Problem 18 - Elementary Algebra Within Reach

Question

You earn \(\$ 12\) per hour working at a grocery store. You receive a \(10 \%\) raise in pay. Show how you can use the Distributive Property to find your new hourly pay rate.

Step-by-Step Solution

Answer

The new hourly wage after a 10% raise is \$13.20.

Step 1: Express the raise as a decimal

Firstly, express the raise of 10% as a decimal. This is done by dividing by 100, turning the 10% into 0.10.

Step 2: Set up and apply the Distributive Property

The Distributive Property, a(b + c) = ab + ac, is set up to calculate the new pay rate. In this scenario, the equation for the new hourly wage rate becomes (hourly rate)*(1+raise) where the hourly rate is $12 and the raise is 0.10. Substituting these values will give (12)(1+0.10).

Step 3: Carry out the operations

After the substitution, perform the operations. First, add 1 + 0.10 to give 1.10. Then multiply 12 by 1.10 to yield \$13.20.

Step 4: Interpret the result

The resulting value from the previous step is \$13.20. This means after the 10% raise, the new hourly pay rate becomes \$13.20.

Key Concepts

Understanding Percentage Increase

A percentage increase is a way to show how much something has grown in size compared to its original amount. Consider an initial amount and a change. By using percentages, we represent this change relative to the original amount, making it easy to understand growth.

To calculate a percentage increase:

• First, determine the original value and the new value.
• Subtract the original value from the new value to find the difference.
• Express this difference as a percentage of the original value by dividing the difference by the original value and then multiplying by 100.

The formula looks like this: \[\text{Percentage Increase} = \left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100\]This method is used when you want to express how much more or less something has become, in terms of percentage. In the context of wages, if you are given a raise, that's calculated as a percentage increase over your current wage this rate helps easily communicate changes in financial terms.

Calculating Hourly Wage After a Raise

When you receive a raise, it directly impacts your hourly wage, increasing your total income. The task is to calculate your new hourly wage by applying the percentage raise. To do this efficiently, you can utilize the Distributive Property.

Start by expressing the percentage raise as a decimal (just divide by 100). Then use the equation:\[\text{New Hourly Wage} = \text{Old Hourly Wage} \times (1 + \text{Decimal Raise})\]For example, earning \(12 per hour with a 10% raise means:

• Convert 10% to a decimal: \(0.10\).
• Plug these into the equation: \((12) \times (1 + 0.10)\).

Using the Distributive Property, where \(a(b + c) = ab + ac\), this setup calculates the updated wage.Perform operations by first calculating inside the parentheses \(1 + 0.10 = 1.10\) and then multiply \(12 \times 1.10\) to get \\)13.20. This adjustment reflects your new earnings for each hour of work.

Using Decimal Representation of Percentages

When working with percentages in mathematical calculations, it's critical to convert percentages into decimals. This conversion simplifies computations and allows equations like those used in financial calculations to be straightforward.

To convert any percentage to a decimal, divide by 100. So for a 10% raise:

• Take 10 and divide it by 100.
• This results in 0.10, which is the decimal representation of 10%.

This method is applicable universally:

• To represent 25%, convert it to \(0.25\).
• For 50%, reduce it to \(0.50\).

Essentially, moving the decimal point two places to the left changes a percentage into a decimal. This practice is especially useful in quick calculations, enabling you to plug decimal values into equations effortlessly. Understanding how to seamlessly convert percentages into decimals can enhance your mathematical efficiency and accuracy.