Problem 7 - Prealgebra

Question

The formula \(G=H \cdot R\) tells us how much gross pay \(G\) a person receives for working \(H\) hours at an hourly rate of pay \(R\).find \(G\). $$H=30 \text { hours and } R=\$ 9 \frac{1}{2}$$

Step-by-Step Solution

Answer

The gross pay \( G \) is 285 dollars.

Step 1: Understand the Formula

The formula given is \( G = H \cdot R \), where \( G \) is the gross pay, \( H \) is the number of hours worked, and \( R \) is the hourly rate. We need to find \( G \) using the values of \( H \) and \( R \) provided.

Step 2: Convert Mixed Number to Improper Fraction

Convert 9\( \frac{1}{2} \) to an improper fraction. \( 9 \frac{1}{2} = 9 + \frac{1}{2} = \frac{18}{2} + \frac{1}{2} = \frac{19}{2} \). So, \( R = \frac{19}{2} \). This will make multiplication with \( H \) easier.

Step 3: Multiply Hours by Hourly Rate

Substitute \( H = 30 \) and \( R = \frac{19}{2} \) into the formula: \( G = 30 \cdot \frac{19}{2} \). Multiply \( 30 \) by \( \frac{19}{2} \).

Step 4: Simplify the Multiplication

Perform the multiplication: \( 30 \cdot \frac{19}{2} = \frac{30 \times 19}{2} \). First, calculate \( 30 \times 19 = 570 \), and then divide by \( 2 \).

Step 5: Final Calculation

Divide 570 by 2 to find \( G \): \( \frac{570}{2} = 285 \). Thus, \( G = 285 \). The gross pay \( G \) is 285 dollars.

Key Concepts

Gross Pay Calculation

Calculating gross pay means figuring out how much money someone earns before any deductions like taxes or insurance. To find the gross pay, you need to know two main things:

• How many hours the person worked, which we call 'H'.
• The hourly rate of pay, which we call 'R'.

The formula for gross pay looks like this: \[G = H \cdot R\]where

• \(G\) is the gross pay,
• \(H\) is the number of hours worked,
• and \(R\) is the hourly rate.

For example, if someone worked 30 hours at a rate of \(\$9.50\) per hour, calculating their gross pay involves multiplying these two numbers.
Simply use the formula: \(G = 30 \cdot 9.5\). The answer, which is 285, gives us their gross pay in dollars.

Multiplication of Fractions

Multiplying fractions might seem tricky at first, but it really involves doing two simple steps:

• Multiply the numerators (the top numbers) together.
• Multiply the denominators (the bottom numbers) together.

When a whole number multiplies a fraction, we can think of the whole number as a fraction over 1.
Example: If you have 30 times \(\frac{19}{2}\), you can rewrite it as:\[\frac{30}{1} \cdot \frac{19}{2} = \frac{30 \times 19}{1 \times 2} = \frac{570}{2}\]This results from multiplying 30 by 19 and 1 by 2, then putting the results over each other.
The final step is often simplifying this new fraction to get your final answer.

Conversion to Improper Fractions

Converting mixed numbers into improper fractions is super useful with multiplication.
A mixed number is a combination of a whole number and a fraction, like \(9 \frac{1}{2}\). To convert it:

• Multiply the whole number by the fraction’s denominator.
• Add this result to the fraction’s numerator.
• Put the result over the original denominator.

For instance: \[9 \frac{1}{2} = \frac{19}{2}\]Here's how:

• Multiply 9 (the whole number) by 2 (the denominator) to get 18.
• Add the initial fraction's numerator, 1, to get 19.
• This gives \(\frac{19}{2}\).

Converting in this way helps keep calculations straightforward, especially in operations like multiplication.