Question

Multiple Choice Which expression can be used to find the total cost of \(b\) bats and \(g\) gloves if a bat costs \(\$ 50\) and a glove costs \(\$ 75\) ? (Algebra Review) (A) \((50+75) \times(b+g)\) (B) \((50 \times 75)+(b \times g)\) (C) \(50 b+75 g\) (D) \(75 b+50 g\)

Short answer

(C) \(50b + 75g\)

Step-by-step solution

Understanding the Problem

We want to find the total cost for buying a certain number of bats and gloves, with each bat costing $50 and each glove costing $75. The problem requires an algebraic expression to represent this total cost.

Identify the Cost of Bats

The cost of buying bats is calculated by multiplying the number of bats by the cost of each bat. If you buy \(b\) bats, each costing \(\$50\), then the total cost for the bats is \(50b\).

Identify the Cost of Gloves

Similarly, the cost of buying gloves is calculated by multiplying the number of gloves by the cost of each glove. If you buy \(g\) gloves, each costing \(\$75\), then the total cost for the gloves is \(75g\).

Formulating the Total Cost Expression

To find the total cost of both bats and gloves, you add the cost expressions for the bats and gloves together: \(50b + 75g\).

Choosing the Correct Option

Review the options provided:(A) \((50+75) \times(b+g)\) is incorrect because it adds the individual costs first and then multiplies by the quantities, which is improper.(B) \((50 \times 75)+(b \times g)\) is incorrect because it multiplies prices together and quantities together.(C) \(50b + 75g\) is correct as it directly adds the costs of the items.(D) \(75b + 50g\) is incorrect because it reverses the costs of bats and gloves.

Key concepts

Cost Calculation

Cost calculation is a fundamental concept in algebra, especially when dealing with real-world scenarios such as shopping or budgeting. In this exercise, we want to calculate the total cost of purchasing certain items. Each item has a fixed price, and the total cost depends on how many items are purchased. The formula to calculate this cost is the product of the unit price and the quantity of each item. For example, if a bat costs \(50, then buying one bat costs 50 times the number of bats, represented algebraically as \(50b\). Similarly, for gloves costing \)75 each, the expression is \(75g\). The total cost of both items is then found by adding these two expressions together, yielding \(50b + 75g\). This method allows us to dynamically find the total cost based on varying numbers of items.

Word Problems

Word problems involve translating descriptive text into mathematical expressions. This can often be challenging but is a valuable skill that helps in real-world problem-solving. In our given problem, interpreting the sentence "a bat costs \(50 and a glove costs \)75" into mathematical terms is crucial.

• First, identify keywords such as 'cost' and 'each' to signify multiplication.
• Recognize quantities, like \(b\) for bats and \(g\) for gloves, which represent variables.
• Establish relationships: the total cost involves summing individual costs, signifying addition.

By translating words into algebraic expressions, students can simplify complex real-world problems into manageable mathematical operations.

Multiplication and Addition in Algebra

The core of this problem rests on two basic arithmetic operations: multiplication and addition, which are foundational to algebra. When dealing with expressions like \(50b\) and \(75g\), each term represents a multiplication between a constant (the cost per item) and a variable (number of items).

• Multiplication: It calculates the total for each item type separately, so \(50b\) means $50 multiplied by the number of bats \(b\).
• Addition: Combines these results to yield a total cost for all items being considered, represented as \(50b + 75g\).

Understanding how to apply multiplication and addition within algebraic expressions is key to solving such word problems. This skill forms the basis for more advanced topics in mathematics and its applications in everyday life.