Problem 11 - 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. $$\begin{array}{|l|c|c|c|c|}\hline \text { Total pay, } p & \$ 18 & \$ 42 & \$ 48 & \$ 30 \\\\\hline \text { Hours worked, } h & 3 & 7 & 8 & 5 \\\\\hline \text { Ratio } & ? & ? & ? & ? \\\\\hline\end{array}$$ If you work 6 hours on the fifth day, what will your total pay be?

Step-by-Step Solution

Answer

The short answer will be found by multiplying 6 (number of hours) by the constant ratio that we obtained since it's confirmed that there's a direct correlation between 'Total pay' and hours worked.

Step 1: Find ratio

Calculate the ratio of 'Total pay' to 'Hours worked'. This can be done by dividing each 'Total pay' by its corresponding 'Hours worked'. For instance, the ratio for the first column is obtained by dividing the total pay, $18, by the hours worked, 3. That is, \(\frac{18}{3} = 6\). Similarly, the other ratios can be calculated.

Step 2: Validate consistent ratio

Check if the ratio calculated in step 1 is maintained in all columns. If all ratios are equivalent, this validates the direct correlation between the 'Total pay' and 'Hours worked'.

Step 3: Compute total pay for 6 hours

Using the consistent ratio identified, compute the total pay for working 6 hours. If the ratio is 'a', then the total pay would be calculated by the formula \(6a\).

Key Concepts

Proportional Relationships

Understanding proportional relationships is essential in algebra, as they represent the concept of having a constant rate or ratio between quantities. Proportionality appears in various forms and scenarios, such as scaling objects in geometry, converting currencies in real-world financial problems, and analyzing rates and densities in science.

As seen in our textbook example, the relationship between total pay and hours worked is proportional if the pay per hour remains the same no matter how long you work. In other words, if you earn a specific amount for one hour, earning for three hours would be thrice that amount, and so on. This is how we detect a proportional relationship in a table format: by dividing the total pay by the hours worked and observing if the resulting ratio is consistent across the board.

Let's highlight a few characteristics of proportional relationships:

• The ratio between corresponding measurements in two quantities remains constant.
• Graphically, proportional relationships are represented by a straight line passing through the origin.
• With proportional relationships, we can predict one quantity when given the other, knowing they change at a constant rate.

Understanding this concept enables students to analyze and solve a wide array of algebraic problems efficiently.

Solving Ratio Problems

Solving ratio problems is a fundamental skill in algebra that involves finding the relation between two comparable quantities. Ratios can be represented in different forms: as fractions, decimals, or using a colon (e.g., 3:1). To solve problems efficiently, the key is to understand the underlying proportionality and use it to find an unknown quantity based on known values.

For instance, if you're faced with a table showing payment and hours worked, like in the provided exercise, you can solve for an unknown payment by first determining the pay per hour. This is done by dividing the total pay by hours worked to get the ratio. Once the consistent ratio is found, it's simply a matter of multiplying that value by the number of hours in question.

Here’s a simplified process to solve ratio problems:

• Identify the two quantities and their units.
• Write down the known ratios.
• Find a consistent ratio, if one exists.
• Use this ratio to find unknown values.

Having a clear step-by-step approach will enable students to tackle varying ratio problems with confidence.

Direct Variation

Direct variation is a concept in algebra where two quantities increase or decrease together at the same rate. When one variable changes, the other changes in a way that maintains a constant ratio between them. This is a specific type of proportional relationship where one variable is a constant multiple of the other.

Direct variation can be expressed mathematically as: \(y = kx\), where \(k\) is the constant of variation, \(y\) is the dependent variable, and \(x\) is the independent variable. In the context of our exercise, the total pay \(p\) varies directly with the hours worked \(h\), hence the equation \(p = kh\) where \(k\) represents the pay per hour.

Several key points to note about direct variation include:

• If one variable goes up, the other goes up proportionally, and vice versa.
• The graph of a direct variation equation is always a straight line passing through the origin.
• An important step in solving problems involving direct variation is identifying the constant ratio or the 'k' value.

Once the 'k' value is known, predicting outcomes for one variable given the other becomes straightforward, a concept which is not only mathematically intriguing but also immensely practical.