- Translate the problem requirements: Initially, Kaye and Alberto have stamps in a \(5:3\) ratio. After Kaye gives Alberto 10 stamps, their new ratio becomes \(7:5\). We need to find how many more stamps Kaye has than Alberto after this exchange.
- Set up variables using the initial ratio: Use the ratio multiplier approach to express both people's initial stamp counts in terms of a single variable.
- Model the exchange and new ratio condition: Express how the stamp counts change after the 10-stamp transfer and set up an equation using the new \(7:5\) ratio.
- Solve for the ratio multiplier: Use the equation from the new ratio condition to find the value of our variable.
- Calculate final stamp counts and difference: Determine each person's final stamp count and find how many more stamps Kaye has than Alberto.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- Initially, Kaye and Alberto have stamps in a \(5:3\) ratio (for every 5 stamps Kaye has, Alberto has 3)
- Kaye gives Alberto exactly 10 stamps
- After this exchange, their new ratio becomes \(7:5\)
- We need to find how many more stamps Kaye has than Alberto after the exchange
This is a classic "before and after" ratio problem where we need to track how the quantities change.
Process Skill: TRANSLATE - Converting the problem language into clear mathematical relationships
2. Set up variables using the initial ratio
Since we know the initial ratio is \(5:3\), we can use a ratio multiplier approach. Think of it this way: if the ratio is \(5:3\), then Kaye has 5 "units" and Alberto has 3 "units" where each unit represents the same number of stamps.
Let's call this unit size 'x':
- Initially, Kaye has \(5\mathrm{x}\) stamps
- Initially, Alberto has \(3\mathrm{x}\) stamps
This ensures their ratio is exactly \(5:3 = 5\mathrm{x}:3\mathrm{x}\)
3. Model the exchange and new ratio condition
Now let's track what happens when Kaye gives Alberto 10 stamps:
- Kaye loses 10 stamps, so she has: \(5\mathrm{x} - 10\) stamps
- Alberto gains 10 stamps, so he has: \(3\mathrm{x} + 10\) stamps
We're told that after this exchange, their new ratio is \(7:5\). In plain English, this means:
"Kaye's new amount" to "Alberto's new amount" = 7 to 5
Setting up the equation: \((5\mathrm{x} - 10) : (3\mathrm{x} + 10) = 7 : 5\)
Converting to equation form: \(\frac{5\mathrm{x} - 10}{3\mathrm{x} + 10} = \frac{7}{5}\)
4. Solve for the ratio multiplier
Cross-multiplying our equation:
\(5(5\mathrm{x} - 10) = 7(3\mathrm{x} + 10)\)
\(25\mathrm{x} - 50 = 21\mathrm{x} + 70\)
\(25\mathrm{x} - 21\mathrm{x} = 70 + 50\)
\(4\mathrm{x} = 120\)
\(\mathrm{x} = 30\)
So each "unit" in our original ratio represents 30 stamps.
5. Calculate final stamp counts and difference
Now we can find the actual numbers:
After the exchange:
- Kaye has: \(5\mathrm{x} - 10 = 5(30) - 10 = 150 - 10 = 140\) stamps
- Alberto has: \(3\mathrm{x} + 10 = 3(30) + 10 = 90 + 10 = 100\) stamps
Let's verify our ratio: \(140:100 = 14:10 = 7:5\) ✓
The difference: Kaye has \(140 - 100 = 40\) more stamps than Alberto
Final Answer
Kaye had 40 more stamps than Alberto after giving him 10 stamps.
Answer: C. 40
Common Faltering Points
Errors while devising the approach
- Misunderstanding what the question is asking for: Students often confuse whether they need to find the difference before or after the exchange. The question asks "As a result of the gift, Kaye had how many more stamps than Alberto" - this clearly indicates we need the difference AFTER the exchange, not before. Some students might calculate the initial difference instead.
- Setting up variables incorrectly: Instead of using the ratio multiplier approach (\(5\mathrm{x}\) and \(3\mathrm{x}\)), students might assign actual numbers too early or use two separate variables (like K and A) without properly connecting them through the ratio relationship. This makes the algebra much more complicated and error-prone.
- Misinterpreting the direction of the gift: Students might incorrectly assume Alberto gives stamps to Kaye instead of Kaye giving stamps to Alberto, which would completely reverse the setup and lead to wrong equations.
