Multiplying Fractions
The logic of multiplying fractions is the exact same as multiplying whole numbers. If you understand how multiplication works with whole numbers, you will find this remarkably easy and natural to do.
🍎 1. The Meaning of Multiplication: Counting Equal Groups
Multiplication was invented as a shortcut for repeated addition. Whether you are counting plates of whole apples or plates of sliced half-apples, the arithmetic rule never changes!
Whole Numbers: 3 Plates with 5 Apples Each
Multiplication is simply counting equal groups: 3 × 5 = 15. When each plate has 5 apples, you count 5, 10, 15...
The Exact Same Logic in Fractions: 6 Plates with12Each
Counting 6 pieces of 12 gives 6 halves. In mathematics, we write this as 62 — because 62 literally means 6 pieces of 12! Since every 2 pieces make 1 full whole (22 = 1), 6 pieces give us:
Next Example: 8 Plates with13Each
Counting 8 pieces of 13 gives 8 thirds. In mathematics, we write this as 83 — because 83 literally means 8 pieces of 13! Since every 3 pieces make 1 full whole (33 = 1), 6 pieces give us 2 full wholes, plus 2 pieces remaining:
🧺 2. Scaling: Finding a Fraction "OF" a Quantity
What does this mean?
Let's see what it means with whole numbers
1 group of 20 = 20
2 groups of 20 = 40
3 groups of 20 = 60
We already know these.
35 × 20 = 35 of a group of 20
"35 groups of 20" is a bit confusing, isn't it?
How does it sound?
"3 groups of 20" makes sense. But what do we understand when we say "35 of a group of 20"? Let's think about apples:
I understand this! So "half a 20" and "half of 20" are the same thing, and "a quarter 20" and "a quarter of 20" are the same thing.
Back to our question
So when I say 35 × 20, it means 35 OF 20.
Now 20 is the whole
Cutting 20 into 5 equal parts (15 each) gives 4 in each part, so 15 of 20 = 4.
35 of 20 is 3 of those parts: 4 + 4 + 4, so 35 of 20 = 12.
The result makes sense, because:
Less than 1 group of 20 gives less than 20.
What does this mean?
What did we just learn?
35 × 20 meant 35 of 20. With the same logic, 34 × 25 means 34 of 25.
How does it sound?
I have 25 of an apple in my hand. I want to take 34 of the piece in my hand. So our whole is no longer the full apple — our whole is the 25 piece.
The apple is cut into 5 equal parts. The 2 colored parts are the 25 in my hand.
Now 25 is the whole
To take 34 of it, we cut only the 25 piece into 4 equal parts.
Look at the thick outline: the 2 side-by-side parts together make one horizontal rectangle. 4 of these rectangles make our 25 piece, so this rectangle is 14 of 25.
Now take 3 of these rectangles together. They are 34 of 25.
How much of the whole apple is it?
To name this part as a fraction of the whole apple, the whole apple must be cut into equal parts too. So we extend the cuts across the whole apple: now it is made of 4 × 5 = 20 equal small squares.
14 of 25 is 2 small squares, so 14 of 25 = 220.
34 of 25 is 2 + 2 + 2 = 6 small squares, so 34 of 25 = 620.
The result makes sense, because:
We took less than the whole 25, so the result is less than 25.
Watch it step by step
This is our whole apple.
🪟 3. The Area Model in Multiplication
Area Model with Whole Numbers: 4 × 6
Let's first see how the area model works with whole numbers.
One unit square
This is 1 unit square. Each side is 1 long, and its area is 1 square unit.
4 rows of 6 unit squares
We put 6 unit squares in each row and make 4 rows. Now we have a rectangle: one side is 6 long, the other side is 4 long.
Count the unit squares
The area of the rectangle is the number of unit squares inside it: 6 + 6 + 6 + 6 = 24 unit squares.
Multiply the side lengths
Area = 4 × 6 = 24
We don't have to count every square. To find the area of a rectangle, we multiply its side lengths.
Area Model with Fractions: 34 × 25
The same idea works with fractions: 34 × 25 is the area of a rectangle whose sides are 34 and 25 long.
One unit square
Again we start with 1 unit square. Each side is 1 long, and its area is 1.
One side: 25
We cut the top side into 5 equal parts (15) and take 2 of them. So one side of our rectangle is 25 long.
Other side: 34
We cut the left side into 4 equal parts (14) and take 3 of them. So the other side of our rectangle is 34 long.
What is the area of our rectangle?
The size of this yellow area is the result of our multiplication. To find the result of 34 × 25, we need to calculate how big this area is.
Put both cuts on the unit square
Now we draw both cuts on the same unit square. To name a part as a fraction, the whole must be cut into equal parts. That is why we cut the whole unit square into equal parts.
The unit square is cut into 20 equal squares
The cuts divide the whole unit square into 4 × 5 = 20 equal small squares. So each small square is 120 of the unit square.
Count the squares in our rectangle
Our rectangle covers 2 + 2 + 2 = 6 small squares out of 20. So its area is 620.
Multiply the side lengths
3 × 2 = 6 → the small squares in our rectangle
4 × 5 = 20 → all the small squares in the unit square
Just like 4 × 6, we find the area by multiplying the side lengths.
🧮 4. Multiplying Fractions: The Standard Method
Now let's calculate without drawing. Here are the 4 kinds of multiplication, step by step.
This is the rule we use every time:
- Numerator × numerator gives the new numerator.
- Denominator × denominator gives the new denominator.
- A whole number is written over 1. A mixed number is changed to an improper fraction first.
The problem
Write the whole number as a fraction (put it over 1): 3 = 31.
Multiply the numerators together and the denominators together.
65 is 6 pieces of 15. 5 pieces of 15 make 1 whole, and 1 piece is left: 115.
The problem
Multiply the numerators: 3 × 2 = 6. Multiply the denominators: 4 × 5 = 20.
Write the new fraction.
Simplify: divide the numerator and the denominator by 2.
The problem
213: each whole is 3 pieces of 13, so 2 wholes are 6 pieces. With 1 more piece, we have 7 pieces of 13: 73. Write the whole number as a fraction: 4 = 41.
Multiply straight across.
283 is 28 pieces of 13. Every 3 pieces of 13 make 1 whole: 27 pieces make 9 wholes, and 1 piece is left: 913.
The problem
112: 1 whole is 2 pieces of 12, plus 1 piece makes 3 pieces: 32. 223: 2 wholes are 6 pieces of 13, plus 2 pieces makes 8 pieces: 83.
Multiply straight across.
246 is 24 pieces of 16. Every 6 pieces of 16 make 1 whole, so 24 pieces make 4 wholes.
✏️ 5. Practice: Your Turn
Solve each problem step by step. Type your answer in the boxes and press Check. If you get stuck, press Hint.
- 1
Write the whole number as a fraction.
4 = - 2
Multiply the numerators together and the denominators together.
- 3
Change the improper fraction to a mixed number.
- 1
Multiply the numerators together and the denominators together.
23 × 38 = - 2
Simplify the fraction as much as you can.
- 1
Change the mixed number to an improper fraction.
125 = - 2
Write the whole number as a fraction.
- 3
Multiply the numerators together and the denominators together.
- 4
Change the improper fraction to a mixed number.
- 1
Change the first mixed number to an improper fraction.
112 = - 2
Change the second mixed number to an improper fraction.
- 3
Multiply the numerators together and the denominators together.
- 4
Simplify: write the answer as a whole number.