Question

A store sells a box of 5 frozen yogurt bars for \(\$ 1.20 .\) The store also sells a box containing 7 of the same frozen yogurt bars for \(\$ 1.59 .\) Which is the better buy? Explain how you decided.

Short answer

The box containing 7 bars is the better buy as its cost per frozen yogurt bar, at \$0.227, is less than the cost per bar of the box containing 5 bars.

Step-by-step solution

Identify the Cost Per Unit for Each Box

First, it's necessary to find out how much each individual bar of frozen yogurt costs in each box. This can be determined by dividing the total cost of each box by the number of bars in that box. For the box priced at \(\$1.20\), there are 5 bars, so the cost per bar is obtained by \(\$1.20\) divided by 5. Similarly, for the box priced at \(\$1.59\), there are 7 bars, so the cost per bar is obtained by \(\$1.59\) divided by 7.

Calculate the Unit Cost

Now perform the division from step 1. This will give the cost per frozen yogurt bar for each box: \(\$1.20\) / 5 = \(0.24\) dollars/bar for the first box and \(\$1.59\) / 7 = \(0.227\) dollars/bar (rounded to three decimal places) for the second box.

Compare the Unit Cost

Comparing the price per bar from each box, one can tell that the second box, which is priced at \$1.59 for 7 bars, is a better buy, since each bar costs about \(0.227\) dollars, which is less than the \(0.24\) dollars per bar in the first box.

Key concepts

Cost Per Unit

Understanding the 'cost per unit' is crucial when trying to find the best deal in any purchasing decision. It refers to the price of a single unit of a product when you purchase a group or set of that product.

Take, for example, frozen yogurt bars sold in boxes with differing quantities. To determine which box offers the more economical choice, you must calculate the cost per unit. This is done by dividing the total price of the box by the number of bars it contains. By comparing the resulting figures — the cost of one frozen yogurt bar from each box — you can discern which one provides more value for your money. If one box's unit cost is lower, you're getting a better deal per bar, stretching your dollars further.

Rate Comparison

Rate comparison is an essential skill that helps consumers compare different offers effectively. After calculating the cost per unit, the next step is to perform a side-by-side analysis of these rates.

Considering the example of frozen yogurt bars, once you have computed the cost for a single bar from each box, you compare these costs. The box with the lower cost per bar represents the better rate, which in this case, indicates it's the better purchase. It's essential to note that price isn't the only factor to consider; quality and quantity also play vital roles.

Mathematical Reasoning

Mathematical reasoning involves using logical thinking to solve problems with numerical information. It's more than just performing operations; it's about understanding the 'why' behind each step.

In choosing the better buy between different boxes of frozen yogurt bars, mathematical reasoning is used to justify the decision. The rationale involves computing the unit rates, understanding value, and making an informed decision based on the comparison. These logical steps grounded in mathematical principles help to arrive at a conclusion that is both economical and reasonable.

Division Operation

The division operation is a foundational mathematical process used to determine how many times a number is contained within another number. It's the operation employed when calculating cost per unit.

In our scenario with frozen yogurt bars, division is used to break down the total cost by the number of items in each box, yielding the cost per individual bar. The division operation allows us to convert a bulk price into a per-unit price, which is then used for a direct comparison. Mastery of the division operation is essential for identifying the most cost-effective option in real-world situations.