Multiplying fractions and mixed numbers

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What is Multiplication?

Multiplication is a quick way to add the same number many times. For example, 3 × 4 means 4 + 4 + 4 = 12.
The numbers you multiply are called factors. The answer is called the product.
The multiplication sign is × (like 3 × 4) or a dot (like 3 ⋅ 4). Both mean the same.
Multiplication is commutative: 3 × 4 = 4 × 3. The order does not change the product.
You can think of multiplication as counting objects in a rectangle. If a rectangle is 3 rows and 4 columns, it has 3 × 4 = 12 squares.

Multiplying a Fraction by a Whole Number

To multiply a fraction by a whole number, multiply the top number (the numerator) by the whole number. The bottom number (the denominator) stays the same.
Example: \frac{2}{5} \times 3 = \frac{2 \times 3}{5} = \frac{6}{5}.
If the product is an improper fraction (top bigger than bottom), you can write it as a mixed number: \frac{6}{5} = 1\frac{1}{5}.
Think of \frac{2}{5} \times 3 as 3 groups of \frac{2}{5}, which is \frac{2}{5} + \frac{2}{5} + \frac{2}{5} = \frac{6}{5}.

Multiplying a Fraction by a Fraction

To multiply two fractions, multiply the numerators together and multiply the denominators together.
Example: \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}.
You can simplify before multiplying if you see a common factor. For example, \frac{2}{3} \times \frac{3}{4} can be simplified to \frac{1}{1} \times \frac{1}{2} = \frac{1}{2}.
Always check if your answer can be simplified. For example, \frac{4}{8} simplifies to \frac{1}{2}.

Why Multiplying by a Fraction Less Than 1 Makes the Answer Smaller

When you multiply a number by a fraction that is less than 1, the product is smaller than the original number.
Example: 6 \times \frac{1}{2} = 3. The answer 3 is less than 6.
This is because multiplying by \frac{1}{2} is like finding half of the number.
Multiplying by \frac{3}{4} is like finding three quarters of the number, so the answer is less than the number.
This is different from multiplying by a whole number greater than 1, which makes the answer bigger.

Using Area Models for Fraction Multiplication

An area model is a rectangle divided into equal parts. It helps you see fraction multiplication.
To multiply \frac{1}{2} \times \frac{1}{3}, draw a rectangle, split it into 2 rows and 3 columns. Shade 1 row and 1 column. The overlapping part is \frac{1}{6} of the whole.
The area of a rectangle is length × width. If the sides are fractions, the area is the product of those fractions.
Area models show why \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}.

Multiplying Mixed Numbers

A mixed number has a whole number and a fraction, like 2\frac{1}{3}.
To multiply mixed numbers, first change them to improper fractions (where the numerator is bigger than the denominator).
To change 2\frac{1}{3} to an improper fraction: multiply the whole number by the denominator (2 × 3 = 6), add the numerator (6 + 1 = 7), and keep the denominator: \frac{7}{3}.
Then multiply the improper fractions as usual.
Example: 1\frac{1}{2} \times 2\frac{1}{3} = \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = 3\frac{1}{2}.

Dividing with Fractions (Unit Fractions)

A unit fraction has 1 as the numerator, like \frac{1}{2}, \frac{1}{3}, \frac{1}{4}.
To divide a whole number by a unit fraction, multiply by the denominator. Example: 4 \div \frac{1}{2} = 4 \times 2 = 8.
To divide a unit fraction by a whole number, multiply the denominator by the whole number. Example: \frac{1}{3} \div 2 = \frac{1}{3 \times 2} = \frac{1}{6}.
Division is the opposite of multiplication. If 3 \times 4 = 12, then 12 \div 3 = 4.

Solving Real-World Problems

You can use fraction multiplication to find a fraction of an amount. For example, \frac{2}{3} of 12 is \frac{2}{3} \times 12 = 8.
If a recipe needs \frac{3}{4} cup of sugar and you make half the recipe, you need \frac{3}{4} \times \frac{1}{2} = \frac{3}{8} cup.
If you run \frac{2}{5} of a 10 km race, you run \frac{2}{5} \times 10 = 4 km.
When solving word problems, look for words like 'of', 'times', or 'product' to know you need to multiply.

Key Vocabulary

Fraction: a number that shows part of a whole, like \frac{1}{2}.
Numerator: the top number of a fraction. It tells how many parts you have.
Denominator: the bottom number of a fraction. It tells how many equal parts the whole is divided into.
Improper fraction: a fraction where the numerator is bigger than or equal to the denominator, like \frac{7}{3}.
Mixed number: a whole number plus a fraction, like 2\frac{1}{3}.
Product: the answer you get when you multiply.
Unit fraction: a fraction with numerator 1, like \frac{1}{5}.

