Problem 86 - Prealgebra

Question

Divide. Round the answers to the nearest thousandth, if necessary. $$\frac{21}{84}$$

Step-by-Step Solution

Answer

The answer is 0.25.

Step 1: Understanding the Problem

We need to divide 21 by 84 and then round the result to the nearest thousandth, if necessary.

Step 2: Perform the Division

We calculate \(\frac{21}{84}\). Performing this division gives us 0.25, since 21 divided by 84 is equal to 0.25.

Step 3: Rounding the Result

Since 0.25 already has no digits beyond the tenths place, rounding it to the nearest thousandth does not change its value.

Key Concepts

Division

Division is a fundamental operation in mathematics where you determine how many times one number is contained within another. In this problem, we are dividing 21 by 84. To perform division, you can think of it as evenly distributing 21 units into 84 groups. The result, which is 0.25 in this case, tells us that if we divide 21 into 84 equal parts, each part would be 0.25.
Understanding division as finding the ratio of two numbers can also be helpful. The ratio of 21 to 84 is the same as 0.25, which means 21 is 25% of 84.

• When dividing, the first number (21 in our case) is called the 'dividend'.
• The second number (84) is known as the 'divisor'.
• The result of this operation is the 'quotient'.

Rounding Numbers

Rounding numbers is the process of adjusting a number to make it simpler or to fit it into a certain level of precision. For example, rounding to the nearest thousandth means you want to have only three digits after the decimal point.
In this exercise, when we divided 21 by 84, we got 0.25. If you need to round 0.25 to the nearest thousandth, you look at the digit in the fourth decimal place. Here, 0.25 only has two decimal places, which is the tenths and hundredths, meaning no rounding is necessary. When rounding:

• If the digit after your target place is 5 or more, you add 1 to the target digit.
• If it is less than 5, you do not alter the target digit.
• This makes 0.250 still 0.25 when rounded to the thousandth place.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common divisors other than 1. In this exercise, the fraction \( \frac{21}{84} \) simplifies. To simplify, find the greatest common divisor (GCD) of the numerator (21) and the denominator (84). Here, both 21 and 84 share a GCD of 21.
You simplify by dividing both the numerator and the denominator by their GCD:

• Divide 21 by 21 to get 1.
• Divide 84 by 21 to get 4.

Thus, the fraction \( \frac{21}{84} \) simplifies to \( \frac{1}{4} \), which perfectly corresponds with the division result of 0.25.