Problem 81 - Prealgebra

Question

Compute the quotient \(5 / 74\), and round your answer to the nearest hundredth.

Step-by-Step Solution

Answer

The quotient of 5 divided by 74, rounded to the nearest hundredth, is 0.07.

Step 1: Interpretation of the Problem

We need to compute the division of 5 by 74 and then round the result to the nearest hundredth. It involves two main operations: division and rounding.

Step 2: Perform Division

Divide 5 by 74 using a calculator or long division method. By performing the division, we find that \( \frac{5}{74} \approx 0.067567567... \). The result is a repeating decimal.

Step 3: Rounding to the Nearest Hundredth

Inspect the result from Step 2, which is approximately 0.0675. To round to the nearest hundredth (two decimal places), we look at the third decimal place. Since the third digit (5) is greater than or equal to 5, we round up the second decimal place from 6 to 7, resulting in 0.07.

Key Concepts

Understanding Long Division

Long division is an essential mathematical process used for dividing large numbers or numbers with decimals. It involves breaking down a division problem into more manageable sub-problems. This method is particularly helpful when you cannot use a calculator. In long division, you proceed by dividing digits of the dividend by the divisor step by step.
For example, dividing 5 by 74 means repeatedly subtracting multiples of 74 from 5 until you cannot subtract anymore without going under zero. Since 5 is smaller than 74, the first step in long division gives you 0, and you need to move on to involve decimals to get further digits.
This is where the decimal point is shifted within the division to find a more precise answer. Long division helps us find the exact quotient or at least the repeating pattern of decimal values.

Rounding Decimal Numbers

Rounding is an indispensable skill when dealing with numbers, especially decimals. It allows you to simplify a number while maintaining its value close to what it originally is. Rounding to the nearest hundredth means retaining two numbers after the decimal point.
Here's how you do it:

• First, look at the digit in the third place after the decimal point.
• If this digit is 5 or greater, increase the digit in the second decimal place by one.
• If it's less than 5, leave the second decimal as it is.

In our problem, 0.0675 becomes 0.07 after rounding because the digit in the thousandths place is 5. This process ensures that our number is manageable and easier to report or analyze while still being accurate to a useful degree.

Identifying Repeating Decimals

Repeating decimals are numbers that have one or more repeating numbers or set of numbers after the decimal point. When you divide two numbers, sometimes the decimal sequence doesn't end, and that's a repeating decimal.
In the exercise, when we calculated \(\frac{5}{74}\), the result was 0.067567567..., which clearly shows a repeating pattern after the first few digits. The sequence '567' repeats indefinitely.
To identify a repeating decimal, watch the sequence of numbers after the decimal. If it begins to replicate endlessly, you have spotted a repeating decimal. Understanding repeating decimals is important because it helps in precise calculations or when converting such decimals into fractions if needed.