Problem 331 - Elementary Algebra

Question

In the following exercises, solve. Alonzo works as a car detailer. He charges \(\$ 175\) per car. He is planning to move out of his parents' house and rent his first apartment. He will need to pay \(\$ 120\) for application fees, \(\$ 950\) for security deposit, and first and last months' rent at \(\$ 1,140\) per month. He has \(\$ 1,810\) in savings. How many cars must he detail to have enough money to rent the apartment?

Step-by-Step Solution

Answer

Alonzo must detail 9 cars.

Step 1 - Calculate Total Cost

Add up all the expenses: application fees, security deposit, and first and last months' rent.Application fees: \$ 120 Security deposit: \$ 950 First month’s rent: \$ 1,140 Last month’s rent: \$ 1,140 Total cost is: \[ 120 + 950 + 1,140 + 1,140 = 3,350 \]

Step 2 - Determine Additional Amount Needed

Subtract the amount of savings from the total cost.Total cost: \[ 3,350 \] Savings: \[ 1,810 \] Additional amount needed: \[ 3,350 - 1,810 = 1,540 \]

Step 3 - Calculate Number of Cars to Detail

Divide the additional amount needed by the charge per car to find out how many cars Alonzo needs to detail.Additional amount needed: \[ 1,540 \] Charge per car: \[ 175 \, per \, car \] Number of cars to detail: \[ \lceil \frac{1,540}{175} \rceil \approx 9 \]

Key Concepts

arithmetic operations

In solving problems involving money, basic arithmetic operations such as addition, subtraction, multiplication, and division are crucial.

In this exercise, you need to:

• **Add** the costs to find out the total expenses.
• **Subtract** the savings from the total expenses to determine the additional amount needed.
• **Divide** the additional amount by the profit per car to find the number of cars Alonzo should detail.

Practicing these operations helps in performing calculations quickly and checking for accuracy.

financial planning

Financial planning involves managing your money to meet your goals. In this case, Alonzo is planning to move out and rent an apartment.

This problem highlights the importance of:

• **Estimating costs**: Knowing all the expenses beforehand will help avoid unexpected financial troubles.
• **Budgeting**: Subtracting savings from the total cost allows understanding how much more money is needed.
• **Income Management**: Calculating how many cars need to be detailed to meet the additional financial requirement.

equation solving

Equation solving is about finding the value of unknowns. Here, we need to find out how many cars Alonzo must detail.

The steps to solve the equations related to this problem include:

• First, **add** all the individual costs to get a total cost.
• Second, **subtract** the amount of savings from the total cost to determine the shortfall.
• Third, **divide** the shortfall by the earning per car detailed to find out how many cars Alonzo should detail.

The equation formed here is \[ \frac { \text{additional amount}}{\text{earnings per car}} \]. When solving for the number of cars, rounding up to the nearest whole number ensures Alonzo has enough funds.

word problems

Word problems require translating a scenario into a mathematical problem to solve. This involves:

**Reading carefully**: Understand what is being asked and identify the known values and what needs to be calculated.
**Breaking down the problem**: Here, break the problem into steps: calculating total cost, determining additional money needed, and finding out the number of cars.
**Setting up equations**: Convert words into mathematical expressions.

In Alonzo's case:

• Total cost = sum of all individual costs
• Additional amount needed = Total cost - savings
• Number of cars to detail = Additional amount needed / earnings per car

Following these steps simplifies solving any word problem.