Question

Change the following fractions to decimals. Carry division three decimal places as necessary. \(\frac{8}{64}\) ______

Short answer

The decimal of \( \frac{8}{64} \) is 0.125.

Step-by-step solution

Simplify the Fraction

First, simplify the fraction \( \frac{8}{64} \) by finding the greatest common divisor (GCD) of 8 and 64. The GCD is 8. Simplify the fraction by dividing both the numerator and the denominator by the GCD: \( \frac{8 \div 8}{64 \div 8} = \frac{1}{8} \).

Convert to Decimal

Now, divide 1 by 8 to convert \( \frac{1}{8} \) into a decimal. Begin by performing long division: 1 divided by 8. The division gives 0.125.

Confirm Three Decimal Places

The quotient is 0.125, which already has three decimal places. There is no need for additional steps since the division is complete and fulfills the decimal places requirement.

Key concepts

Simplifying Fractions

Simplifying fractions is a crucial step when converting fractions to decimals. It involves reducing the fraction to its smallest equivalent form. This is done by finding the greatest common divisor (GCD) of the numerator and the denominator.
For example, let's take the fraction \( \frac{8}{64} \). Here, both 8 and 64 share a GCD of 8. This means the largest number that can evenly divide both 8 and 64 is 8.
To simplify the fraction, divide both the numerator (8) and the denominator (64) by this GCD. This results in:

• Numerator: \(8 \div 8 = 1\)
• Denominator: \(64 \div 8 = 8\)

After simplification, the fraction \(\frac{8}{64}\) reduces to \(\frac{1}{8}\). Simplifying fractions makes calculations easier and helps avoid unnecessary steps in mathematical problems.

Long Division

Long division is a method used to divide larger numbers. When converting fractions to decimals, long division is often applied to the simplified fraction.
For \(\frac{1}{8}\), you need to divide 1 by 8 using long division.
The steps for long division are:

• Set up the division by writing 1 (the dividend) inside the division bracket and 8 (the divisor) outside.
• Since 8 does not fit into 1, add a decimal point and a zero to the dividend. This changes 1 to 10.
• Now, 8 divides into 10 one time. Write 1 above the division bracket.
• Multiply 1 (quotient) by 8 to get 8; subtract this from 10 to get a remainder of 2.
• Add another zero to the remainder to make it 20.
• 8 divides into 20 two times. Write 2 next to the 1.
• Multiply 2 by 8 to get 16; subtract from 20 to get a remainder of 4.
• Repeat by adding another zero to get 40.
• 8 goes into 40 five times evenly. Write 5 next to the quotient.

The final result of this division is 0.125.

Decimal Places

Decimal places represent precision in a decimal number. Each digit to the right of the decimal point is a decimal place.
In mathematical problems, sometimes you are asked to round or carry out calculations to a certain number of decimal places. For example, in this conversion, we need to have three decimal places.
The decimal result of \(\frac{1}{8}\) is 0.125, which already has three decimal places.
It's important to understand the placement as it indicates how precise your conversion or calculation is:

• The first digit after the decimal is the tenths place.
• The second digit is the hundredths place.
• The third digit is the thousandths place.

For the decimal 0.125:

• 1 is in the tenths place, equivalent to \(\frac{1}{10}\).
• 2 is in the hundredths place, equivalent to \(\frac{2}{100}\).
• 5 is in the thousandths place, equivalent to \(\frac{5}{1000}\).

Understanding and accurately calculating decimal places ensures your answers are precise and meet mathematical requirements.