 50, etc. The more you internalize these, the less you’ll have to think about it.

• Use the “double‑and‑half” trick when the other factor is even. Example: 14 × 6 → think of 6 as 2 × 3, so (10 + 4) × 6 = (10 + 4) × 2 × 3 = (20 + 8) × 3 = 28 × 3 = 84. It’s a tiny shortcut that sometimes feels smoother Easy to understand, harder to ignore..

• Write down the steps once, then erase. If you’re nervous, jot “(10‑k)×n = 10n – kn” on a scrap piece of paper, fill in the numbers, then cross it out. The act of writing reinforces the pattern.

• Turn it into a game. Challenge a friend: “What’s 7 × 8? Use factoring the tens.” The fastest correct answer wins a coffee. Competition cements the habit.

• Apply it to real‑world scenarios. Need to calculate 9 × 12 hours for a project estimate? Do (10 – 1) × 12 = 120 – 12 = 108 hours. You’ll impress the boss and keep the spreadsheet tidy.

FAQ

Q: Can I use this method for three‑digit numbers?
A: Absolutely, but you’ll usually factor out a hundred instead of a ten. For 108 × 7, think (100 + 8) × 7 = 700 + 56 = 756.

Q: What if both numbers are far from a ten, like 23 × 47?
A: Break each into a round ten plus a remainder: (20 + 3) × (40 + 7). Then use the distributive property: 20×40 + 20×7 + 3×40 + 3×7 = 800 + 140 + 120 + 21 = 1,081 Worth keeping that in mind..

Q: Is this the same as the “difference of squares” trick?
A: Not exactly. The difference of squares works for pairs like (a + b)(a – b). Factoring the tens is about rewriting one factor as a round ten plus/minus a small number Most people skip this — try not to..

Q: How fast can I get with practice?
A: Most people can answer a 9‑times‑table problem in under two seconds after a few minutes of rehearsal. For mixed numbers, 5–7 seconds is realistic It's one of those things that adds up..

Q: Does this replace the need to learn multiplication tables?
A: It complements them. Knowing the tables speeds up the “big part” (10 × n, 20 × n, etc.). The trick is a shortcut, not a replacement for basic facts.

So there you have it. Factoring the tens isn’t a secret formula locked away in a math textbook; it’s a mindset shift. Spot the nearest round ten, split the problem, adjust, and you’ve got the answer before you even finish the sentence.

Give it a try next time you’re at the grocery checkout, balancing a budget, or just trying to impress a friend with quick mental math. So you’ll find the product, and you’ll find a little more confidence in the process, too. Happy calculating!

Common Pitfalls and How to Avoid Them

1. Forgetting the Adjustment
   It’s easy to stop at the “10 × n” part and forget to subtract or add the small tweak. A quick mental checklist works: “Did I add the extra 4? Did I subtract the 1?” A brief pause before you speak the final number can catch this Surprisingly effective..

2. Mis‑reading the Remainder
   When you write a number as (10 – k) or (10 + k), double‑check that k is the correct difference. For 17, 10 + 7 is correct; for 23, it’s 20 + 3, not 30 – 7. A quick mental “is the remainder smaller than 10?” test helps Most people skip this — try not to..

3. Over‑Complicating with Extra Steps
   If the numbers are both near a multiple of 10, you might be tempted to break both down. Sometimes, keeping only one factor in the (10 ± k) form is simpler. For 13 × 37, you could do (10 + 3) × 37 or 13 × (40 – 3), whichever feels smoother.

4. Not Practicing the Reverse Trick
   The “reverse” approach (breaking the larger factor) can be faster for numbers like 9 × 27. Practice both ways so you can pick the fastest path on the fly Simple as that..

Quick‑Reference Cheat Sheet

Keep this sheet in a pocket calculator app or a sticky note on your desk—quick visual cues can shave milliseconds off your calculation time Not complicated — just consistent..

Beyond the Tens: Expanding the Technique

While the “factor the tens” trick shines for single‑digit and low‑double‑digit products, you can extend the same idea to larger numbers:

• Hundreds – Think of 345 × 7 as (300 + 45) × 7 = 2100 + 315 = 2415.
• Multiples of 5 – When one factor ends in 5, split it as (10 – 5) or (10 + 5) to keep the adjustment small.
• Multiplying by 9 – 9 × n = 10 × n – n. This is a special case of the “10 – 1” split and works for any n.

The key is always the same: find the nearest round base, decompose, compute the simple part, then correct with a tiny adjustment And that's really what it comes down to..

