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

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4. Find the GCF of 40 and 120.

Find the prime factorization of 40.
$40 = 2 \times 20 = 2 \times 2 \times 10 = 2 \times 2 \times 2 \times 5 = 2^3 \times 5^1$
Find the prime factorization of 120.
$120 = 2 \times 60 = 2 \times 2 \times 30 = 2 \times 2 \times 2 \times 15 = 2 \times 2 \times 2 \times 3 \times 5 = 2^3 \times 3^1 \times 5^1$
Identify the common prime factors and their lowest powers.
The common prime factors are 2 and 5.
The lowest power of 2 is $2^3$ .
The lowest power of 5 is $5^1$ .
Multiply the common prime factors with their lowest powers.
$\text{GCF}(40, 120) = 2^3 \times 5^1 = 8 \times 5 = 40$
Answer:
The GCF of 40 and 120 is $\boxed{\mathbf{40}}$ .

5. Write the decimal number 0.8333333... as a rational number.
Let $x$ equal the decimal.
$x = 0.8333...$
Multiply by 10 to move the non-repeating part to the left of the decimal.
$10x = 8.333... \quad (1)$
Multiply by 100 to move one full repeating block to the left of the decimal.
$100x = 83.333... \quad (2)$
Subtract equation (1) from equation (2) to eliminate the repeating part.
$100x - 10x = 83.333... - 8.333...$ $90x = 75$
Solve for $x$ and simplify the fraction.
$x = \frac{75}{90}$
Divide both the numerator and the denominator by their greatest common divisor, which is 15.
$x = \frac{75 \div 15}{90 \div 15} = \frac{5}{6}$
Answer:
The rational number is $\boxed{\mathbf{\frac{5}{6}}}$ .

6. Complete the equivalent fraction by filling in the box. $\frac{19}{7} = \frac{76}{\square}$
Determine the factor by which the numerator 19 was multiplied to get 76.
$19 \times k = 76$ $k = \frac{76}{19} = 4$
Multiply the denominator 7 by the same factor.
$7 \times 4 = 28$
Answer:
The completed equivalent fraction is $\frac{19}{7} = \frac{76}{\boxed{\mathbf{28}}}$ .

7. Express 16/3 as a decimal correct to 4 decimal places.
Divide 16 by 3.
$\frac{16}{3} = 5.333333...$
Identify the first 5 decimal places to determine rounding.
The fifth decimal place is 3.
Round to 4 decimal places.
Since the fifth decimal place (3) is less than 5, we round down (keep the fourth decimal place as it is).
$5.3333$
Answer:
$16/3$ as a decimal correct to 4 decimal places is $\boxed{\mathbf{5.3333}}$ .

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

Step-by-step solution

Find the prime factorization of 40.

Find the prime factorization of 120.

Identify the common prime factors and their lowest powers.

Multiply the common prime factors with their lowest powers.

Let $x$ equal the decimal.

Multiply by 10 to move the non-repeating part to the left of the decimal.

Multiply by 100 to move one full repeating block to the left of the decimal.

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

Solve for $x$ and simplify the fraction.

Determine the factor by which the numerator 19 was multiplied to get 76.

Multiply the denominator 7 by the same factor.

Divide 16 by 3.

Identify the first 5 decimal places to determine rounding.

Round to 4 decimal places.

Still stuck on this one?

More Mathematics Questions

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

Step-by-step solution

Find the prime factorization of 40.

Find the prime factorization of 120.

Identify the common prime factors and their lowest powers.

Multiply the common prime factors with their lowest powers.

Let xxx equal the decimal.

Multiply by 10 to move the non-repeating part to the left of the decimal.

Multiply by 100 to move one full repeating block to the left of the decimal.

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

Solve for xxx and simplify the fraction.

Determine the factor by which the numerator 19 was multiplied to get 76.

Multiply the denominator 7 by the same factor.

Divide 16 by 3.

Identify the first 5 decimal places to determine rounding.

Round to 4 decimal places.

Still stuck on this one?

More Mathematics Questions

Let $x$ equal the decimal.

Solve for $x$ and simplify the fraction.