Problem 72 - Prealgebra

Question

Express each ratio as a unit rate. Round to the nearest hundredth, if necessary. \(\$ 22\) in 5 hours

Step-by-Step Solution

Answer

The unit rate is \( \$4.4 \) per hour.

Step 1: Understand the Problem

We are given a ratio: \( \$22 \) in 5 hours. A unit rate expresses this ratio with 1 as the denominator. In this problem, the unit rate will be the amount of money per hour.

Step 2: Set Up the Equation

To find the unit rate, we need to divide the total amount of money, which is \(\\(22\), by the number of hours, which is 5. Therefore, we will calculate \(\frac{\\)22}{5\,\text{hours}}\).

Step 3: Perform the Division

Divide 22 by 5 to convert the amount of money (\\(22) in 5 hours into a rate per hour. \[ \frac{\\)22}{5} = 4.4 \]

Step 4: Present as a Unit Rate

The result from the division is 4.4, so the unit rate is \( \$4.4 \) per hour. There's no need to round further as it is already at the hundredth place.

Key Concepts

Understanding Ratios

Ratios are a mathematical way of comparing two quantities. They show the relative size of one quantity in relation to another. In the real world, ratios are everywhere: from mixing recipes to comparing speeds. For instance, when we say "22 dollars in 5 hours," we are essentially providing a ratio. This ratio tells us how 22 dollars spread over 5 hours. To work with ratios effectively, understanding their representations is key:

• Ratios can be written as fractions, such as \( \frac{22}{5} \), or with a colon, like 22:5.
• The numbers in a ratio tell us the proportion of the quantities they represent.

Learning to manipulate ratios enables you to convert them into a form that might be more useful, such as a unit rate.

The Role of Division in Unit Rates

Division is the mathematical operation we use to transform a ratio into a unit rate. A unit rate helps us understand the cost or quantity of one single unit, allowing for easier comparison. In our problem, division is used to determine how much money is earned in one hour. The process is straightforward:

• Take the total quantity and divide it by the associated number of units.
• Use division to solve for one unit, such as finding "money per hour" in our exercise.
• Division helps simplify ratios to make them accessible and meaningful.

Once divided, the unit rate provides a clear picture of the situation and allows you to make decisions based on single units.

Fractions in Mathematics

Fractions are a vital part of mathematics and have an important role in solving real-world problems. They represent parts of a whole and are integral in calculations involving division and ratios. In our example:

• We express the ratio \( \frac{22}{5} \) as a fraction, which is then simplified through division.
• Fractions provide a precise representation of the ratio and allow arithmetic operations to yield concrete answers.

Fractions enable complex numerical situations to be broken down into simpler, understandable parts, making them a preferred method for mathematicians when dealing with unit rates.

Mathematics Education and Real-World Applications

Understanding how to calculate unit rates is more than just an academic exercise. In mathematics education, it bridges the gap between numbers and their practical applications. Calculating unit rates is seen in various scenarios:

• Determining the price per item when shopping.
• Calculating speeds, such as kilometers per hour, in science or travel.
• Budgeting and financial planning in economics and personal finance.

Learning these skills empowers students to apply mathematical concepts in daily life, enhancing their problem-solving abilities and decision-making skills. Education that incorporates these concepts encourages critical thinking and prepares learners for the challenges of real-world scenarios.