Question

Jay is purchasing gifts for his four friends' high school graduation. He has a budget of at most \(\$ 150 . \mathrm{He}\) is purchasing a restaurant gift card of \(\$ 25\) for one friend, a tool set that costs \(\$ 40\) for another friend, and a \(\$ 35\) college sweatshirt for a third friend. For his fourth friend, he wants to see how many \(\$ 0.25\) quarters \((Q)\) he can give for the friend to use for laundry money. What expression gives the range of quarters Jay can acquire given his budgetary restrictions? (A) \(1 \leq Q \leq 300\) (B) \(1 \leq Q \leq 200\) (C) \(10 \leq Q \leq 120\) (D) \(40 \leq Q \leq 60\)

Short answer

The expression is (B) \(1 \leq Q \leq 200\).

Step-by-step solution

Understand Jay's Total Budget

Jay has a total budget of $150 to spend on gifts for his four friends.

Calculate Total Cost of First Three Gifts

Jay is purchasing three gifts: a restaurant gift card for \(25, a tool set for \)40, and a college sweatshirt for \(35. Calculate the total cost: \[ 25 + 40 + 35 = 100 \] So, the total cost of the first three gifts is \)100.

Determine Remaining Budget for Quarters

Subtract the \(100 spent on the first three gifts from Jay's total budget:\[ 150 - 100 = 50 \] Jay has \)50 remaining for the quarters.

Calculate Maximum Number of Quarters

Each quarter costs \(0.25, so calculate the maximum number of quarters Jay can purchase with his remaining \)50:\[ \frac{50}{0.25} = 200 \] Thus, Jay can purchase a maximum of 200 quarters with the remaining budget.

Set the Range Expression for Quarters

The expression for the range of quarters Jay can give must account for at least one quarter but no more than 200. Thus, the expression is:\[ 1 \leq Q \leq 200 \]

Key concepts

Understanding Budget Calculation

A budget is simply the total amount of money you plan to spend. In Jay's case, he has set a budget of \(\\(150\) specifically for purchasing gifts for his friends' graduation. This means that the total cost of all gifts must not exceed this amount. Budget calculation often involves several steps to ensure every part of your spending fits within this limit.
To start with, Jay calculates the amount he needs for the first three gifts combined:

• A restaurant gift card for \(\\)25\)
• A tool set for \(\\(40\)
• A college sweatshirt for \(\\)35\)

Adding these amounts gives the total expenditure of \(25 + 40 + 35 = \\(100\).
By subtracting the \(\\)100\) from the overall budget of \(\\(150\), Jay finds that he has \(\\)50\) left to spend. This remaining budget is incredibly important, as it determines how much he can spend on the fourth gift for his friend. Budget calculation helps you to manage your finances effectively and prevents overspending. It's about making your money go as far as possible, while still achieving your goals.

Exploring Range Expression

Range expressions are used in math to describe a series of numbers that fit within certain boundaries. They are helpful in understanding what is possible when dealing with financial constraints, such as a fixed budget.
In Jay's quest to give his friend quarters, he must figure out just how many he can provide without surpassing his remaining budget. Since each quarter is worth \(\\(0.25\), Jay needs to divide his leftover \(\\)50\) by \(\$0.25\) to find out how many quarters he can actually afford. This calculation provides \(\frac{50}{0.25} = 200\) quarters.
Thus, Jay can buy anywhere from 1 quarter to a maximum of 200 quarters. The range expression capturing this situation is \(1 \leq Q \leq 200\). Range expressions are valuable in setting realistic limits based on initial calculations and ensuring that constraints, like budgets, are not exceeded.

Applying a Step-by-Step Solution

Step-by-step solutions make complex problems seem simpler by breaking them into manageable parts. This approach helps in understanding and solving problems without missing crucial information. Let's see how this applies to Jay's situation:
Firstly, Jay calculates the total budget available to him, which is \(\\(150\). Then, he adds up the costs of the three gifts he has already planned, totaling \(\\)100\). These steps ensure he remains within his budget for the first portion of his spending.
Next, Jay determines how much money he has left for the quarters: \(150 - 100 = \\(50\). With this information, he calculates the maximum number of \(\\)0.25\) quarters: \(\frac{50}{0.25} = 200\) quarters.
Finally, Jay needs to express how many quarters he can buy, which is effectively a range from the smallest possible number, 1, to the maximum, 200. This step-by-step process not only aids in problem-solving but also ensures that every layer of detail is considered in crafting the solution. Through this methodical approach, students can explore the application of budgeting and mathematical expressions in real-world scenarios.