Problem 43 - Prealgebra

Question

Express each rate as a unit rate. Round to the nearest tenth, if necessary. 183.4 miles in 3.2 hours

Step-by-Step Solution

Answer

The unit rate is 57.3 miles per hour.

Step 1: Identify the given values

We are given that 183.4 miles is covered in 3.2 hours. Our task is to express this rate as a unit rate, which means finding the rate per 1 hour.

Step 2: Set up the rate as a division

To find the unit rate, we set up the given values as a division. We divide the total miles by the total number of hours to find how many miles are traveled in one hour. This will be \( \frac{183.4 \, \text{miles}}{3.2 \, \text{hours}} \).

Step 3: Perform the division

Calculate the division: \( 183.4 \div 3.2 \). First, move the decimal to convert \(3.2\) into a whole number (32), and do the same to the numerator, making it \(1834\). Now perform the division: \( 1834 \div 32 = 57.3125\).

Step 4: Round to the nearest tenth

Round the result from step 3 to the nearest tenth. The number 57.3125 rounds to 57.3 when rounded to one decimal place, since the digit in the hundredths place (1) is less than 5.

Key Concepts

Rounding Decimals

Rounding decimals is an essential skill that's often used to simplify numbers and make them easier to work with, especially in real-world scenarios. When you round a decimal, you are adjusting it to a simpler, approximate value, while trying to maintain its original data's significance. Consider the process of rounding to the nearest tenth.

• Identify the digit in the tenths place. In our example, that's the digit immediately after the decimal point.
• Now, look at the digit to the right of the tenths place, which is in the hundredths place.
• If the hundredths digit is 5 or more, increase the tenths digit by one. If it's less than 5, keep the tenths digit the same.

For instance, in the number 57.3125, the tenths digit is 3, and the hundredths digit is 1. Since 1 is less than 5, we leave the tenths digit as 3, resulting in the rounded number 57.3. This simplification is crucial for easier communication and calculation, especially when handling large data sets or presenting results.

Division in Math

Division in Math is a fundamental operation that is used to determine how many times one number is contained within another, essentially distributing a number into equal parts. Here’s how to approach division when a decimal is involved, like in the given exercise:

• First, make the divisor a whole number if it's a decimal. Do this by shifting the decimal point to the right.
• Apply the same operation to the dividend to maintain the equation's balance.
• Perform the division as you usually would with whole numbers.
• If necessary, use long division for dividing larger numbers or to reach a specific level of precision.

In our example, we initially have to change 3.2 into 32 by moving the decimal point. Consequently, adjust the dividend from 183.4 to 1834 to match the change. Performing the long division of 1834 by 32, you reach the quotient 57.3125. This method helps to simplify the division process and ensures your result is both precise and comprehensible in everyday applications.

Real-World Math Application

Math is not just an abstract concept; it's incredibly applicable in the real world, influencing various fields and everyday tasks. One such example is calculating unit rates, which is crucial for understanding speed, efficiency, and rates in activities ranging from travel to cooking. Consider traveling as in our unit rate exercise, where knowing the speed (miles per hour) can help you plan your journey better. Understanding how far you can travel within a given period helps optimize routes, manage fuel consumption, or prepare for weather conditions.

• In business, unit rates help in pricing strategies, where knowing the cost per unit of production can inform pricing and profit calculations.
• In shopping, calculating the price per item or per ounce helps consumers make informed decisions.

Being adept at calculating and understanding unit rates, rounding, and division, empowers individuals to make efficient and intelligent decisions in their personal and professional lives. Thus, these math skills have a broad application making them indispensable in our daily routines.