Rational Numbers Understanding for 7th Grade

1. Which of the following is not a rational number?

2

3/4

1.3

0.612452727

A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero. While 2, 3/4, and 1.3 can all be represented as fractions (2/1, 3/4, and 13/10 respectively), the number 0.612452727 is a non-terminating, non-repeating decimal. Such decimals cannot be expressed as a simple fraction of two integers, making them irrational. Thus, 0.612452727 is not a rational number.

Explanation

A rational number is defined as any number that can be expressed as the quotient of two integers, where the denominator is not zero. While 2, 3/4, and 1.3 can all be represented as fractions (2/1, 3/4, and 13/10 respectively), the number 0.612452727 is a non-terminating, non-repeating decimal. Such decimals cannot be expressed as a simple fraction of two integers, making them irrational. Thus, 0.612452727 is not a rational number.

2. Order the following numbers from least to greatest: 0.5, 3/4, 60%.

0.5, 3/4, 60%

3/4, 0.5, 60%

60%, 3/4, 0.5

3/4, 60%, 0.5

To order the numbers from least to greatest, we first convert them to a common format. The decimal 0.5 is equivalent to 50%, and 3/4 is equal to 75%. Thus, we have 0.5 (50%), 3/4 (75%), and 60%. When comparing these values, 0.5 is the smallest, followed by 60% (which is 0.6 in decimal), and then 3/4. Hence, the correct order from least to greatest is 0.5, 3/4, and 60%.

Explanation

To order the numbers from least to greatest, we first convert them to a common format. The decimal 0.5 is equivalent to 50%, and 3/4 is equal to 75%. Thus, we have 0.5 (50%), 3/4 (75%), and 60%. When comparing these values, 0.5 is the smallest, followed by 60% (which is 0.6 in decimal), and then 3/4. Hence, the correct order from least to greatest is 0.5, 3/4, and 60%.

3. What is 1/2 + 3/4?

1/4

5/4

10/8

1/2

To add the fractions 1/2 and 3/4, first convert 1/2 to a fraction with a common denominator of 4. This gives us 2/4. Now, add 2/4 and 3/4:

2/4 + 3/4 = (2 + 3)/4 = 5/4.

Thus, the sum of 1/2 and 3/4 is 5/4, which can also be expressed as 1 and 1/4.

Explanation

To add the fractions 1/2 and 3/4, first convert 1/2 to a fraction with a common denominator of 4. This gives us 2/4. Now, add 2/4 and 3/4:

2/4 + 3/4 = (2 + 3)/4 = 5/4.

Thus, the sum of 1/2 and 3/4 is 5/4, which can also be expressed as 1 and 1/4.

4. What is the cube root of 27?

3

6

9

27

The cube root of a number is a value that, when multiplied by itself three times, equals the original number. In this case, 3 multiplied by itself three times (3 × 3 × 3) equals 27. Therefore, the cube root of 27 is 3, as it is the number that satisfies this condition.

Explanation

The cube root of a number is a value that, when multiplied by itself three times, equals the original number. In this case, 3 multiplied by itself three times (3 × 3 × 3) equals 27. Therefore, the cube root of 27 is 3, as it is the number that satisfies this condition.

5. Which number can be both a perfect square and a perfect cube?

27

36

49

64

64 is both a perfect square and a perfect cube because it can be expressed as 8^2 (a perfect square) and also as 4^3 (a perfect cube). A perfect square is a number that can be expressed as the square of an integer, while a perfect cube is a number that can be expressed as the cube of an integer. Since 64 meets both criteria, it qualifies as both a perfect square and a perfect cube.

Explanation

64 is both a perfect square and a perfect cube because it can be expressed as 8^2 (a perfect square) and also as 4^3 (a perfect cube). A perfect square is a number that can be expressed as the square of an integer, while a perfect cube is a number that can be expressed as the cube of an integer. Since 64 meets both criteria, it qualifies as both a perfect square and a perfect cube.

6. What is 0.6 × 0.5?

0.30

0.35

0.25

0.20

To find the product of 0.6 and 0.5, you multiply the two numbers together. First, convert them into fractions: 0.6 is 6/10 and 0.5 is 5/10. Multiplying these gives (6/10) × (5/10) = 30/100. Simplifying 30/100 results in 0.30. Thus, the product of 0.6 and 0.5 is 0.30.

Explanation

To find the product of 0.6 and 0.5, you multiply the two numbers together. First, convert them into fractions: 0.6 is 6/10 and 0.5 is 5/10. Multiplying these gives (6/10) × (5/10) = 30/100. Simplifying 30/100 results in 0.30. Thus, the product of 0.6 and 0.5 is 0.30.

7. The number 121 is:

Neither

Perfect cube

Perfect square

Both perfect square and cube

The number 121 is a perfect square because it can be expressed as 11 multiplied by itself (11 × 11 = 121). A perfect square is defined as the product of an integer with itself. However, 121 is not a perfect cube since there is no integer that, when multiplied by itself three times, equals 121. Therefore, it qualifies only as a perfect square.

Explanation

The number 121 is a perfect square because it can be expressed as 11 multiplied by itself (11 × 11 = 121). A perfect square is defined as the product of an integer with itself. However, 121 is not a perfect cube since there is no integer that, when multiplied by itself three times, equals 121. Therefore, it qualifies only as a perfect square.

8. How many tiles are on each side of a square deck with 16,384 square tiles?

128

256

64

32

To determine the number of tiles on each side of a square deck with 16,384 square tiles, we need to find the square root of 16,384. Since a square has equal sides, if we let the number of tiles on each side be \( x \), then \( x^2 = 16,384 \). Calculating the square root gives us \( x = 128 \). Therefore, each side of the square deck has 128 tiles.

Explanation

To determine the number of tiles on each side of a square deck with 16,384 square tiles, we need to find the square root of 16,384. Since a square has equal sides, if we let the number of tiles on each side be \( x \), then \( x^2 = 16,384 \). Calculating the square root gives us \( x = 128 \). Therefore, each side of the square deck has 128 tiles.