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1.What is the result of a multiplication operation called?
Easy
- Asum
- Bdifference
- Cproduct
- Dquotient
2.Which symbol is commonly used to denote multiplication?
Easy
- A+
- B×
- C−
- D÷
3.Multiplying a fraction less than 1 by a whole number always gives a smaller answer than the whole number.
Easy
True or false?
4.Which expression gives the same value as 5 × 2/3?
Medium
- A10/3
- B5/6
- C2/15
- D7/3
5.Which of the following are correct ways to write 2 times 3? (select all that apply)
Medium
- A2×3
- B2⋅3
- C2+3
- D23
- E2×3
6.Match each multiplication expression with its product.
Medium
- 1/2 × 1/3
- 2/5 × 3/4
- 3 × 2/9
- 1/6
- 3/10
- 2/3
7.Arrange these steps in the correct order to multiply 1 1/2 by 3.
Medium
- Convert 1 1/2 to an improper fraction: 3/2
- Multiply 3/2 by 3 to get 9/2
- Write the answer as a mixed number: 4 1/2
8.A rectangle has a length of 2/3 meter and a width of 3/4 meter. What is its area?
Medium
- A1/2 square meter
- B5/7 square meter
- C6/12 square meter
- D1/2 meter

Thêm câu hỏi

9.Put these steps in the correct order to multiply 1 1/2 by 2/3.
Medium
- Convert 1 1/2 to an improper fraction: 3/2
- Multiply the numerators: 3 × 2 = 6
- Multiply the denominators: 2 × 3 = 6
- Simplify 6/6 to 1
10.Which of the following are correct ways to multiply 2/3 by 3/4? (select all that apply)
Medium
- A(2×3)/(3×4)
- B(2×4)/(3×3)
- C(2/3) × (3/4)
- D(2×3)/(3×4) = 6/12
- E(2/3) × (4/3)
11.When you multiply a whole number by a fraction less than 1, the product is always less than the whole number.
Easy
True or false?
12.What is 2 1/3 × 1 1/2?
Medium
- A3 1/2
- B3 1/3
- C2 1/6
- D3 5/6
13.Which of the following are true? (select all that apply)
Medium
- A3 × 2/5 = 6/5
- B2/5 × 3 = 6/5
- C3 × 2/5 = 5/6
- D2/5 × 3 = 6/15
- E3 × 2/5 = 1 1/5
14.Arrange these steps in the correct order to find the area of a rectangle with length 2/3 m and width 3/4 m.
Medium
- Write the formula: area = length × width
- Substitute the numbers: area = 2/3 × 3/4
- Multiply numerators: 2 × 3 = 6
- Multiply denominators: 3 × 4 = 12
- Simplify 6/12 to 1/2
15.The product of two fractions less than 1 is always less than 1.
Easy
True or false?
16.Match each mixed number multiplication with its product.
Medium
- 1 1/2 × 2/3
- 2 1/4 × 1/3
- 3 1/5 × 5/8
- 1
- 3/4
- 2
17.A recipe needs 3/4 cup of sugar. You want to make 1/2 of the recipe. How much sugar do you need?
Hard
- A3/8 cup
- B1 1/2 cups
- C3/2 cups
- D6/4 cups
18.What is 1 \frac{1}{2} \times 2 \frac{1}{3}?
Medium
- A2 \frac{1}{6}
- B3 \frac{1}{2}
- C3 \frac{1}{3}
- D2 \frac{5}{6}
19.Which of the following is the correct first step to multiply \frac{2}{5} \times \frac{3}{4}?
Easy
- AMultiply the numerators: 2 \times 3 = 6
- BMultiply the denominators: 5 \times 4 = 20
- CAdd the numerators: 2 + 3 = 5
- DCross-multiply: 2 \times 4 = 8
20.\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}
Easy
True or false?
21.What is \frac{3}{5} \times 10?
Medium
- A6
- B5
- C15
- D30
22.Which of the following are correct ways to multiply 1 \frac{1}{2} \times 2? (select all that apply)
Medium
- A1.5 \times 2 = 3
- B\frac{3}{2} \times 2 = 3
- C1 \times 2 + \frac{1}{2} \times 2 = 3
- D\frac{1}{2} \times 2 = 1
- E1 \times 2 = 2
23.Arrange these steps in the correct order to multiply 2 \frac{1}{4} by \frac{2}{3}.
Medium
- Convert 2 \frac{1}{4} to \frac{9}{4}
- Multiply numerators: 9 \times 2 = 18
- Multiply denominators: 4 \times 3 = 12
- Simplify \frac{18}{12} to 1 \frac{1}{2}
24.A recipe calls for \frac{2}{3} cup of flour. If you want to make 1 \frac{1}{2} times the recipe, how much flour do you need?
Medium
- A1 cup
- B\frac{3}{4} cup
- C1 \frac{1}{3} cups
- D1 \frac{1}{2} cups
25.Which of the following expressions have the same value as \frac{3}{5} \times \frac{5}{6}? (select all that apply)
Medium
- A\frac{1}{2}
- B\frac{15}{30}
- C\frac{3}{6}
- D\frac{5}{10}
- E\frac{1}{3}
26.Multiplying a number by \frac{3}{4} always gives a smaller result than the original number.
Easy
True or false?
27.A jug contains \frac{3}{4} liter of juice. How much juice is in 2 \frac{2}{3} jugs?
Medium
- A2 liters
- B1 \frac{1}{2} liters
- C\frac{3}{2} liters
- D2 \frac{1}{4} liters
