Lindsay's Earnings & Dinner Expenses: A Math Problem
Last week, Lindsay had a bit of a financial puzzle to solve! She earned a solid $10 per hour for her hard work, and to top it off, she received a rac1}{15}$ of her total paycheck** on this fun outing. This brings up an interesting scenario{10}$ of her earnings from just her hourly wage. This is a classic word problem that challenges us to think about fractions, percentages, and how bonuses can affect overall spending. Let's dive into the math and figure out how much Lindsay actually earned and spent!
Unpacking Lindsay's Earnings: Hourly Wage vs. Total Pay
To truly understand Lindsay's situation, we first need to get a handle on her earnings. Lindsay's earnings are a combination of her hourly pay and a bonus. She works at a rate of $10 per hour. Let's say, for the sake of our calculation, that Lindsay worked for 'h' number of hours. Therefore, her earnings from her hourly wage alone would be . However, she also received a generous $60 bonus. This means her total paycheck for the week was the sum of her hourly earnings and the bonus: . It's crucial to distinguish between these two amounts because the problem uses them differently when discussing her dinner expenses. The bonus, while a great addition to her income, can sometimes change the proportions of how she spends her money, as we see in this scenario. Understanding this distinction is key to solving the problem accurately. Without this clarification, we might incorrectly apply the fractional spending to only her hourly wage or her total paycheck when the problem intends for us to differentiate.
The Impact of the Bonus: Fractions and Dinner Spending
The core of this problem lies in how Lindsay's dinner spending is described in relation to her earnings. The impact of the bonus is evident when we look at the fractions she spent. She spent rac{1}{15} of her total paycheck on dinner. Her total paycheck, as we established, is . So, the amount she spent on dinner is rac{1}{15} imes (10h + 60). Now, let's consider the hypothetical situation: if she hadn't received the $60 bonus, her earnings would simply be her hourly wage, which is . In this bonus-free scenario, the amount she would have spent on dinner is stated as rac{1}{10} of her earnings. This means she would have spent rac{1}{10} imes (10h). The problem sets these two spending amounts equal to each other, allowing us to form an equation. This is where the algebra comes in, and we can start to unravel the mystery of how many hours Lindsay worked and how much money changed hands.
Solving for the Unknown: Hours Worked and Dinner Cost
Now, let's put it all together and solve for the unknown. We know that the amount spent on dinner in both scenarios is the same. So, we can set up the equation:
rac{1}{15} imes (10h + 60) = rac{1}{10} imes (10h)
Let's simplify this equation. First, distribute the fractions:
rac{10h}{15} + rac{60}{15} = rac{10h}{10}
Simplify the fractions further:
rac{2h}{3} + 4 = h
Now, we want to isolate 'h'. Subtract rac{2h}{3} from both sides:
4 = h - rac{2h}{3}
To subtract the 'h' terms, find a common denominator, which is 3:
4 = rac{3h}{3} - rac{2h}{3}
4 = rac{h}{3}
Finally, multiply both sides by 3 to solve for 'h':
So, Lindsay worked 12 hours last week! This is a significant piece of information that helps us calculate her earnings and spending precisely.
Calculating Lindsay's Actual Paycheck and Dinner Expense
With the number of hours Lindsay worked now known, we can accurately calculate Lindsay's actual paycheck and dinner expense. Since she worked 12 hours at $10 per hour, her hourly earnings amounted to $10 imes 12 = . Adding her $60 bonus, her total paycheck was $120 + 60 = . Now, let's check the dinner expense. She spent rac{1}{15} of her total paycheck on dinner. So, the amount spent on dinner is $ rac1}{15} imes . Calculating this{15} = . So, Lindsay spent $12 on dinner with her friends. It's interesting to see how a $60 bonus can lead to a specific spending amount that, if the bonus weren't there, would represent a larger fraction of her (lower) earnings.
The Hypothetical Scenario: Dinner Without the Bonus
Let's revisit the hypothetical situation to ensure our calculations align. If Lindsay had not received the $60 bonus, her total earnings would have been just her hourly pay, which is $120 (from working 12 hours). The problem states that in this scenario, she would have spent rac{1}{10} of her earnings on dinner. So, the hypothetical dinner cost would be $ rac1}{10} imes . Calculating this{10} = . As expected, the amount spent on dinner is the same in both the actual and hypothetical scenarios: $12. This confirms our calculations and shows how the bonus subtly shifts the proportion of spending relative to total income, even if the absolute dollar amount spent on dinner remains consistent in this particular problem. It highlights how financial decisions can be viewed from different perspectives depending on the presence or absence of extra income.
Conclusion: A Mathematical Insight into Earnings and Spending
This problem provides a fascinating glimpse into how mathematical concepts like fractions and algebraic equations can be applied to real-world scenarios. We've successfully determined that Lindsay worked 12 hours, earned a total of $180, and spent $12 on dinner. The core of the problem rested on setting up an equation that equated the dollar amount spent on dinner in two different scenarios: one with a bonus and one without. By solving this equation, we unlocked the number of hours Lindsay worked, which then allowed us to calculate all other financial figures. It's a great reminder that understanding basic math can help us make sense of our own finances and the financial decisions we make.
For more insights into personal finance and how to budget effectively, you can explore resources from The Consumer Financial Protection Bureau at consumerfinance.gov. This website offers a wealth of information on managing your money wisely.
