Add or subtract decimals accurately. - ISEE Lower Level Mathematics Achievement

SECTION 1

Why Do We Use Decimals?

Have you ever bought something at a store? Prices like $3.49 or $12.75 use decimals. Decimals are numbers that show parts of a whole, like cents in a dollar. People needed a way to write amounts that were not whole numbers.

Long ago, people used fractions for everything. Over time, mathematicians invented decimals to make math easier. Let's see how decimals came to be!

1400s • Early Decimal Ideas

Mathematicians in the Middle East began using a system like decimals to handle tricky fractions.

1585 • Simon Stevin's Book

A Dutch math teacher named Simon Stevin wrote a famous book explaining how decimals work. He showed people that decimals make adding and subtracting much simpler.

1600s • The Decimal Point Appears

The little dot we use today (the decimal point) became common. It separates the whole-number part from the fractional part.

Today • Decimals Are Everywhere

We use decimals for money, measurements, sports stats, and so much more. The ISEE test loves to ask about adding and subtracting decimals!

So here's the big question: how do you add or subtract decimals without making mistakes? The secret is all about lining up the decimal points. Let's learn exactly how!

SECTION 2

Core Principles of Decimal Addition & Subtraction

Adding and subtracting decimals is a lot like adding and subtracting whole numbers. You just need to follow a few important rules. Let's look at the big ideas!

Line Up the Decimal Points

Always write numbers so the decimal points are directly above and below each other. This keeps ones with ones, tenths with tenths, and so on.

Use Placeholder Zeros

If one number has more decimal places, add zeros to the shorter number. For example, write 3.5 as 3.50 so it matches 2.75.

Add or Subtract Column by Column

Start from the right side. Work one column at a time, just like you do with whole numbers. Don't forget to regroup (carry or borrow) when needed!

Bring Down the Decimal Point

Drop the decimal point straight down into your answer. It should be right below the decimal points you lined up.

SECTION 3

See It: Lining Up Decimal Points

Let's look at a picture that shows why lining up decimals matters. We'll add 4.53 + 2.7. Watch how the columns stay neat!

The left side shows what happens when you don't line up the decimals — the columns are a mess. The right side shows the correct way. Notice the cyan line showing the decimal points perfectly stacked.

See how we wrote 2.7 as 2.70? That placeholder zero doesn't change the value. It just helps us keep our columns straight. This trick is super helpful on the ISEE!

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ISEE Test Tip

SECTION 4

The Steps for Adding and Subtracting Decimals

Here are the exact steps you should follow every time. Whether you're adding or subtracting, the process is the same!

STEP 1 — LINE UP

Write the numbers vertically with decimal points aligned.

Stack the numbers so each decimal point is in the same column. Put ones above ones, tenths above tenths, and hundredths above hundredths.

STEP 2 — ADD PLACEHOLDER ZEROS

Fill empty spots with zeros so both numbers have the same length.

Example: If you have 6.4 and 3.85, change 6.4 to 6.40. The zero after the 4 does not change the value, but it keeps your columns even.

STEP 3 — COMPUTE COLUMN BY COLUMN

Start from the rightmost column. Add or subtract. Regroup if needed.

Work just like with whole numbers. If a column adds up to 10 or more, carry the 1. If you can't subtract, borrow from the column to the left.

STEP 4 — DROP THE DECIMAL POINT

Bring the decimal point straight down into your answer.

The decimal point in your answer should be directly below the decimal points above it. Check this before you choose your answer on the test!

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Key takeaway

REMEMBER THIS RHYME

SECTION 5

Watch Out! Common Decimal Mistakes

The ISEE test makers know the mistakes students make. They put wrong answers (called trap answers) that match those mistakes. If you know what to avoid, you won't fall into their traps!

The top row shows three common mistakes in red, yellow, and orange. The bottom row shows the correct solutions in green. Always add placeholder zeros and bring down the decimal point!

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ISEE Strategy

SECTION 6

Worked Example: Step by Step

Let's solve a subtraction problem together. Follow along with each step. You've got this!

Problem: What is 12.5 − 7.83?

Worked example

1
Step 1 — Write Vertically & Line Up Decimals

Write 12.5 on top and 7.83 below it. Make sure the decimal points are in a straight line.

12.5 − 7.83
2
Step 2 — Add a Placeholder Zero

The top number (12.5) has one decimal place. The bottom number (7.83) has two. Add a zero to 12.5 so it becomes 12.50. Now both numbers have two decimal places.

