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J.3 Multiply 2-digit numbers by 2-digit numbers: word problems

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Multiply 2-digit numbers by 2-digit numbers

When we multiply a two-digit number by another two-digit number, we are finding the total of equal groups. For example, 14 × 23 means "14 groups of 23" or "23 groups of 14." Because the numbers are larger, we break the problem into smaller parts and then add those parts together. This process is sometimes called multi-digit multiplication.

Real-world example:

A school has 24 classrooms. Each classroom has 32 desks. How many desks are there in all?
We need to solve 24 × 32. The answer will be the total number of desks.

Note

In fourth grade, you will learn several different ways to solve these problems. All the strategies are correct. You can choose the one that makes the most sense to you.

Strategy 1: Break apart by place value (partial products)

This strategy uses what you know about expanded form. You break each two-digit number into tens and ones. Then you multiply each part and add the results. It is also called the "partial products method."

Example: 21 × 34

Step 1: Write each number in expanded form: 21 = 20 + 1, and 34 = 30 + 4.
Step 2: Multiply each part of 21 by each part of 34:
20 × 30 = 600
20 × 4 = 80
1 × 30 = 30
1 × 4 = 4
Step 3: Add all the partial products: 600 + 80 = 680, 680 + 30 = 710, 710 + 4 = 714.
So, 21 × 34 = 714.

Stacked method (partial products):

× 34

(21 × 4) 84

(21 × 30) 630

714

Note

When you write the partial products in a stack, make sure to line up the digits correctly. The 630 is actually 63 tens, so you write it shifted one place to the left.

Strategy 2: The standard algorithm (traditional method)

The standard algorithm is a shortcut for the partial products method. You multiply the ones first, then the tens. You must remember to "carry" when a product is ten or more.

Example: 45 × 27

Step 1: Multiply the ones: 5 × 7 = 35. Write 5 in the ones place, carry the 3 (which means 3 tens).
Step 2: Multiply the top tens digit by the bottom ones digit: 4 (tens) × 7 = 28 tens. Add the carried 3 tens: 28 + 3 = 31 tens. Write 31 in front of the 5, so you have 315. This is the first partial product (45 × 7).
Step 3: Now multiply by the tens digit of the bottom number (2). Because this 2 really means 20, we put a 0 in the ones place as a placeholder. Then multiply 45 × 2 = 90, and write it in front of the zero: 900. This is the second partial product (45 × 20).
Step 4: Add the partial products: 315 + 900 = 1,215.

Stacked standard algorithm:

× 27

(45 × 7) 315

(45 × 20) 900

1,215

Note

Some students forget the placeholder zero. Always remember: when you multiply by the tens digit, you are really multiplying by a multiple of ten, so your answer will end in a zero.

Strategy 3: Area model (box method)

The area model uses a rectangle split into smaller rectangles. The length and width of the big rectangle are the two numbers you are multiplying. The area of each small rectangle is a partial product. The total area is the sum of the partial products.

Example: 52 × 38

Step 1: Draw a rectangle. Divide it into two rows and two columns (because we have two parts for each number).
Step 2: Label the top with 50 and 2 (the expanded form of 52). Label the left side with 30 and 8 (the expanded form of 38).
Step 3: Find the area of each small rectangle:
- Top-left: 50 × 30 = 1,500
- Top-right: 2 × 30 = 60
- Bottom-left: 50 × 8 = 400
- Bottom-right: 2 × 8 = 16
Step 4: Add: 1,500 + 60 = 1,560; 1,560 + 400 = 1,960; 1,960 + 16 = 1,976.
So, 52 × 38 = 1,976.

Note

The area model is helpful because it shows why multiplication works: you are really finding the area of a rectangle. It also helps you keep track of all the partial products.

Estimating to check your answer

Before you multiply, you can round the numbers to the nearest ten to make an estimate. An estimate tells you about how big the answer should be. It helps you check if your final answer is reasonable.

Example: 61 × 48

Round 61 to 60. Round 48 to 50.
60 × 50 = 3,000. So the actual answer should be close to 3,000.
Now solve exactly: 61 × 48 = 2,928. (Check with a strategy!)
2,928 is close to 3,000, so our answer is reasonable.

Note

If your answer is very different from your estimate, you may have made a mistake. Go back and check your work.

Strategy 4: Lattice multiplication

Lattice multiplication is a visual method that uses a grid. Each digit is multiplied separately, and the products are recorded in the grid. Then you add along the diagonals.

