➖ Lesson 1.3: Subtraction & Making Change

Addition combines amounts; subtraction tells you the difference — what's left, how much more, or how much change is coming back to you. In this lesson we'll line up the columns, learn to borrow (even across a stubborn zero), check our work by adding back, and pick up one of the most useful everyday skills there is: making change.

📚 What You'll Learn

By the end of this lesson, you will be able to:

Subtract whole numbers by lining up the columns and working right to left
Borrow (regroup) when a top digit is smaller than the one below it — including across a zero
Check a subtraction by adding the answer back
Make change by counting up from the price to the money handed over

⏱️ Estimated Time: 45–60 minutes (take two sittings if you like — steady beats rushed)

🎯 Project: Work out the change for three real purchases as if you were the cashier — count up out loud, then check each one by adding back.

In This Lesson

What Subtraction Really Asks

Subtraction answers three everyday questions that are really the same question in different clothes:

How much is left? You had $50 and spent $18 — what remains?
How much more? You need $200 and have $145 — how much further to go?
What's the difference? One bill is $88, another is $62 — how much bigger is one than the other?

We write subtraction with the minus sign: 50 − 18 = 32. The big number you start with is called the total (or minuend), the amount you take away is the part (subtrahend), and the answer is the difference. You don't need those fancy names — just remember: you always take the smaller amount out of the bigger one, and the columns must line up.

🧠 Mindset

If borrowing tied your stomach in knots back in school, you are in very good company — it's the single step most adults tell me they dreaded. Here's the reframe: borrowing isn't a trick that some people "just get." It's a small, logical move you can learn in an afternoon, and every time it doesn't click yet, that's just a step you haven't practiced yet. You're not "bad at math." You're an adult learning a skill properly, at your own pace. Marathon, not sprint. 🌱

Column Subtraction & Borrowing

Just like addition, we stack the numbers with the bigger number on top, line up ones under ones and tens under tens, and work right to left. When the top digit in a column is big enough, you just subtract. Let's warm up with an easy one, 68 − 25:

68 − 25 43

Ones: 8 − 5 = 3. Tens: 6 − 2 = 4. Answer: 43. No trouble there. The interesting case is when the top digit is smaller than the one below it — then we borrow.

🧮 Worked Example: 52 − 27 (borrowing once)

Stack them and start in the ones column: 2 − 7. You can't take 7 from 2 — so borrow.
Borrow one ten from the tens column. The 5 tens become 4 tens, and that borrowed ten joins the ones: the 2 becomes 12.
Ones now: 12 − 7 = 5.
Tens now: 4 − 2 = 2.
Difference: 25.

52 − 27 25

That's the whole idea of borrowing: when a column comes up short, you break one unit down from the next column to the left — one ten becomes ten ones, one hundred becomes ten tens — so you always have enough to subtract. Why does it matter for real life? This is how you work out how much is left in your wallet, how much a bill dropped from last month, or how many hours remain in a shift.

💡 Order matters in subtraction. With addition, 3 + 5 and 5 + 3 give the same answer. Subtraction is different: 7 − 3 = 4, but 3 − 7 is not the same. Always take the smaller amount out of the bigger one, and keep the bigger number on top.

Borrowing Across a Zero

The one that trips people up is subtracting from a number with a zero in the middle, like 400 − 168. The ones column needs to borrow, but the tens column is a 0 — there's nothing there to borrow from directly. The fix: borrow from the next column that does have something, and let the zero "pass the borrow along."

🧮 Worked Example: 400 − 168 (borrowing across zero)

Ones: 0 − 8 — can't do it. Look left to borrow, but the tens digit is also 0.
So borrow from the hundreds first. The 4 hundreds become 3 hundreds, and the tens column jumps from 0 to 10 tens.
Now borrow one of those ten tens for the ones. Tens drop from 10 to 9, and the ones become 10.
Ones: 10 − 8 = 2.
Tens: 9 − 6 = 3.
Hundreds: 3 − 1 = 2.
Difference: 232.

400 − 168 232

A simple way to picture it: think of 400 as four $100 bills. To pay out, you break one hundred into ten $10 bills, then break one of those tens into ten $1 coins. Now you have coins to hand over. Nothing was lost — you just made change with yourself. That's exactly what "borrowing across a zero" is doing.

💡 A quick shortcut for round numbers

When you subtract from a round number like 100, 200, or 1,000, you can borrow all the way across in one mental sweep: every 0 becomes a 9, and the last one becomes 10. For 1000 − 258, think "9, 9, 10" across the top: 9 − 2 = 7, 9 − 5 = 4, 10 − 8 = 2 → 742. Check: 742 + 258 = 1000. ✅

Checking by Adding Back

Here's the reassuring part: subtraction has a built-in answer key. Because taking away and adding back are opposites, the difference plus the part you subtracted should equal the total you started with. If it does, you got it right. If it doesn't, you've caught a mistake before it costs you anything.