28.Match each expression with its simplified product.
Medium
- \frac{2}{3} \times \frac{3}{4}
- \frac{1}{2} \times \frac{4}{5}
- \frac{3}{7} \times \frac{7}{9}
- \frac{1}{2}
- \frac{2}{5}
- \frac{1}{3}
29.Arrange these steps in the correct order to find the area of a rectangle with length 2 \frac{1}{3} m and width \frac{3}{5} m.
Hard
- Convert 2 \frac{1}{3} to \frac{7}{3}.
- Multiply the numerators: 7 \times 3 = 21.
- Multiply the denominators: 3 \times 5 = 15.
- Simplify \frac{21}{15} to \frac{7}{5}.
30.A square has a side length of \frac{3}{4} meter. What is its area?
Hard
- A\frac{9}{16} square meter
- B\frac{3}{4} square meter
- C1 \frac{1}{2} square meters
- D\frac{3}{8} square meter
31.Arrange these steps in the correct order to multiply 3/4 by 2/5.
Medium
- Multiply the numerators: 3 × 2 = 6
- Multiply the denominators: 4 × 5 = 20
- Simplify 6/20 to 3/10
32.The product of a fraction and a whole number can be found by multiplying the numerator by the whole number and keeping the denominator.
Easy
True or false?
33.A rectangle has a length of 2 1/2 meters and a width of 3/4 meter. What is its area in square meters?
Medium
- A1 7/8
- B2 1/8
- C3 1/8
- D3 3/4
34.Arrange these steps in the correct order to multiply 1 1/2 by 2 1/3.
Medium
- Convert 1 1/2 to 3/2
- Convert 2 1/3 to 7/3
- Multiply: (3/2) × (7/3) = 21/6
- Simplify 21/6 to 3 1/2
35.To multiply mixed numbers, you should first convert them to improper fractions.
Easy
True or false?
36.What is \frac{2}{5} \times 3?
Easy
- A\frac{6}{5}
- B\frac{5}{6}
- C\frac{6}{15}
- D\frac{2}{15}
37.What is \frac{3}{8} \times \frac{2}{3}? Give your answer in simplest form.
Medium
- A\frac{1}{4}
- B\frac{6}{24}
- C\frac{1}{3}
- D\frac{2}{8}
38.What is 1 \frac{1}{2} \times 4?
Medium
- A6
- B4 \frac{1}{2}
- C5
- D8
39.Which of the following are correct ways to multiply \frac{2}{3} by \frac{5}{7}? (select all that apply)
Medium
- AMultiply numerators: 2 × 5, and denominators: 3 × 7
- BMultiply denominators: 3 × 7, and numerators: 2 × 5
- CAdd the numerators and add the denominators
- DSimplify before multiplying if possible
40.When you multiply a fraction by a whole number, you multiply only the numerator by the whole number and keep the denominator the same.
Easy
True or false?
41.What is 3/4 of 20?
Easy
42.Complete the sentence.
Easy
The two numbers being multiplied are called ____.
43.What is the reciprocal of 2/5?
Medium
44.A rectangle has a length of 1 1/2 meters and a width of 3/4 meter. What is its area in square meters?
Medium
45.Fill in the blank:
Easy
The product of 2/5 and 3 is ____.
46.Write the reciprocal of 3/4.
Medium
47.What is \frac{2}{3} \times 6?
Easy
48.Fill in the blank to make the statement true.
Medium
\frac{3}{4} \times \frac{2}{5} = \frac{6}{20}, which simplifies to ____.
49.A rectangle has a length of \frac{3}{4} meter and a width of \frac{2}{3} meter. What is its area in square meters?
Medium
50.What is the product of \frac{2}{7} and \frac{3}{5}?
Easy
51.What is \frac{2}{3} \times 9?
Easy
52.What is \frac{3}{4} \times \frac{2}{5}? Give your answer as a fraction in simplest form.
Medium
53.Explain why multiplying by a fraction less than 1 gives a smaller result than the original number.
Hard
54.Explain why multiplying a number by a fraction less than 1 gives a smaller result than the original number. Use an example.
Hard
55.Explain why the area of a rectangle is the same even if you multiply length × width or width × length. Use the commutative property in your explanation.
Hard
56.Explain why the product of \frac{2}{3} and \frac{4}{5} is less than both \frac{2}{3} and \frac{4}{5}.
Hard
57.What is 3/4 of 20?
Easy
- A15
- B14
- C17
- D16
58.A rectangle has a length of 1 1/2 meters and a width of 3/4 meter. What is its area in square meters?
Medium
- A2.125 m²
- B1.125 m²
- C0.125 m²
- D3.125 m²
59.What is \frac{2}{3} \times 6?
Easy
- A5
- B3
- C4
- D6
60.A rectangle has a length of \frac{3}{4} meter and a width of \frac{2}{3} meter. What is its area in square meters?
Medium
- A2.5 m²
- B0.5 m²
- C1.5 m²
- D-0.5 m²
61.What is \frac{2}{3} \times 9?
Easy
- A7
- B5
- C8
- D6
62.What is \frac{3}{4} \times \frac{2}{5}? Give your answer as a fraction in simplest form.
Medium
- A0.3
- B1.3
- C-0.7
- D2.3

