Question

Two workers have different pay scales. Worker A receives $\mathrm{\$}50$ for any day worked plus $\mathrm{\$}15$ per hour. Worker $B$ receives $\mathrm{\$}27$ per hour. Both workers may work for a fraction of an hour and be paid in proportion to their respective hourly rates. If worker A arrives at 9: 21 a.m. and receives the $\mathrm{\$}50$ upon arrival and Worker $\mathrm{B}$ arrives at 10: 09 a.m. assuming both work continuously, at what time would their earnings be identical? a. $11:09\mathrm{a}.\mathrm{m}.$ b. $12:01\mathrm{p}.\mathrm{m}.$ c. $2:31\mathrm{p}.\mathrm{m}.$ d. $3:19\mathrm{p}.\mathrm{m}.$ e. $3:36\mathrm{p}.\mathrm{m}.$

Short answer

3:19 p.m.

Step-by-step solution

- Define Earnings for Worker A

Worker A earns a base pay of $\mathrm{\$}50$ plus $\mathrm{\$}15$ per hour worked. Let $t$ be the number of hours worked after arriving at 9:21 a.m. Hence, Worker A's earnings as a function of time $t$ in hours can be written as: $$E_A=50+15t$$

- Define Earnings for Worker B

Worker B earns $\mathrm{\$}27$ per hour worked. Worker B arrives at 10:09 a.m., which is 48 minutes or 0.8 hours after Worker A. Let $t$ be the number of hours Worker A has worked. Worker B's earnings as a function of time $t$ can be written as: $$E_B=27(t-0.8)$$ because Worker B starts working 0.8 hours after Worker A.

- Set Earnings Equal

To find the time when their earnings are identical, set $E_A=E_B$: $$50+15t=27(t-0.8)$$

- Solve for t

Solve the equation $50+15t=27(t-0.8)$: $$50+15t=27t-21.6$$ $$\Rightarrow 50+15t=27t-21.6$$ $$\Rightarrow 50+21.6=27t-15t$$ $$\Rightarrow 71.6=12t$$ $$\Rightarrow t=\frac{71.6}{12}\approx 5.967$$ Therefore, Worker A has worked for approximately 5.967 hours since 9:21 a.m.

- Convert Hours to Time

Convert 5.967 hours to hours and minutes: $$0.967\times 60\approx 58$$ minutes. Thus, Worker A has worked for 5 hours and 58 minutes. Add this time to Worker A's start time of 9:21 a.m.: $$9:21\text{ a.m.}+5:58\text{ hours}=3:19\text{ p.m.}$$ Therefore, the time when both workers' earnings are identical is 3:19 p.m.

Key concepts

Earnings Calculation

Understanding how to calculate earnings is crucial in many real-life scenarios, especially when dealing with pay rates that involve both a base rate and an hourly wage. In this problem, Worker A has a base pay of $\mathrm{\$}50$ plus an additional $\mathrm{\$}15$ per hour worked. Worker B, on the other hand, earns $\mathrm{\$}27$ per hour with no base pay. To find the total earnings for each worker, you need to account for both parts in Worker A's case and just the hourly rate for Worker B. Understanding and being able to set up these calculations correctly can significantly help when solving complex work and pay rate problems.

Work Rate

The concept of work rate deals with how much work is done or how much money is earned within a given period, usually per hour. In this exercise, Worker A works from 9:21 a.m. and earns $\mathrm{\$}15$ per hour after the initial $\mathrm{\$}50$ base pay. Worker B starts working 48 minutes later, at 10:09 a.m., earning $\mathrm{\$}27$ per hour. Calculating the work rate can involve different start times and adjusting the formulas accordingly to integrate the delayed starting time. This helps determine who earns more over the same period and can show the exact time when their earnings match.

Linear Equations

Linear equations are used to represent relationships between quantities. They are especially useful in problems involving rates and time. In this example, we set the earnings of Worker A equal to the earnings of Worker B to find the specific point in time when both workers earn the same amount. The equation $50+15t=27(t-0.8)$ is at the heart of this calculation. Solving this linear equation involves combining like terms and isolating the variable $t$. Learning how to manipulate and solve these equations is fundamental in algebra and applies directly to financial calculations involving salaries and hours worked.

Time Conversion

Converting time from hours and minutes to a decimal format and vice versa is essential in solving problems with different starting points and work durations. In this case, we need to convert 48 minutes to its decimal form (0.8 hours) and later, the calculated 5.967 hours back into hours and minutes. Understanding these conversions allows students to accurately add or subtract times, ensuring the final answer is precise. For instance, converting 0.967 hours to minutes involves multiplying by 60, resulting in approximately 58 minutes, which is then added to the initial time of 9:21 a.m. to find the final time of 3:19 p.m.