We checked 52 − 27 = 25. Add the answer back to the amount removed:

25 + 27 52

It lands back on 52 — the number we started with — so the subtraction is correct. ✅ Let's check the tricky one too, 400 − 168 = 232:

232 + 168 400

Back to 400. ✅ This "add it back" check takes ten seconds and catches almost every borrowing slip. Make it a habit — especially with money, where a small error can matter.

Making Change: Counting Up

When you pay $20.00 for something that costs $6.40, the change is 20.00 − 6.40 = 13.60. You can do that with column subtraction — but there's an easier, more natural way that good cashiers use, and it needs almost no borrowing. It's called counting up: start at the price and add money in easy jumps until you reach what was handed over. The jumps you added are the change.

🧮 Worked Example: $6.40 item, paid with $20

Start at the price: $6.40.
Jump up to the next round amount — to $6.50. That's 10¢.
Up to $7.00. That's another 50¢.
Up to $10.00. That's $3.00.
Up to $20.00 (what they paid). That's $10.00.
Add the jumps: 0.10 + 0.50 + 3.00 + 10.00 = 13.60. Change = $13.60.

Notice how friendly the steps are — you climb to the next "nice" number (the next dime, the next dollar, the next ten) each time. Counting up in the direction of bigger keeps you out of the messy borrowing that 20.00 − 6.40 would force. It's the same logic behind handing back coins first, then bills, at a register.

🧮 Worked Example: $13.45 item, paid with $20

Start at $13.45.
Up to $13.50 — 5¢.
Up to $14.00 — 50¢.
Up to $20.00 — $6.00.
Add them: 0.05 + 0.50 + 6.00 = 6.55. Change = $6.55.

And you can always double-check with the "add it back" rule: 13.45 + 6.55 = 20.00. ✅ Whether you count up or subtract in columns, the two should agree.

graph LR
    A["Price: $13.45"] -->|"+5¢"| B["$13.50"]
    B -->|"+50¢"| C["$14.00"]
    C -->|"+$6.00"| D["Paid: $20.00"]
    D --> E["Change = $6.55"]

Estimating a Difference

Before you do the exact math, a quick estimate tells you roughly what to expect — so a wildly wrong answer stands out. Round each number first (from Lesson 1.1), then subtract the round numbers in your head.

🧮 Worked Example: About how much is 812 − 289?

Round 812 to 800 (the ones/tens are small, round down).
Round 289 to 300 (89 is close to 100, round up).
Estimate: 800 − 300 = 500. So the answer should be around 500.
Exact answer: 812 − 289 = 523 — nicely close to our estimate, so it passes the sniff test. ✅

Estimating is your safety net at the register too: if a screen says your change is $50 when you expected about $6, you'll know instantly to speak up. A rough number in your head is a small habit that saves real money.

🔊 Hear It & Read Along — Key Sentences

Press 🔊 Listen on a sentence and follow the words with your eyes. Hearing and seeing a sentence at the same time builds reading fluency and confidence. Play each one as many times as you like.

Subtraction means taking away.
The minus sign means subtract.
Ten minus four equals six.
I subtract to find how much is left.
I can count my change at the store.

Practice & Project

🏋️ Exercise 1: Subtract with borrowing

Goal: Line up the columns, borrow where needed, and find the difference.

Work out 634 − 247.

✅ Answer

634 − 247 387

Ones: 4 − 7 can't, so borrow a ten → 14 − 7 = 7 (the 3 tens become 2). Tens: 2 − 4 can't, so borrow a hundred → 12 − 4 = 8 (the 6 hundreds become 5). Hundreds: 5 − 2 = 3. Difference = 387.

Check by adding back: 387 + 247 = 634 ✅

🏋️ Exercise 2: Borrow across a zero

Goal: Handle the zero in the middle.

Work out 500 − 236.

✅ Answer

500 − 236 264

Ones: 0 − 6 can't, and the tens are 0 too. Borrow from the hundreds first (5 → 4), making the tens 10; then borrow one ten for the ones (tens 10 → 9, ones become 10). Ones: 10 − 6 = 4. Tens: 9 − 3 = 6. Hundreds: 4 − 2 = 2. Difference = 264.

Check by adding back: 264 + 236 = 500 ✅

🏋️ Exercise 3: Make change from $20

Goal: Count up from the price to the money paid.

A purchase comes to $7.75 and the customer pays with a $20 bill. What's the change?

✅ Answer

Count up from the price:

$7.75 → $8.00 = 25¢
$8.00 → $10.00 = $2.00
$10.00 → $20.00 = $10.00

Add the jumps: 0.25 + 2.00 + 10.00 = 12.25. Change = $12.25.

Check by adding back: 7.75 + 12.25 = 20.00 ✅

🎯 Your Project: Be the Cashier

The best way to lock in making change is to do it out loud, the way a cashier does. Grab paper and a pen.

(3 min) Write down three purchases — real or made up — each with a price and an amount paid (for example, a $12.60 item paid with a $20).
(6 min) For each one, count up from the price to the amount paid, writing down every jump, and total the jumps to get the change.
(4 min) Check each answer by adding the change back to the price — it should equal what was paid.
(3 min) Redo one of them with column subtraction instead, and confirm you get the same change both ways.
(2 min) Note in your journal which method felt more natural to you, and why.