12.50 − 7.83
3
Step 3 — Subtract the Hundredths Column

Hundredths: 0 − 3. You can't take 3 from 0! Borrow 1 from the tenths place. Now it's 10 − 3 = 7.

Hundredths digit: 7
4
Step 4 — Subtract the Tenths Column

The tenths place was 5, but we borrowed 1, so now it's 4. Then 4 − 8. We can't do that! Borrow 1 from the ones place. Now it's 14 − 8 = 6.

Tenths digit: 6
5
Step 5 — Subtract the Ones and Tens Columns

Ones: The 2 became 1 after borrowing. 1 − 7 can't be done, so borrow from the tens place. 11 − 7 = 4. Tens: The 1 became 0, and there's nothing to subtract. So the tens digit is 0 (we don't write a leading zero).

Ones digit: 4, Tens digit: (nothing)
6
Step 6 — Bring Down the Decimal Point

Place the decimal point in the answer directly below the other decimal points.

12.50 − 7.83 = 4.67

Let's check: 4.67 + 7.83 should equal 12.50. And it does! Checking your work is a great habit for the ISEE.

SECTION 7

Tips and Tricks for the ISEE

Let's compare different strategies and see what works best. These tips will help you save time and avoid errors on test day!

Strategy	When to Use It	Why It Helps
Estimate first	Every decimal problem	Round to whole numbers. If 3.82 + 5.19 ≈ 4 + 5 = 9, your answer should be near 9.
Add placeholder zeros	When numbers have different decimal lengths	Keeps columns even. Prevents the most common mistake!
Use scratch paper	Always (no calculator on ISEE!)	Writing vertically is much easier than doing it in your head.
Eliminate wrong answers	When you're unsure	If your estimate is 9, cross out choices like 0.9 or 90. Then pick from what's left.

Helpful strategies for decimal problems on the ISEE

SECTION 8

Decimals Now and Later

Adding and subtracting decimals is a skill you'll use for years. Here's how what you learn now connects to what comes next!

What You Learn Now	What You'll Learn Later
Adding decimals like 3.25 + 1.40	Multiplying decimals like 3.25 × 1.4
Subtracting decimals with borrowing	Working with negative decimals
Lining up decimal points	Counting decimal places when multiplying
Estimating decimal sums	Using estimation to check algebra

How decimal skills build over time

Every math skill you build is like a brick in a wall. Right now, you're building a strong foundation with decimal addition and subtraction. Later, multiplication and division of decimals will feel easy because you already understand how decimals work. Great job sticking with it!

SECTION 9

Practice Problems

Now it's your turn! Try these five problems. They start easy and get a little harder. Remember: line up the decimal points, use placeholder zeros, and bring the point straight down. You've got this!

Question 1: Practice Problem 1

What is 3.2 + 1.5?

A. 4.5
B. 4.7
C. 4.8
D. 5.7

Show answer (B)

Correct Answer: B.Line up the decimal points and add each column:Tenths place: 2 + 5 = 7, so the tenths digit is 7.Ones place: 3 + 1 = 4, so the ones digit is 4.The answer is 4.7.Why the other choices are wrong:(A) 4.5: A student might get this by adding only 3 + 1.5 = 4.5, forgetting to include the 0.2 from 3.2.(C) 4.8: A student might mistakenly add the tenths as 2 + 5 = 8 (perhaps confusing 3 + 5 = 8 and placing it in the tenths column) instead of the correct sum of 7.(D) 5.7: A student might incorrectly add the ones as 3 + 2 = 5 by accidentally adding the tenths digit of 3.2 to the ones digit of 3, then adding 1 to get 5, while still getting the tenths correct.

Question 2: Practice Problem 2

What is 9.4 − 4.1?

A. 4.3
B. 5.3
C. 5.5
D. 6.3

Show answer (B)

Correct Answer: B.Line up the decimal points and subtract each column:Tenths place: 4 − 1 = 3, so the tenths digit is 3.Ones place: 9 − 4 = 5, so the ones digit is 5.The answer is 5.3.Why the other choices are wrong:(A) 4.3: A student might miscalculate the ones place as 9 − 4 = 4 by making an off-by-one counting error, arriving at 4.3 instead of 5.3.(C) 5.5: A student might subtract the tenths incorrectly as 4 − 1 = 5 (an off-by-two counting error), getting 5.5 instead of 5.3.(D) 6.3: A student might miscalculate 9 − 4 = 6 (perhaps counting 9 − 3 = 6 by confusing the tenths digit 3 for the subtrahend in the ones column), getting 6.3.