Errors while executing the approach
- Cross-multiplication errors: When solving \(\frac{5\mathrm{x} - 10}{3\mathrm{x} + 10} = \frac{7}{5}\), students frequently make algebraic mistakes during cross-multiplication. A common error is writing \(7(5\mathrm{x} - 10) = 5(3\mathrm{x} + 10)\) instead of the correct \(5(5\mathrm{x} - 10) = 7(3\mathrm{x} + 10)\), essentially swapping the denominators.
- Sign errors during algebraic manipulation: When expanding and collecting terms, students often lose track of positive and negative signs. For example, they might write \(25\mathrm{x} - 50 = 21\mathrm{x} + 70\) correctly but then incorrectly get \(25\mathrm{x} - 21\mathrm{x} = 70 - 50\) instead of \(25\mathrm{x} - 21\mathrm{x} = 70 + 50\).
- Arithmetic errors in final calculations: Even with the correct value \(\mathrm{x} = 30\), students might make simple arithmetic mistakes when calculating \(5(30) - 10\) or \(3(30) + 10\), leading to wrong final stamp counts.
Errors while selecting the answer
- Calculating the wrong difference: Students might correctly find that Kaye has 140 stamps and Alberto has 100 stamps after the exchange, but then calculate \(140 + 100 = 240\) instead of \(140 - 100 = 40\). This type of error occurs when students lose focus on what the question is actually asking for in the final step.
- Reporting intermediate values: Some students might report \(\mathrm{x} = 30\) (the ratio multiplier) as their final answer, or report the initial difference before the exchange, instead of the final difference of 40 stamps.
Alternate Solutions
Smart Numbers Approach
Step 1: Choose smart numbers for the initial ratio
Since Kaye and Alberto initially have stamps in a \(5:3\) ratio, let's choose convenient numbers that satisfy this ratio. Let's say:
- Kaye initially has 50 stamps
- Alberto initially has 30 stamps
This gives us the \(5:3\) ratio (\(50:30 = 5:3\)) and uses round numbers that are easy to work with.
Step 2: Model the exchange
After Kaye gives Alberto 10 stamps:
- Kaye has: \(50 - 10 = 40\) stamps
- Alberto has: \(30 + 10 = 40\) stamps
Step 3: Check if this satisfies the new ratio condition
The problem states that after the exchange, the ratio should be \(7:5\).
Our current ratio is \(40:40 = 1:1\), which is NOT \(7:5\).
This means our initial choice of 50 and 30 stamps doesn't work.
Step 4: Scale our smart numbers
Since we need the final ratio to be \(7:5\), let's work backwards. If the final amounts are in ratio \(7:5\), let's try:
- Final amounts: 70 stamps for Kaye, 50 stamps for Alberto (\(7:5\) ratio)
Working backwards to initial amounts:
- Kaye initially: \(70 + 10 = 80\) stamps
- Alberto initially: \(50 - 10 = 40\) stamps
Step 5: Verify the initial ratio
Initial ratio: \(80:40 = 2:1\), which is NOT \(5:3\).
Let's try another approach with the final ratio \(7:5\). Let the final amounts be \(7\mathrm{k}\) and \(5\mathrm{k}\):
- Kaye finally: \(7\mathrm{k}\) stamps
- Alberto finally: \(5\mathrm{k}\) stamps
Initial amounts:
- Kaye initially: \(7\mathrm{k} + 10\)
- Alberto initially: \(5\mathrm{k} - 10\)
For initial ratio \(5:3\): \((7\mathrm{k} + 10):(5\mathrm{k} - 10) = 5:3\)
Cross multiply: \(3(7\mathrm{k} + 10) = 5(5\mathrm{k} - 10)\)
\(21\mathrm{k} + 30 = 25\mathrm{k} - 50\)
\(80 = 4\mathrm{k}\)
\(\mathrm{k} = 20\)
Step 6: Calculate final difference
Final amounts:
- Kaye: \(7\mathrm{k} = 7(20) = 140\) stamps
- Alberto: \(5\mathrm{k} = 5(20) = 100\) stamps
Difference: \(140 - 100 = 40\) stamps
Therefore, Kaye has 40 more stamps than Alberto after the exchange.
