Problem 17 - Thinking Mathematically

Question

A television sells for \(\$ 750\). Instead of paying the total amount at the time of the purchase, the same television can be bought by paying \(\$ 100\) down and \(\$ 50\) a month for 14 months. How much is saved by paying the total amount at the time of the purchase?

Step-by-Step Solution

Answer

By paying the total amount at the time of the purchase, a total of $50 is saved.

Step 1: Calculate the total cost of the installment plan

The installment plan involves paying $100 down payment and then $50 a month for 14 months. Therefore, the total payment with the installment plan can be calculated using the formula: \(installment cost = down payment + (monthly payment \times months)\). Substituting the given values we get: \(installment cost = $100 + ($50 \times 14) = $100 + $700 = $800.\)

Step 2: Determine the savings

To determine the savings if the television was bought by paying the full amount at the time of the purchase, subtract the upfront cost from the total cost when using the installment plan. This can be represented with the formula \(savings = installment cost - upfront cost\). Substituting the given values, we get: \(savings = $800 - $750 = $50.\)

Key Concepts

Installment Plan Calculation

Understanding how to calculate an installment plan is essential for making informed purchasing decisions. In the exercise, a television can be bought either for \(\$750\)in a single payment or through an installment plan that requires a down payment of \(\$100\) followed by monthly payments of \(\$50\) for 14 months.

The key to calculating the total cost of the installment plan is to consider both the initial down payment and the cumulative monthly payments. In this case, you simply need to add the down payment to the total of all monthly payments. Mathematically, the formula is expressed as:
\[\text{installment cost} = \text{down payment} + (\text{monthly payment} \times \text{number of months})\]
By plugging in the values provided:\[\text{installment cost} = \$100 + (\$50 \times 14) = \$100 + \$700 = \$800.\]
This calculation shows that through the installment plan, you end up paying a total of \(\$800\), which is more than the upfront cost of the television.

Cost Comparison

Cost comparison is a practical skill that involves analyzing different purchasing options to identify the most cost-effective choice. When comparing the cost of buying an item outright versus purchasing it through an installment plan, the overall expenses and potential savings must be considered.

In the example of the television, a straightforward subtraction shows the difference in cost between the two payment methods. The formula for figuring out the savings is:
\[\text{savings} = \text{installment cost} - \text{upfront cost}\]
Substituting the relevant figures from our problem:\[\text{savings} = \$800 - \$750 = \$50.\]
Therefore, if you choose to pay the full amount at the time of purchase, you save \(\$50\), which might be directed to other expenses or savings.

Simple Arithmetic

Simple arithmetic forms the foundation of various financial calculations including installment plans and cost comparisons. It involves basic operations such as addition, subtraction, multiplication, and division.

In the discussed problem, all three operations come into play: Multiplication calculates the total of the monthly payments, addition sums up the down payment and the total monthly payments, and subtraction determines the savings. Getting comfortable with these operations is crucial since they are frequently used in daily transactions. For instance:

• Multiplication: \(\$50 \times 14 = \$700\) calculates the total of 14 monthly payments.
• Addition: \(\$100 + \$700 = \$800\) gives the total cost of the installment.
• Subtraction: \(\$800 - \$750 = \$50\) figures out how much is saved by paying upfront.

Solidifying your arithmetic skills will not only assist you in solving textbook problems but also in managing real-life financial decisions.