Find the GCF of 40 and 120. Write the decimal number 0.8333333... as a rational number.

Here are the solutions to the questions:

4. Find the GCF of 40 and 120.

Working out:
We can find the Greatest Common Factor (GCF) by listing the factors or using prime factorization.
Using prime factorization:
First, find the prime factorization of each number.

$$40 = 2 \times 20 = 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1$$ $$120 = 2 \times 60 = 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 15 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3^1 \times 5^1$$

To find the GCF, we take the lowest power of each common prime factor.
Common prime factors are 2 and 5.
The lowest power of 2 is $2^3$.
The lowest power of 5 is $5^1$.

$$\text{GCF}(40, 120) = 2^3 \times 5^1 = 8 \times 5 = 40$$

Answer:

$$\boxed{40}$$

5. Write the decimal number 0.8333333... as a rational number.

Working out:
Let $x = 0.8333...$
Multiply by 10 to move the non-repeating digit to the left of the decimal:

$$10x = 8.3333... \quad (1)$$

Multiply by 100 to move one repeating block and the non-repeating digit to the left of the decimal:

$$100x = 83.3333... \quad (2)$$

Subtract equation (1) from equation (2) to eliminate the repeating part:

$$100x - 10x = 83.3333... - 8.3333...$$ $$90x = 75$$

Now, solve for $x$:

$$x = \frac{75}{90}$$

Simplify the fraction by dividing both the numerator and the denominator by their greatest common factor, which is 15:

$$x = \frac{75 \div 15}{90 \div 15} = \frac{5}{6}$$

Answer:

$$\boxed{\frac{5}{6}}$$

6. Complete the equivalent fraction by filling in the box. $\frac{19}{7} = \frac{76}{\square}$

Working out:
To find the missing number, we first determine what factor the numerator 19 was multiplied by to get 76.

$$19 \times \text{factor} = 76$$ $$\text{factor} = \frac{76}{19} = 4$$

Since the numerator was multiplied by 4, the denominator must also be multiplied by 4 to keep the fraction equivalent.

$$7 \times 4 = 28$$

So, the equivalent fraction is $\frac{76}{28}$.

Answer:

$$\boxed{28}$$

7. Express 16/3 as a decimal correct to 4 decimal places.

Working out:
Divide 16 by 3:

$$\frac{16}{3} = 5.333333...$$

To express this correct to 4 decimal places, we look at the fifth decimal place. If it is 5 or greater, we round up the fourth decimal place. If it is less than 5, we keep the fourth decimal place as it is.
The fifth decimal place is 3, which is less than 5. So, we keep the fourth decimal place as 3.

$$5.3333$$

Answer:

$$\boxed{5.3333}$$