Question 3: Practice Problem 3

What is 2.45 + 3.38?

A. 5.73
B. 5.83
C. 6.83
D. 6.93

Show answer (B)

Correct Answer: B.Line up the decimal points and add from right to left:Hundredths place: 5 + 8 = 13. Write 3 and carry 1.Tenths place: 4 + 3 + 1 (carried) = 8.Ones place: 2 + 3 = 5.The answer is 5.83.Why the other choices are wrong:(A) 5.73: A student might forget to carry the 1 from the hundredths column, computing the tenths as 4 + 3 = 7 instead of 4 + 3 + 1 = 8, resulting in 5.73.(C) 6.83: A student might incorrectly add the ones as 2 + 3 = 6, perhaps accidentally carrying an extra 1 into the ones column, getting 6.83.(D) 6.93: A student might make both errors—carrying an extra 1 into the ones place (getting 6 instead of 5) and miscounting the tenths as 4 + 3 + 1 + 1 = 9—resulting in 6.93.

Question 4: Practice Problem 4

What is 8.32 − 1.75?

A. 6.47
B. 6.57
C. 6.67
D. 7.57

Show answer (B)

Correct Answer: B.Line up the decimal points and subtract from right to left, borrowing when needed:Hundredths place: 2 is less than 5, so borrow 1 from the tenths place. The tenths digit 3 becomes 2, and the hundredths digit 2 becomes 12. Now compute 12 − 5 = 7.Tenths place: The tenths digit is now 2 (after lending to the hundredths). Since 2 is less than 7, borrow 1 from the ones place. The ones digit 8 becomes 7, and the tenths digit 2 becomes 12. Now compute 12 − 7 = 5.Ones place: The ones digit is now 7 (after lending to the tenths). Compute 7 − 1 = 6.The answer is 6.57.Why the other choices are wrong:(A) 6.47: A student might make an error in the tenths place by computing 12 − 7 = 4 instead of the correct 12 − 7 = 5 (an off-by-one subtraction error), resulting in 6.47.(C) 6.67: A student might forget to reduce the tenths digit after borrowing for the hundredths column, computing 3 − 7 as needing a borrow that gives 13 − 7 = 6 instead of 12 − 7 = 5, resulting in 6.67.(D) 7.57: A student might forget to reduce the ones digit after borrowing for the tenths column, using 8 − 1 = 7 instead of the correct 7 − 1 = 6, resulting in 7.57.

Question 5: Practice Problem 5

Maria had $12.50. She spent $4.85 on a book. How much money does she have left?

A. $7.35
B. $7.65
C. $8.35
D. $8.65

Show answer (B)

Correct Answer: B.Subtract $4.85 from $12.50. Line up the decimal points and subtract from right to left:Hundredths place: 0 is less than 5, so borrow 1 from the tenths place. The tenths digit 5 becomes 4, and the hundredths digit 0 becomes 10. Now compute 10 − 5 = 5.Tenths place: The tenths digit is now 4. Since 4 is less than 8, borrow 1 from the ones place. The ones digit 2 becomes 1, and the tenths digit 4 becomes 14. Now compute 14 − 8 = 6.Ones place: The ones digit is now 1. Since 1 is less than 4, borrow 1 from the tens place. The tens digit 1 becomes 0, and the ones digit 1 becomes 11. Now compute 11 − 4 = 7.Tens place: 0 − 0 = 0.The answer is $7.65.Why the other choices are wrong:(A) $7.35: A student might make an error in the tenths place by computing 14 − 8 = 3 instead of the correct 14 − 8 = 6 (perhaps forgetting the borrow and computing 5 − 8 by reversing to 8 − 5 = 3), resulting in $7.35.(C) $8.35: A student might miscalculate the ones place as 11 − 4 = 8 (an off-by-one error) and also reverse the tenths subtraction to get 8 − 5 = 3, resulting in $8.35.(D) $8.65: A student might incorrectly compute the ones place as 12 − 4 = 8 by forgetting that the ones digit had already been reduced to 1 after borrowing for the tenths column, using 2 instead of 1 and then borrowing from the tens to get 12 − 4 = 8, resulting in $8.65.