Example: 37 × 42

Step 1: Draw a 2-by-2 grid. Draw diagonal lines through each square from top right to bottom left.
Step 2: Write 3 and 7 above the grid. Write 4 and 2 along the right side of the grid.
Step 3: Multiply each digit on top with each digit on the side. Write the tens digit above the diagonal and the ones digit below the diagonal.
- 3 × 4 = 12 → write 1 above diagonal, 2 below in top-left cell.
- 7 × 4 = 28 → write 2 above, 8 below in top-right cell.
- 3 × 2 = 6 → write 0 above (since 6 is just 6, we write 0 in the tens place), 6 below in bottom-left cell.
- 7 × 2 = 14 → write 1 above, 4 below in bottom-right cell.
Step 4: Add along the diagonals from bottom right to top left. The first diagonal (ones): 4. The second diagonal (tens): 8 + 6 + 1 = 15, write 5, carry 1 to next diagonal. The third diagonal (hundreds): 2 + 2 + 0 + the carried 1 = 5. The last diagonal (thousands): 1.
So the digits from left to right are 1, 5, 5, 4 → 1,554.
Check: 37 × 42 = 1,554.

Note

Lattice multiplication is a great way to keep numbers organized. It works for larger numbers, too, like 3-digit by 3-digit multiplication.

Choosing a strategy that works for you

All the methods we have learned—partial products, standard algorithm, area model, and lattice—will give you the same correct answer if you do them carefully. The best strategy is the one you understand and can use accurately.

Same problem, different methods: 23 × 45

Partial products: 20×40=800, 20×5=100, 3×40=120, 3×5=15 → total = 1,035
Standard algorithm:
23
×45
115
920
1,035
Area model: 20×40=800, 20×5=100, 3×40=120, 3×5=15 → total = 1,035
Lattice: Also gives 1,035.

Note

Whichever method you use, always check your multiplication facts and addition. One small mistake can change the whole answer.

More worked-out examples

Let’s look at a few more problems to make sure we understand all the steps.

Example A: 86 × 54

Area model: 80×50=4,000; 80×4=320; 6×50=300; 6×4=24. Sum: 4,000+320=4,320; 4,320+300=4,620; 4,620+24=4,644.
Standard algorithm check:
86
×54
(86×4) 344
(86×50) 4,300
4,644

Example B: 93 × 71

Estimate: 90×70=6,300.
Partial products: 90×70=6,300; 90×1=90; 3×70=210; 3×1=3. Total: 6,300+90=6,390; 6,390+210=6,600; 6,600+3=6,603.
6,603 is close to 6,300, so our answer is reasonable. (The estimate was a bit low because we rounded down.)

Note

When you estimate, if you round one factor up and the other down, your estimate might be a little off. But it will still give you a ballpark number.

Special cases: numbers with zero digits

Sometimes you will multiply numbers like 30 × 40 or 50 × 60. These are multiples of ten. You can multiply the non-zero digits first, then count the total number of zeroes and attach them to the product.

Example: 70 × 80

Multiply 7 × 8 = 56.
There is one zero in 70 and one zero in 80, so there are two zeroes total.
Attach two zeroes to 56 → 5,600.
Check: 70 × 80 = 5,600.

Example: 24 × 30

One way: 24 × 3 = 72, then attach one zero → 720.
Or use place value: 30 = 3 tens, so 24 × 3 tens = 72 tens = 720.

Note

Be careful with numbers like 101 × 20. The shortcut only works when the whole number is a multiple of ten (like 20, 30, 40) not just a number with a zero inside (like 101).

Common errors to avoid

Even fourth graders who understand multiplication can make small mistakes. Here are some things to watch out for.

Mistake 1: Forgetting the placeholder zero in the standard algorithm.

Wrong:
42
× 35
210
126
336

(This is wrong because 42 × 30 should be 1,260, not 126.)
Correct:
42
× 35
210
1,260
1,470

Mistake 2: Adding instead of multiplying.

When you see 24 × 13, do not add 24 + 13. Always multiply.
Wrong: 24 × 13 is not 37. It is 312.

Mistake 3: Misplacing digits in the area model.

Make sure you put the correct number of zeros. 40 × 60 = 2,400, not 24 and not 240.

Note

Always double-check your work. Use a different strategy to verify your answer. For example, if you used the standard algorithm, check with the area model.

Using 2-digit multiplication in real life

People use 2-digit multiplication in many jobs and everyday situations. Knowing how to multiply quickly and accurately is an important skill.

Examples:

Baking: A baker makes 24 batches of cookies. Each batch uses 15 chocolate chips. How many chocolate chips in all? 24 × 15 = 360 chips.
Travel: A car travels 65 miles per hour for 14 hours. How far does it go? 65 × 14 = 910 miles.
School supplies: There are 36 students, and each needs 25 pencils. Total pencils = 36 × 25 = 900 pencils.

Note

When you solve word problems, look for key words like "each" and "total" to know that you need to multiply.

Go to the practice →

Common Core alignment: CCSS.MATH.CONTENT.4.NBT.B.5 – Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Notes for teachers

This lesson is aligned with CCSS.MATH.CONTENT.4.NBT.B.5. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework. The study section provides detailed explanations of partial products, standard algorithm, area model, lattice multiplication, estimation, and common errors. Each section includes a definition, examples, and a helpful note.