Problem 22 - Algebra 1: Concepts and Skills

Question

Find the unit rate. Earn 126 dollars for working 18 hours

Step-by-Step Solution

Answer

The unit rate is $7 per hour.

Step 1: Understand the problem

In this problem, a person earns $126 for 18 hours of work. The unit rate is the amount earned for each single unit of work - in this case, each single hour.

Step 2: Apply the formula for unit rate

The formula for unit rate is \(Unit\:Rate = \frac{Total\:Amount}{Number\:of\:Units}\). In this case, 'Total Amount' is the total pay ($126), and 'Number of Units' is the hours worked (18). Substituting these values in, we have \(Unit\:Rate = \frac{126}{18}\).

Step 3: Calculate the unit rate

By dividing 126 by 18, we calculate the unit rate value.

Key Concepts

Division

Division is a fundamental mathematical operation that allows us to distribute a quantity into equal parts. When solving problems involving division, you split a whole into smaller parts or groups. In the context of calculating a unit rate, division helps us determine the value of a single unit.

Let's think of division in this way: if you have 126 apples that you want to share equally among 18 friends, you would use division to find out how many apples each friend would get. The total number of apples (126) is divided by the number of friends (18).

• Dividend: The total amount or whole, here it is 126 dollars.
• Divisor: The number of equal parts to divide the dividend into, here it is 18 hours.
• Quotient: The result of the division, which is the amount earned per hour in this example.

Using division, you can easily solve for the unit rate by computing the quotient of the total earnings divided by the total hours worked, which gives you the pay per hour.

Ratios

Ratios are a way to compare two quantities, showing the relative size of one to the other. They can help us understand how much of one thing exists in relation to another. In our problem, we are assessing the amount of money earned in relation to time spent working.

In a ratio, the unit rate is a particular case where one part of the ratio is reduced to one unit. For example, in the problem, we start with the ratio of 126 dollars to 18 hours or 126:18.

• Simplified Ratios: Ratios can be simplest by dividing both terms by the greatest common divisor.
• Unit Ratios: When the second term is one, it's called a unit ratio. For unit rate calculation, we simplify 126:18 to find the rate per 1 hour.

Ratios are essential in converting a total quantity into a rate of one unit, which makes understanding comparisons much easier.

Word Problems

Word problems in mathematics are scenarios explained in sentences that describe relationships between numbers and require problem-solving skills. Often, they involve real-life situations that need to be translated into mathematical operations, like in the given earnings and hours problem.

When approaching a word problem, it's crucial to:

• Comprehend the information: Recognize what is given and what you need to find out.
• Identify operations: Decide which mathematical operations are required for solving the problem. Here, we need division.
• Translate into numbers: Convert the words into numerical expressions or equations.

Once you understand the problem and have determined the needed operations, solving becomes straightforward. Word problems help develop critical thinking as they might not always present the numbers in a straightforward manner, requiring strategic planning to reach a solution.