Ghi chú ôn tập

## Multiplying Fractions and Mixed Numbers ### Multiplying a Fraction by a Whole Number - **Multiply the numerator (top number) by the whole number.** Keep the denominator (bottom number) the same. - Example: \(\frac{2}{5} \times 3 = \frac{2 \times 3}{5} = \frac{6}{5}\). - If you can, simplify the answer. \(\frac{6}{5}\) is an improper fraction (top bigger than bottom). You can write it as a mixed number: \(1\frac{1}{5}\). ### Multiplying a Fraction by a Fraction - **Multiply the numerators together, and multiply the denominators together.** - Example: \(\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}\). - **Why is the answer smaller?** When you multiply by a fraction less than 1, you are taking a part of something. For example, \(\frac{1}{2} \times \frac{3}{4}\) means half of three-quarters, which is less than three-quarters. ### Area Models - You can draw a rectangle and split it into parts to show fraction multiplication. - Example: To show \(\frac{1}{2} \times \frac{3}{4}\), draw a rectangle, split it into 4 equal columns and shade 3 columns (that's \(\frac{3}{4}\)). Then split it into 2 equal rows and shade 1 row (that's \(\frac{1}{2}\)). The double-shaded part is \(\frac{3}{8}\) of the whole rectangle. ### Multiplying Mixed Numbers - **Convert each mixed number to an improper fraction first.** - A mixed number has a whole number and a fraction, like \(2\frac{1}{3}\). - To convert: multiply the whole number by the denominator, then add the numerator. Put that over the denominator. - Example: \(2\frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}\). - Then multiply the improper fractions as usual. - Example: \(1\frac{1}{2} \times 2\frac{1}{3} = \frac{3}{2} \times \frac{7}{3} = \frac{21}{6} = 3\frac{1}{2}\). ### Dividing with Unit Fractions - A **unit fraction** has 1 as the numerator, like \(\frac{1}{2}\), \(\frac{1}{3}\), \(\frac{1}{4}\). - **Dividing a unit fraction by a whole number:** Multiply the denominator by the whole number. - Example: \(\frac{1}{3} \div 2 = \frac{1}{3 \times 2} = \frac{1}{6}\). - **Dividing a whole number by a unit fraction:** Multiply the whole number by the denominator. - Example: \(2 \div \frac{1}{3} = 2 \times 3 = 6\). ### Real-World Problems - **Finding a fraction of an amount:** Multiply the amount by the fraction. - Example: Find \(\frac{2}{3}\) of 12. \(\frac{2}{3} \times 12 = \frac{2 \times 12}{3} = \frac{24}{3} = 8\). - Always check if your answer makes sense. If you multiply by a fraction less than 1, the answer is smaller than the original number. ### Key Tips - **Simplify** your answers when possible. - **Convert mixed numbers** to improper fractions before multiplying. - Use **area models** to see what is happening. - Practice with real-life examples like recipes, sharing, and measuring.