✅ Project Completion Checklist

☐ I wrote down three purchases with a price and an amount paid
☐ I counted up to find the change for each one
☐ I checked each answer by adding the change back to the price
☐ I redid one using column subtraction and matched the answer
☐ I noted which method I prefer in my journal

👥 Working with a tutor or group?

Play "shop." One person is the customer and names a price and the bill they're paying with; the other is the cashier and counts the change back out loud, coins-and-bills first ("...that's twenty-five cents makes eight dollars, and ten makes twenty"). Then swap. If you have real or play coins, hand them across — the physical motion makes counting up click. Check every transaction by adding back, together.

🎯 Quick Quiz

Question 1: What is 803 − 47?

Question 2: An item costs $12.60 and the customer pays with a $20 bill. How much change?

Tips & Common Mix-Ups

✅ Do's

Put the bigger number on top and line up ones under ones, tens under tens.
Work right to left, one column at a time, and borrow the moment a column comes up short.
Check by adding back — the difference plus the part removed should equal the total.
For change, count up to the next dime, dollar, and ten; add the jumps.

❌ Common Mix-Ups

⚠️ Watch Out

Flipping a column: when the top digit is smaller, don't just subtract "the small from the big" (like doing 7 − 2 in the ones of 52 − 27). You must borrow — the answer is 5, not 5-by-accident.
Forgetting to drop the borrowed column: after you borrow a ten, that tens digit is now one less. Cross it out and write the smaller number so you don't forget.
The zero freeze: a 0 you need to borrow from isn't a dead end — borrow from the next non-zero column to its left, and the 0 becomes a 9 as the borrow passes through.
Change cents slip: from an amount like $12.60, it takes 40¢ to reach the next dollar, not 60¢. Count up to the next whole dollar carefully.

✅ Affirmation

Every borrow you practice makes the next one easier. If one didn't work out, you didn't fail — you found the exact spot to slow down next time. That's how skill is built, one honest attempt at a time.

📓 Learning Journal

Keep your math journal going — a notebook or a note on your phone. After this lesson, take five minutes to write down:

What you learned — a rule, a method, an idea
What clicked for you
What's still confusing, so you know what to revisit
Where you'll use it in real life this week
How you feel about your progress

✍️ This lesson's prompt: Think of a moment when getting change back or figuring out "how much is left" mattered to you — at a store, paying a bill, or budgeting to payday. How would counting up or a quick add-back check help you feel more sure of yourself next time? Write a few sentences. Naming a real use gives the skill somewhere to live.

📝 Lesson Summary

🎓 Key Takeaways

Subtraction finds the difference — what's left, how much more, or how far apart two amounts are.
Stack numbers (bigger on top), line up columns, and work right to left; borrow when a top digit is too small.
To borrow across a zero, borrow from the next non-zero column; the zero becomes a 9 as the borrow passes through.
Check by adding back: difference + part removed = the original total.
Make change by counting up from the price to the money paid — the jumps add up to the change.

🎉 What You've Accomplished

You can now subtract whole numbers with confidence — including the borrowing-across-zero case that stops a lot of people — and you can prove your answer is right by adding it back. On top of that, you've learned a genuinely practical register skill: counting up to make change. These are the everyday tools behind checking a bill, tracking what's left in a budget, and never getting shortchanged. Real progress. 🎉

❓ Common Questions at This Stage

Do I have to borrow, or can I subtract another way?

Column subtraction with borrowing is the reliable written method, and it's worth owning. But for money especially, counting up often feels easier and avoids borrowing altogether. Use whichever fits the moment — and let them check each other.

What if I accidentally put the smaller number on top?

For the everyday "how much is left / how much change" questions, always put the bigger amount on top. If you ever genuinely need the smaller minus the bigger, the answer goes into negative numbers — a topic for later. For now, bigger on top.

Is it cheating to check with a calculator?

Not at all — a calculator is a fine tool. But do the thinking first, then use the calculator to confirm. The goal is that you understand the steps, so you'd catch it if the calculator (or a cashier's screen) were wrong.

The borrowing still feels shaky. Should I move on?

Practice a few more with the add-back check until it feels routine — but you don't have to be perfect to continue. Skills firm up as you revisit them across later lessons. Short, regular practice wins.

🔭 Looking Ahead

In Lesson 2.1, we move to multiplication — a fast way to add the same amount many times. It's how you find the cost of several identical items, your pay from hours × wage, or the area of a room. Your solid grip on place value and columns will make it feel familiar.

✅ Before the Next Lesson

Do two subtraction problems that require borrowing, and check each by adding back.
The next time you buy something, work out the change in your head by counting up before you get it.
Write your journal entry for this lesson.

🌟 Encouragement for the Journey

Borrowing across a zero is one of the classic "I could never do that" moments — and you just did it. That's exactly the kind of small win that stacks up into real confidence with money and numbers. Keep showing up for yourself; you're building something that lasts. See you in Lesson 2.1! 👋