Problem 9 - Algebra 1

Question

You work a different number of hours each day. The table shows your total pay p and the number of hours h you worked. $$\begin{array}{|l|c|c|c|c|}\hline \text { Total pay, } p & \$ 18 & \$ 42 & \$ 48 & \$ 30 \\\\\hline \text { Hours worked, } h & 3 & 7 & 8 & 5 \\\\\hline \text { Ratio } & ? & ? & ? & ? \\\\\hline\end{array}$$ Copy and complete the table by finding the ratio of your total pay each day to the number of hours you worked that day.

Step-by-Step Solution

Answer

The ratio is \$6 per hour for each day.

Step 1: Understanding ratios

A ratio is a way of comparing two or more quantities. In this case, we're comparing total pay to hours worked, so the ratio will be obtained by dividing the total pay by the number of hours worked.

Step 2: Calculating the ratio for the first day

For the first day, the total pay is \$18 and the number of hours worked is 3. Divide \$18 by 3 to find the ratio. Thus, the ratio is 18/3 = \$6 per hour.

Step 3: Calculating the ratio for the second day

For the second day, the total pay is \$42 and the number of hours worked is 7. Divide \$42 by 7 to find the ratio. Thus, the ratio is 42/7 = \$6 per hour.

Step 4: Calculating the ratio for the third day

For the third day, the total pay is \$48 and the number of hours worked is 8. Divide \$48 by 8 to find the ratio. Thus, the ratio is 48/8 = \$6 per hour.

Step 5: Calculating the ratio for the fourth day

For the fourth day, the total pay is \$30 and the number of hours worked is 5. Divide \$30 by 5 to find the ratio. Thus, the ratio is 30/5 = \$6 per hour.

Key Concepts

Proportional Reasoning

When working with ratios, proportional reasoning becomes a powerful tool. If two quantities are proportional, they change at a constant rate with one another. This means that if you double one quantity, the other also doubles. In this exercise, proportional reasoning allows us to identify a consistent relationship between pay and hours worked.
This relationship is expressed in the form of a ratio. A ratio compares two numbers, showing how much of one there is compared to the other. Here, for each day, we divide total pay by hours worked to get dollars per hour. The beauty of proportional reasoning is that when the ratios are equal across different sets of data, it confirms that the relationship remains consistent
For example:

• On the first day: Ratio is 18/3 = 6
• On the second day: Ratio is 42/7 = 6

These constant ratios tell us that for every extra hour worked, you're earning the same amount of money each time. This is what makes the relationship proportional.

Division

Understanding division within ratios can make seemingly complex ideas much more approachable. Division is key to calculating ratios. It breaks a larger total into equal parts. In the context of pay versus hours, each division tells you how much is earned per hour—fundamental for calculating pay efficiency.
To find how much you earn per hour, you divide the amount earned by the number of hours worked. This division means splitting your earnings into equal hourly amounts. Let's take a look at how this works:

• For a total pay of $48 in 8 hours, divide: 48 ÷ 8 = 6
• For a total pay of $30 in 5 hours, divide: 30 ÷ 5 = 6

The result, "6," represents dollars per hour. Each division helps you grasp how much you earn 'per one' unit of time. This understanding of division is a basic yet crucial element of ratio calculation.

Problem Solving

Problem solving with ratios can empower your daily decision-making, especially in practical scenarios. It allows you to evaluate efficiency and compare results effectively. Like in our exercise, we used ratios to determine the consistency of earnings across different days.
Steps to solve problems using ratios include:

• Identifying the quantities you want to compare (here, total pay and hours worked).
• Calculating the ratio (or rate) by dividing one quantity by the other.
• Checking if the calculated ratio remains consistent throughout different scenarios.

This kind of problem solving is broadly applicable. By understanding the process, you can apply it not just in hourly wages, but in various real-life situations, like calculating speeds, prices per item, or costs per distance. Recognizing these scenarios helps enhance your analytical skills and problem-solving abilities.