- Translate the problem requirements: We have 2,420 stocks total. Each stock either went up or down (no stock stayed the same). We need to find how many went up, given that the number that went up is 20% greater than the number that went down.
- Set up variables for the two groups: Define variables for stocks that went up and down, ensuring we account for the total of 2,420 stocks.
- Express the 20% relationship mathematically: Translate "20 percent greater than" into a mathematical relationship between our variables.
- Solve using the total constraint: Use the fact that up + down = 2,420 to find the actual numbers.
Execution of Strategic Approach
1. Translate the problem requirements
Let's break down what we know in plain English:
- We have exactly 2,420 stocks total
- Every single stock either went up OR went down in price (none stayed the same)
- The number that went up is "20 percent greater than" the number that went down
- We need to find how many went up
Think of this like dividing a group of 2,420 people into two teams - the "winners" (stocks that went up) and the "losers" (stocks that went down). We know the winner team has 20% more people than the loser team.
Process Skill: TRANSLATE - Converting the business language about stock prices into a mathematical relationship
2. Set up variables for the two groups
Let's use simple letters to represent our two groups:
- Let D = number of stocks that went DOWN in price
- Let U = number of stocks that went UP in price
Since every stock either went up or down, we know:
\(\mathrm{U + D = 2,420}\)
This is our constraint equation - it captures the fact that all stocks must be accounted for in one of our two groups.
3. Express the 20% relationship mathematically
Now let's translate "20 percent greater than" into math.
If something is "20% greater than" another amount, it means:
Original amount + (20% of original amount) = New amount
So if the number that went UP is 20% greater than the number that went DOWN:
\(\mathrm{U = D + (20\% \text{ of } D)}\)
\(\mathrm{U = D + 0.20D}\)
\(\mathrm{U = 1.20D}\)
This tells us that the "up" group is 1.2 times as large as the "down" group.
4. Solve using the total constraint
Now we substitute our relationship into our constraint equation:
\(\mathrm{U + D = 2,420}\)
Since U = 1.20D, we can replace U:
\(\mathrm{1.20D + D = 2,420}\)
\(\mathrm{2.20D = 2,420}\)
Dividing both sides by 2.20:
\(\mathrm{D = 2,420 ÷ 2.20 = 1,100}\)
So 1,100 stocks went down in price.
Therefore, the number that went up:
\(\mathrm{U = 1.20D = 1.20 × 1,100 = 1,320}\)
Let's verify: \(\mathrm{1,320 + 1,100 = 2,420}\) ✓
And 1,320 is indeed 20% greater than 1,100 ✓
5. Final Answer
The number of stocks that closed at a higher price today than yesterday is 1,320.
This matches answer choice (D) 1,320.
Common Faltering Points
Errors while devising the approach
- Misinterpreting "20 percent greater than": Students often confuse this phrase and set up the equation as \(\mathrm{D = 1.20U}\) (stocks that went down = 1.20 times stocks that went up) instead of the correct \(\mathrm{U = 1.20D}\). This fundamental misreading of the relationship leads to completely wrong equations and answers.
- Missing the constraint that all stocks must be accounted for: Students may focus only on the percentage relationship and forget that the total number of stocks (2,420) is a crucial constraint. Without setting up \(\mathrm{U + D = 2,420}\), they cannot solve the problem systematically.
- Assuming some stocks stayed the same price: The problem states that "all different from today's closing prices" means every stock either went up OR down, but students might incorrectly think some stocks remained unchanged and try to account for a third category.
Errors while executing the approach
- Arithmetic error in division: When calculating \(\mathrm{D = 2,420 ÷ 2.20}\), students may make computational mistakes, especially since 2.20 is a decimal. Some might incorrectly calculate this as \(\mathrm{2,420 ÷ 2.2 = 1,100}\) but then make errors in the subsequent multiplication step.
- Forgetting to convert percentage to decimal: Students might use 20 instead of 0.20 when setting up \(\mathrm{U = D + 0.20D}\), leading to \(\mathrm{U = D + 20D = 21D}\), which gives completely wrong results.
Errors while selecting the answer
- Selecting the value for stocks that went down instead of up: After correctly calculating D = 1,100 and U = 1,320, students might accidentally select answer choice (C) 1,100, which represents the number of stocks that went down, rather than the number that went up (which is what the question asks for).
