💵 Lesson 3.2: Decimals

💡 Decimals Are Just Money, Broken Down

In the last lesson you worked with fractions — parts of a whole written as 12 or 14. A decimal is simply another way to write those same parts, using our familiar place-value columns and a dot called the decimal point.

Here's the friendliest way in: think about a dollar. A dollar is one whole. It splits into 10 dimes and into 100 pennies. That is exactly what decimals do:

• One dime is one tenth of a dollar → we write it $0.10
• One penny is one hundredth of a dollar → we write it $0.01
• So $0.75 means 7 dimes and 5 pennies — seven tenths and five hundredths of a dollar

You already understand decimals in your body — every time you count out change. This lesson just puts the pencil-and-paper name to something you do all the time. The digits to the left of the point are the whole dollars; the digits to the right are the cents (parts of a dollar).

🔢 Place Value After the Point

You already know the columns to the left of the point: ones, tens, hundreds. The columns to the right of the point keep the same pattern — each one is ten times smaller than the one before it:

Let's break the amount $3.47 into its columns, the same way we broke apart whole numbers:

Add the parts back together: 3 + 0.40 + 0.07 = 3.47. So $3.47 is "three dollars and forty-seven cents." The 4 means four tenths (forty cents), and the 7 means seven hundredths (seven cents).

A helpful picture: the decimal point never moves — the digits take their meaning from which side of it they sit on. Left of the point = whole things. Right of the point = pieces of one thing.

🗣️ Reading & Writing Decimals

Reading a money decimal out loud is simple: say the dollars, say "and," then say the cents.

• $0.50 — fifty cents (five tenths — the same as half a dollar)
• $0.05 — five cents (five hundredths — five pennies)
• $2.30 — two dollars and thirty cents
• $12.09 — twelve dollars and nine cents
• $100.00 — one hundred dollars even

When you write money, keep two digits after the point, even when the cents are zero: write $8.00, not $8 on a form; write $5.40, not $5.4. That trailing zero doesn't change the value, but it keeps your columns lined up and your meaning crystal clear.

➕ Adding & Subtracting Decimals

Here is the single most important rule in this whole lesson, and it's an easy one:

Let's total two prices from a shopping trip: a sandwich at $12.45 and a drink at $7.80.

The total is $20.25. Read it back to sanity-check: about twelve-and-a-half plus about eight is around twenty — and 20.25 is right in that neighborhood. Good.

Now the other direction. You pay for that $20.25… but first, how much change from a $20.00 bill if the total were $13.65? Subtracting works the same way — line up the points and borrow when you need to.

The difference is $6.35 — that's your change. Check it by adding back (the trick from Lesson 1.3): 13.65 + 6.35 = 20.00. It matches, so we're confident.

🎯 Rounding Money

Rounding decimals uses the same rule you learned for whole numbers: look at the digit just to the right of where you're rounding. 5 or more rounds up; 4 or less rounds down.

Rounding to the nearest cent (hundredth)

Sometimes a calculation gives you a third digit after the point — say a per-item price of $4.278. Money only goes to the penny, so we round to two places. Look at the third digit (8): it's 5 or more, so round the cents up.

Rounding to the nearest dollar

For a quick estimate — "roughly how much is this cart?" — round to the nearest whole dollar. Look at the tenths digit (the first one after the point).

This is how you glance at a receipt and think "that's about thirteen dollars" before the exact total even registers. Rounding is your fast, close-enough friend at the register.

🔗 Fractions ↔ Decimals

Fractions and decimals are two languages for the same idea: part of a whole. Many everyday amounts you already know as fractions have a simple decimal twin.

graph LR
    A["1/2 (half)"] --> B["0.5"]
    C["1/4 (quarter)"] --> D["0.25"]
    E["3/4 (three quarters)"] --> F["0.75"]
                

Notice 12 = 0.5 = $0.50 — a half-dollar is 50 cents, and 50 cents is five tenths. It all agrees. So when a store says "½ off" a $18.00 item, that's the same as taking off 0.5 of it — $9.00.

You don't need to memorize a big table. Just remember the four everyday ones — ½ = 0.5, ¼ = 0.25, ¾ = 0.75 — and you can figure out the rest by dividing when you need to.

🔊 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.

✍️ Practice & Project

🧭 Tips & Common Mix-Ups

✅ Do's

• Line up the decimal points before you add or subtract — points under points, not just ends under ends.
• Fill in trailing zeros so both numbers have the same digits after the point (5 becomes 5.00).
• Check subtraction by adding back — the change plus the total should equal the cash you paid.
• Keep money to two decimal places and put the point straight down into your answer.

❌ Common Mix-Ups

📓 Learning Journal

📝 Lesson Summary

🎉 What You've Accomplished

You just took the mystery out of that little dot. You can read a price, name every digit after the point, add and subtract amounts with the columns lined up, round for a fast estimate, and swap between simple fractions and decimals. Those are the exact skills behind checking a receipt, making change, and reading a paycheck — real money math you'll use this week. Be proud of that. 🎉

❓ Common Questions at This Stage

🔭 Looking Ahead

In Lesson 3.3 — Percents: Sales, Tax & Tips, we'll use everything you just learned about decimals to handle percents — "out of 100." You'll find a discount on a sale price, add sales tax to a total, and figure a fair tip at a restaurant. Since 25% = ¼ = 0.25, you're already halfway there.

✅ Before the Next Lesson

• Add two prices from a real receipt by hand, points lined up.
• Round three prices you see this week to the nearest dollar in your head.
• Say one amount out loud as dollars and cents (not "point"), and write your journal entry.