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Last year a certain bond with a face value of \(\$5{,}000\) yielded \(8\%\) of its face value in interest. If that interest was approximately \(6.5\%\) of the bond's selling price, approximately what was the bond's selling price?
Let's break this down in everyday language first. Imagine you have a bond - think of it like an investment certificate. This bond has a "face value" printed on it of \(\$5,000\). Think of face value as the official amount written on the bond, like the denomination on a dollar bill.
Now, this bond earned interest last year. The interest was \(8\%\) of that face value amount. But here's the key insight: the bond's actual selling price (what someone paid for it in the market) was different from its face value.
We know that the same interest amount represents \(6.5\%\) of whatever the actual selling price was. Our job is to find that selling price.
Process Skill: TRANSLATE - Converting the relationship between face value, interest earned, and market price into a clear mathematical setup
Let's start with what we know for certain. The bond's face value is \(\$5,000\), and it earned \(8\%\) of this face value as interest.
In plain English: If the face value is \(\$5,000\) and we earned \(8\%\) of that as interest, then:
Interest earned = \(8\% \text{ of } \$5,000\)
Interest earned = \(0.08 \times \$5,000 = \$400\)
So the bond earned exactly \(\$400\) in interest last year. This is a concrete dollar amount we can work with.
Now here's the crucial insight: that same \(\$400\) in interest represents \(6.5\%\) of the bond's actual selling price.
Let's think about this relationship in plain English first. If \(\$400\) represents \(6.5\%\) of some unknown selling price, then we can think of it this way:
- \(6.5\% \text{ of selling price} = \$400\)
- This means: \(0.065 \times \text{selling price} = \$400\)
We now have a clear equation where we know the interest amount (\(\$400\)) and the percentage it represents (\(6.5\%\)), and we need to find the selling price.
From our equation: \(0.065 \times \text{selling price} = \$400\)
To find the selling price, we divide both sides by \(0.065\):
Selling price = \(\$400 \div 0.065\)
Let's calculate this step by step:
\(\$400 \div 0.065 = \$400 \div \frac{65}{1000} = \$400 \times \frac{1000}{65}\)
Simplifying: \(\$400 \times \frac{1000}{65} = \frac{\$400,000}{65}\)
Now: \(\$400,000 \div 65 = \$6,153.85\)
Rounding to the nearest dollar: approximately \(\$6,154\)
The bond's selling price was approximately \(\$6,154\).
Looking at our answer choices:
Our calculated value of \(\$6,154\) matches exactly with choice E.
The answer is E: \(\$6,154\)
1. Confusing face value with selling price
Students often assume that the bond's face value (\(\$5,000\)) is the same as its selling price. They fail to recognize that bonds can trade at prices different from their face value. This leads them to incorrectly set up the problem by using \(\$5,000\) as both the basis for interest calculation AND as the selling price.
2. Misinterpreting which percentage applies to which value
Students may confuse the two different percentage relationships given in the problem. They might incorrectly think that \(8\%\) applies to the selling price instead of the face value, or that \(6.5\%\) applies to the face value instead of the selling price. This fundamental misunderstanding leads to setting up the wrong equations.
3. Not recognizing that the same interest amount has two different percentage representations
Students may fail to understand that the \(\$400\) interest earned is simultaneously \(8\%\) of one amount (face value) and \(6.5\%\) of another amount (selling price). They might try to solve these as separate, unrelated pieces of information rather than understanding they represent the same dollar amount expressed as different percentages.
1. Arithmetic errors when dividing by decimals
When calculating \(\$400 \div 0.065\), students often make computational mistakes. They may incorrectly convert \(6.5\%\) to \(0.65\) instead of \(0.065\), or struggle with the division itself, leading to answers that are off by a factor of 10 or more.
2. Setting up the equation incorrectly
Even if students understand the relationship conceptually, they may write the equation backwards. For example, writing \(\text{selling price} \div 400 = 0.065\) instead of \(400 \div \text{selling price} = 0.065\), which leads to drastically different numerical results.
1. Selecting an answer that seems 'reasonable' without verification
Students might calculate \(\$6,153.85\) but then select answer choice D (\(\$6,000\)) because it's a 'round number' that seems close enough, rather than looking for the closest match (choice E: \(\$6,154\)). They may not trust their precise calculation and default to what appears to be a simpler answer.