Question

Hannah placed an online order for shirts that cost \(\$ 24.50\) per shirt. A tax of \(7 \%\) is added to the cost of the shirts, before a flat, untaxed shipping rate of \(\$ 6\) is charged. Which of the following represents Hannah's total cost for \(s\) shirts, in dollars? A) \(0.07(24.50 s+6)\) B) \(1.07(24.50+6) s\) C) \(1.07(24.50 s)+6\) D) \(1.07(24.50+s)+6\)

Short answer

C) \(1.07(24.50 s)+6\)

Step-by-step solution

Calculate Initial total cost of the shirts

Multiply the cost per shirt by the total number of shirts ordered (\(s\)): \(24.50s\)

Calculate the tax amount

Multiply the initial total cost (\(24.50s\)) by the tax rate (7%): \(0.07(24.50s)\)

Add the tax amount to the initial total cost

Add the initial total cost (\(24.50s\)) to the tax amount (\(0.07(24.50s)\)): \((1 + 0.07)(24.50s) = 1.07(24.50s)\)

Add the untaxed shipping rate

Add the untaxed shipping rate ($6) to the cost of shirts plus tax to find the final total cost for s shirts: \((1.07)(24.50s) + 6\) So, the correct answer is C) \(1.07(24.50 s)+6\).

Key concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like s in our example), and operators (such as +, −, ×, ÷). The purpose of these expressions is to represent general rules about relationships between numbers without specifying what those numbers are. In the case of Hannah's shirt order, the cost can be represented as an algebraic expression involving both the number of shirts she orders (the variable s) and constants like the cost per shirt and tax rate.

This approach allows the use of a single expression, such as \(1.07(24.50 s)+6\), to calculate the total cost for any number of shirts ordered. When working with algebraic expressions, especially in SAT Math problems, it's crucial to understand the order of operations and how to combine like terms. For example, we first multiplied the cost per shirt by the number of shirts, then correctly handled the percentage by converting it into a decimal to incorporate the tax, before finally adding the flat shipping rate.

Remember, clarity in translating a word problem into an algebraic expression is key to solving many types of math problems.

Percentages in Algebra

Percentages represent a ratio or fraction out of 100. In algebra, they are often used to describe relative changes, such as increases or decreases in quantities. The potency of percentages in algebra lies in their ability to compactly modify an initial quantity.

When Hannah's shirt order is taxed, the tax is a percentage (in this case, 7%) of the order's initial cost (\(24.50s\)). Converting this percent to a decimal (by dividing by 100) allows it to be used algebraically to find the tax amount: \(0.07 \times 24.50s\). To include this increase in the initial amount, we simply add 1 (representing 100%) to the tax rate expressed as a decimal, giving us \(1.07(24.50s)\).

Understanding how to manipulate and calculate percentages is indispensable, especially when dealing with financial matters such as taxes, discounts, and interest rates, which are common in both real-life and standardized tests like the SAT.

Math Problem Solving

Math problem solving is a fundamental skill, encompassing the ability to understand the problem, translate it into mathematical language, perform calculations, and interpret the results. When faced with a problem like Hannah's, a structured approach is vital.

First, identify the key elements and what you are solving for; in this exercise, it's the total cost for a number of shirts, s. Next, break down the problem into smaller, manageable steps, like finding the initial cost, calculating the tax, and adding shipping. This step-by-step approach transforms complex problems into a series of simpler tasks.

Always look for keywords in the problem that indicate mathematical operations, such as 'total cost' implying addition. Lastly, translate your solution back into the context of the problem to check its accuracy. It’s this disciplined methodology that can enhance performance in math tasks, homework, and critical exams like the SAT.