Putting It All Together: A Mini‑Case Study

Scenario: You’re a freelance designer estimating the time for a client. The project requires 9 hours of design work for each of 12 pages.

1. Identify the base: 9 × 12.
2. Choose the split: 10 – 1 for the 9.
3. Compute: 10 × 12 = 120.
4. Adjust: – 1 × 12 = –12.
5. Result: 108 hours total.

You’ve just turned a potentially tedious spreadsheet entry into a one‑minute mental calculation—perfect for quick client emails or on‑the‑spot budgeting.

Final Thoughts

Factoring the tens is more than a shortcut; it’s a mindset that turns multiplication from rote memorization into a flexible, problem‑solving skill. By:

• Rewriting numbers around familiar anchors (tens, hundreds, etc.),
• Separating the easy part from the small tweak, and
• Practicing until the adjustment becomes automatic,

you’ll find that numbers no longer feel like obstacles but rather puzzles waiting for a quick, elegant solution.

So next time you see a multiplication problem, pause for a moment, locate the nearest round ten, split, adjust, and speak the answer with confidence. Your mental math will not only get faster—it’ll get sharper. Happy calculating!

Advanced Applications: Real-World Scenarios

The true test of any mental math technique is how well it performs under pressure. Here are some everyday situations where factoring the tens becomes your secret weapon:

Shopping discounts – A 20% off sign on a $45 item? That's 45 × 0.8. Split it as (8 × 45) ÷ 10. First, 8 × 45 = (8 × 40) + (8 × 5) = 320 + 40 = 360. Divide by 10 → $36. You just calculated the sale price in seconds.

Tipping at restaurants – 15% tip on a $68 bill? Multiply by 10 (=$6.80), add half of that (=$3.40), and you have $10.20. The "10 + half" trick is just another flavor of the base-10 split.

Time calculations – Need to know how many minutes are in 7 × 45-minute intervals? 7 × 45 = (10 – 3) × 45 = 450 – 135 = 315 minutes. Convert to hours: 5 hours and 15 minutes Still holds up..

Building Your Practice Routine

Like any skill, fluency comes from deliberate practice. Try incorporating these micro-exercises into your daily routine:

• Morning coffee: Estimate totals while shopping—round prices to the nearest ten, then adjust.
• Commute challenges: Multiply license plate numbers or street addresses as you walk or drive.
• Game nights: Turn board games into mental math workouts by calculating scores without calculators.

Start with numbers you encounter naturally. The more you use the technique, the faster your brain will recognize patterns and execute the split-adjust sequence automatically.

Common Pitfalls to Avoid

Even the best tricks have their traps. Watch out for these:

1. Over‑splitting – If a number is already close to a round base (like 98 × 7), don't force a 10-split on the 98. Instead, treat 98 as (100 – 2) and multiply by 7: 700 – 14 = 686. Choose the path of least resistance Worth keeping that in mind. Nothing fancy..

2. Sign errors – When splitting as (10 – n), remember the adjustment is subtractive. A quick mental check: "10 times is too much, so I need to take some away."

3. Ignoring the obvious – Sometimes the simplest method wins. Multiplying 6 × 6 doesn't need factoring the tens; you just know it's 36. Save the technique for where it actually saves effort It's one of those things that adds up..

The Bigger Picture

Mental math isn't about showing off or replacing calculators—it's about building a deeper relationship with numbers. When you factor the tens, you're not just computing; you're decomposing, analyzing, and reconstructing. These are the same cognitive muscles you use for budgeting, planning, and problem-solving in countless other areas of life Simple, but easy to overlook. Practical, not theoretical..

Each time you choose a round base, split, and adjust, you're reinforcing a growth mindset: there's always a way to make the hard thing easier. That confidence spills over beyond math Simple, but easy to overlook..

Your Next Step

Pick one multiplication problem from your day—perhaps calculating ingredients for a recipe, splitting a bill, or estimating travel time. Apply the factoring the tens method. Notice how it feels. Then try it again tomorrow, and the day after.

Within a week, you'll catch yourself automatically reaching for the nearest ten without thinking. That's when you know the technique has become part of your mental toolkit.

Conclusion

Mastering the art of factoring the tens transforms multiplication from a chore into a quick, satisfying puzzle. By anchoring numbers to familiar bases, separating the easy from the complex, and applying small corrections, you access a level of mental agility that serves you in shopping, work, and everyday life.

The beauty of this method lies in its simplicity: anyone can learn it, and with practice, anyone can make it their own. So the next time numbers appear, remember—you hold the key to turning multiplication into something effortless, elegant, and even enjoyable. Start small, stay consistent, and watch as your mental math confidence grows one split at a time.