Problem 8 - 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 \(G\). $$H=20 \text { hours and } R=\$ 6 \frac{3}{4}$$

Step-by-Step Solution

Answer

The gross pay \( G \) is 135 dollars.

Step 1: Understand the Formula

First, identify what the given formula represents. In this case, the formula \( G = H \cdot R \) represents the gross pay \( G \), which is calculated by multiplying the number of hours worked \( H \) by the hourly rate \( R \).

Step 2: Convert Mixed Number to an Improper Fraction

Convert the hourly rate \( R = 6 \frac{3}{4} \) into an improper fraction for easier multiplication. Multiply the whole number (6) by the denominator (4) and add the numerator (3): \[ 6 \frac{3}{4} = \frac{6 \times 4 + 3}{4} = \frac{27}{4} \]

Step 3: Substitute the Values into the Formula

Substitute \( H = 20 \) hours and \( R = \frac{27}{4} \) into the formula \( G = H \cdot R \): \[ G = 20 \times \frac{27}{4} \]

Step 4: Perform the Multiplication

Multiply \( 20 \) by \( \frac{27}{4} \). You can simplify the multiplication by first performing the division: \[ G = \frac{20 \times 27}{4} = \frac{540}{4} \]

Step 5: Simplify the Fraction

Simplify the fraction \( \frac{540}{4} \) by dividing the numerator by the denominator: \[ \frac{540}{4} = 135 \]

Key Concepts

Multiplying Fractions

Multiplying fractions might seem tricky at first, but it's actually quite simple once you get the hang of it. The process involves two main steps: multiplying the numerators (top numbers) and then multiplying the denominators (bottom numbers). If they can be simplified, that's even better, as simpler fractions are easier to understand and use.

• To multiply two fractions, such as \( \frac{a}{b} \) and \( \frac{c}{d} \), multiply them straightforwardly: \( \frac{a \times c}{b \times d} \).
• There's no need to find a common denominator, which is one of the common confusions with adding or subtracting fractions.
• Always remember to simplify the resulting fraction if possible, by dividing both the numerator and denominator by their greatest common factor (GCF).

When dealing with fractions in equations, like finding gross pay by multiplying an hourly rate by hours worked, exact results matter. So, after calculations, check if your fraction looks nice and clean, as a simplified fraction is much easier to understand.

Mixed Numbers

A mixed number is a whole number combined with a fraction, like \( 6 \frac{3}{4} \). When working with mixed numbers in multiplication, it's essential to convert them into improper fractions first. Here's why and how to do it:

• Mixed numbers can make multiplication and division cumbersome; improper fractions streamline the process.
• For conversion: multiply the whole number by the denominator of the fraction part, then add the numerator. Place this result over the original denominator. For \( 6 \frac{3}{4} \), it becomes \( \frac{27}{4} \), since \( 6 \times 4 + 3 = 27 \).
• This conversion is crucial for performing operations involving mixed numbers, like calculating gross pay in our scenario.

Once converted, you can multiply just as you would with any other fraction. Converting back to a mixed number form after calculations might be necessary if the answer needs to be presented in that format.

Prealgebra Concepts

Prealgebra introduces foundational math skills necessary for solving real-world problems, such as calculating gross pay. These basics involve understanding and manipulating numbers and operations. Here’s how prealgebra concepts can help:

• Focus on basic arithmetic, simple algebraic expressions, and understanding fractions and decimals.
• Learn how to convert numbers, like mixed numbers to improper fractions, to ease calculations.
• Apply formulas in practical scenarios; for example, using \( G = H \cdot R \) teaches how abstract numbers apply to everyday situations like earning money.
• Number sense forms a big part of prealgebra, enabling you to estimate and check the reasonableness of answers.

By mastering these prealgebra concepts, students build a solid base for further studies in algebra, geometry, and beyond, where these basic principles will always be in frequent use.