Problem Solving

Answer: C

There are three general approaches to this word problem: conceptual, algebraic, and plugging-in numbers. Let’s discuss each in turn.

The conceptual approach, likely to be taken by advanced students, is the fastest. Since the question focuses on the price of a stethoscope, we can convert the total tongue depressor cost into an equivalent stethoscope cost. The price of each stethoscope is nine times the price of each box of tongue depressors (side note – we can safely assume that all stethoscopes are equally-priced and that all tongue depressor boxes are equally-priced). Thus nine tongue depressor boxes cost the same as one stethoscope.

Divide 270 (the number of tongue depressor boxes) by nine to calculate that the cost of these 270 boxes is equivalent to the price of 30 (270 ÷ 9) stethoscopes. Therefore, the total bill is equivalent to the cost of 80 stethoscopes: the 50 stethoscopes bought plus 30 more (representing the 270 tongue depressor boxes). As a result, one stethoscope is \(\frac{1}{80}\) of the total bill.

We now need to convert this into a percent. But first let’s review the other two approaches, to make the conceptual approach more understandable.

This approach, likely to be taken by intermediate students, puts the conceptual approach into algebraic form. For this algebra word problem, let’s assign variables to the unknowns:

        S = price of one stethoscope
        B = price of one box of tongue depressors

To calculate the total bill, multiply Price × Quantity for each item and then add the results. We are given the quantities, and can use our variables for the prices:

        Total bill = 50S + 270B

We are told that the price of each stethoscope is nine times the price of each box of tongue depressors. Using our variables, we can write an equation to express this relationship:

        S = 9B

A common mistake is to write this equation backwards, as B = 9S. Since stethoscopes are more expensive, however, we need to make the bigger value S equal to nine times the smaller value B. A small number B cannot equal a big number S times 9.

Since the question asks about stethoscopes and not tongue depressors, we want to get rid of B, the variable that we don’t care about. To do this, isolate B and substitute it away. Divide both sides of our equation by 9 (same as multiplying both sides by \(\frac{1}{9}\)).

        \(\frac{1}{9}\)S = \(\frac{1}{9}\)(9B) = B

Now let’s do an algebraic substitution into the total bill equation:

        Total bill = 50S + 270(\(\frac{1}{9}\)S) = 50S + 30S = 80S

The question asks: the price of one stethoscope is what percent of the total bill? We are looking for a percent, as further verified by the format of the answers. When calculating a percent, a good approach is to form a fraction with the “is” number on top and the “of” number on the bottom:

        Percent = \(\frac{\text{is}}{\text{of}}\) = \(\frac{\text{stethoscope price}}{\text{total bill}}\) = \(\frac{S}{80S}\) = \(\frac{1}{80}\) (since the S variable cancels out)

We’ll convert this into a percent after reviewing the plugging-in numbers approach.

This approach is likely to be taken by less-advanced students, but is actually a great approach for this problem. When the answers represent a ratio or percent, and we don’t have specific numbers provided within the problem, then a very good technique is to pick numbers to work through the math. The price of each stethoscope is nine times the price of each box of tongue depressors, so let’s pick easy numbers. We do not need to worry about whether the numbers are accurate in the real world, just whether the numbers meet the relationship described in the problem.

        Tongue depressor box = $1
        Stethoscope = $9

Now calculate the total bill, using the price numbers we made up and the quantities provided in the problem.

        Total bill = 50 × $9 + 270 × $1 = $450 + $270 = $720

The question asks: the price of one stethoscope is what percent of the total bill? As mentioned in the algebraic approach, we can calculate the percent by forming a fraction with the “is” number on top and the “of” number on the bottom:

        Percent = \(\frac{\text{is}}{\text{of}}\) = \(\frac{\text{stethoscope price}}{\text{total bill}}\) = \(\frac{$9}{$720}\) = \(\frac{1}{80}\) (since 9 goes into 72 eight times)

There are several approaches to convert \(\frac{1}{80}\) into a percent. Worst case, we could do long-hand division. But there are a couple faster approaches.

Notice that \(\frac{1}{80}\) = \(\frac{1}{8}\) × \(\frac{1}{10}\). We recommend that students memorize the decimal equivalents of common fractions. So we should ideally know that \(\frac{1}{8}\) = 0.125. Multiplying by \(\frac{1}{10}\) is the same as moving the decimal one place to the left, resulting in 0.0125. Converting a decimal into a percent is done by moving the decimal two places to the right, so 0.0125 = 1.25%.

Another good shortcut takes advantage of the fact that a percent is equivalent to a fraction with a denominator of 100. How can we turn our denominator of 80 into 100? We can increase 80 by 25%, the same as multiplying 80 by 1.25. To leave the value of our fraction unchanged, we must multiply numerator and denominator by the same number.

        \(\frac{1}{80}\) = \(\frac{1(1.25)}{80(1.25)}\) = \(\frac{1.25}{100}\) = 1.25%

Finally, if we are very short on time and don’t see an easy way to do the conversion, we could at least quickly eliminate three answers by recognizing that:

        \(\frac{1}{100}\) < \(\frac{1}{80}\) < \(\frac{1}{50}\)  →  \(\frac{1}{100}\) < \(\frac{1}{80}\) < \(\frac{2}{100}\)  →  1% < \(\frac{1}{80}\) < 2%

Our correct answer has to be somewhere between 1% and 2%, leaving only answers C and D. As we have seen above, C is the correct answer.