fraction: A number that shows part of a whole. It has a top number (numerator) and a bottom number (denominator).
numerator: The top number of a fraction. It tells how many parts you have.
denominator: The bottom number of a fraction. It tells how many equal parts the whole is divided into.
mixed number: A number made of a whole number and a fraction, like 2 1/3.
improper fraction: A fraction where the numerator is bigger than or equal to the denominator, like 7/3.
unit fraction: A fraction with 1 as the numerator, like 1/2 or 1/4.
product: The answer you get when you multiply two numbers.
area model: A drawing that uses a rectangle split into parts to show multiplication of fractions.
simplify: To make a fraction smaller by dividing the top and bottom by the same number. For example, 2/4 simplifies to 1/2.
commutative property: The rule that says you can multiply numbers in any order and get the same answer. For example, 2 × 3 = 3 × 2.

Thẻ ghi nhớ

What does 3 × 4 mean as repeated addition?

3 × 4 means 4 + 4 + 4, which is 12. It is like having 3 groups of 4.

What is the result of a multiplication called?

The result is called the product. For example, in 3 × 4 = 12, 12 is the product.

What are the two numbers you multiply together called?

They are called factors. In 3 × 4, both 3 and 4 are factors.

How do you multiply a fraction by a whole number? Give an example.

Multiply the numerator by the whole number and keep the denominator the same. For example, 3 × 1/4 = 3/4.

What is 2 × 3/5?

2 × 3/5 = 6/5, which is 1 1/5.

How do you multiply two fractions? Give an example.

Multiply the numerators together and the denominators together. For example, 1/2 × 3/4 = 3/8.

What is 2/3 × 4/5?

2/3 × 4/5 = 8/15.

When you multiply a number by a fraction less than 1, is the product larger or smaller than the original number?

Smaller. For example, 6 × 1/2 = 3, which is less than 6.

Why is 8 × 1/4 less than 8?

Because 1/4 is less than 1, so you are taking only a part of 8. 8 × 1/4 = 2, which is less than 8.

What is 5 × 2/3? Is the product greater or less than 5?

5 × 2/3 = 10/3 = 3 1/3, which is less than 5 because 2/3 is less than 1.

How can you use an area model to show 1/2 × 1/3?

Draw a rectangle, divide it into 2 equal parts vertically and 3 equal parts horizontally. Shade 1/2 and 1/3. The overlapping part is 1/6 of the whole, so 1/2 × 1/3 = 1/6.

What is 1/4 × 1/2? Use an area model idea.

1/4 × 1/2 = 1/8. The overlapping part of 1/4 and 1/2 is 1/8 of the whole.

How do you multiply mixed numbers? Give the first step.

First, change each mixed number to an improper fraction. For example, 1 1/2 becomes 3/2.

What is 1 1/2 × 2 1/3?

Change to improper fractions: 3/2 × 7/3 = 21/6 = 3 1/2.

What is 2 1/4 × 1 1/3?

2 1/4 = 9/4, 1 1/3 = 4/3, so 9/4 × 4/3 = 36/12 = 3.

How do you divide a unit fraction by a whole number? Give an example.

Keep the numerator 1 and multiply the denominator by the whole number. For example, 1/3 ÷ 2 = 1/6.

What is 1/4 ÷ 3?

1/4 ÷ 3 = 1/12.

How do you divide a whole number by a unit fraction? Give an example.

Multiply the whole number by the denominator of the fraction. For example, 2 ÷ 1/3 = 2 × 3 = 6.

What is 5 ÷ 1/2?

5 ÷ 1/2 = 5 × 2 = 10.

A recipe needs 2/3 cup of sugar. You make 1/2 of the recipe. How much sugar do you need?

You need 2/3 × 1/2 = 2/6 = 1/3 cup of sugar.

A rectangle is 3/4 m long and 2/5 m wide. What is its area?

Area = 3/4 × 2/5 = 6/20 = 3/10 square meters.

What is the product of 7 and 3/5?

7 × 3/5 = 21/5 = 4 1/5.

What is 3/8 × 4/9? Simplify if possible.

3/8 × 4/9 = 12/72 = 1/6.

What is 2 1/2 × 3/4?

2 1/2 = 5/2, so 5/2 × 3/4 = 15/8 = 1 7/8.

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