Question

Working for 4 hours a day, a typist earns $$\$ 65.40$$ a day after taxes. At the same rate of pay, what would he earn per day if he worked for 7 hours a day? (Let \(N\) represent after-tax earnings.) A. \(N=\frac{4}{7}(65.40)\) B. \(N=\frac{7}{4}(65.40)\) C. \(N=4(65.40)\) D. \(N=7(65.40)\)

Short answer

B. \(N=\frac{7}{4}(65.40)\)

Step-by-step solution

Find the typist's hourly rate

The typist earns $65.40 a day after working for 4 hours. To find the typist's hourly rate, divide their daily earnings by the number of hours worked: Hourly rate = \(\frac{Total \ earning}{Hours \ worked}\) \[Hourly \ rate = \frac{65.40}{4}\]

Calculate the typist's earnings for 7 hours

Now, multiply the hourly rate by the new number of hours worked (7 hours) to find the typist's new daily earnings: \(N = Hourly \ rate \times Hours \ worked\) \[N = \frac{65.40}{4} \times 7\]

Simplify the expression

Simplify the expression by multiplying 7 with the top value in the fraction and 4 with the bottom value in the fraction: \[N = \frac{7 \times 65.40}{4}\] Now, we can rewrite this expression as: \[N = \frac{7}{4}(65.40)\] So, the answer is B. \(N=\frac{7}{4}(65.40)\).

Key concepts

Ratio and Proportion

When we talk about ratio and proportion, we are discussing the relationship between two numbers. In this problem, we need to determine how a change in one value affects another related value.

A ratio is a way to compare two quantities by division. In our exercise, the ratio is between the number of hours worked and the earnings. The proportion reflects how this ratio can be applied to different numbers of hours to find corresponding earnings.

• To find the full day earnings for 7 hours, we use the ratio of earnings to time, as calculated by dividing the original earnings by the original hours worked, and then applying this to the new number of hours.
• Using ratios is essential in problems where comparisons are made between time, work, and income or any quantities that change together proportionally.

In simpler terms, if something increases or decreases, the related quantity will adjust proportionately.

Mathematical Reasoning

Mathematical reasoning is the thought process required to approach and solve problems logically. In this exercise, we use mathematical reasoning to figure out how the earnings change with different hours worked.

First, we identify that the typist's earnings are based on time worked. From there, we calculate the hourly rate of earnings, which involves dividing the overall earnings by the number of work hours. This gives us an understanding of how much the typist earns per hour.

• This hourly rate forms the basis from which we can extend or reduce the earnings calculation to different amounts of work hours.
• By knowing the hourly rate, we can multiply this by any number of work hours to extrapolate the total earnings for that period.

This application of mathematical reasoning helps us logically extend given data to solve for unknowns in a structured manner.

Problem-solving Strategies

Problem-solving strategies involve identifying patterns, setting clear steps for finding a solution, and verifying our final answer. Here, step-by-step calculation and clear reasoning offer a solid path to the correct solution.

• Firstly, determining necessary information to solve for the hourly wage is crucial. Without this, calculating earnings for any other number of hours accurately wouldn’t be possible.
• Next, using multiplication with the known hourly rate spreads out the earnings over different hours, precisely giving us what we're asked to find – the rate for 7 hours.
• Lastly, rechecking our calculations ensures consistency and correctness, reinforcing our solution pathway. This confirms that both the arithmetic and logical reasoning employed are sound.

Engaging these problem-solving techniques allows students to not only solve problems but to understand the mechanics behind their methods, ensuring deeper mastery for future problems.