A company's inventory of a certain product decreases by a constant amount each month. In the 3rd month of the...
GMAT Algebra : (Alg) Questions
A company's inventory of a certain product decreases by a constant amount each month. In the 3rd month of the year, the company had 250 units of the product in its inventory. In the 7th month of the year, the inventory was down to 170 units. Which of the following equations models the number of units, \(\mathrm{U}\), in the inventory after m months into the year, where \(\mathrm{m}\) is a positive integer?
- \(\mathrm{U = -20m + 170}\)
- \(\mathrm{U = -20m + 250}\)
- \(\mathrm{U = -20m + 310}\)
- \(\mathrm{U = 20m + 190}\)
1. TRANSLATE the problem information
- Given information:
- Month 3: 250 units in inventory
- Month 7: 170 units in inventory
- Inventory decreases by a constant amount each month
- What this tells us: We have two coordinate points (3, 250) and (7, 170), and we're looking for a linear equation since the change is constant.
2. INFER the mathematical approach
- Since the inventory decreases by a constant amount, this is a linear relationship
- We need the slope-intercept form: \(\mathrm{U = mx + b}\)
- Strategy: Find the slope first, then use one point to find the y-intercept
3. SIMPLIFY to find the slope
- Slope = \(\frac{\mathrm{change\ in\ inventory}}{\mathrm{change\ in\ months}}\)
- \(\mathrm{m = \frac{170 - 250}{7 - 3}}\)
\(\mathrm{= \frac{-80}{4}}\)
\(\mathrm{= -20}\) - The negative slope confirms inventory is decreasing by 20 units per month
4. SIMPLIFY to find the y-intercept
- Use the equation \(\mathrm{U = -20m + b}\) with point (3, 250):
- \(\mathrm{250 = -20(3) + b}\)
- \(\mathrm{250 = -60 + b}\)
- \(\mathrm{b = 310}\)
5. Verify with the second point
- Check with (7, 170): \(\mathrm{U = -20(7) + 310 = -140 + 310 = 170}\) ✓
Answer: C (\(\mathrm{U = -20m + 310}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what the y-intercept represents and try to force one of the given points to be the y-intercept.
They might think "at month 3 there were 250 units, so the equation should be \(\mathrm{U = -20m + 250}\)" without realizing that 250 is the inventory at month 3, not month 0. This leads them to select Choice B (\(\mathrm{U = -20m + 250}\)).
Second Most Common Error:
Poor INFER reasoning about the slope sign: Students correctly calculate |slope| = 20 but forget that decreasing inventory means negative slope.
They create the equation \(\mathrm{U = 20m + something}\), immediately leading to Choice D (\(\mathrm{U = 20m + 190}\)) as the only positive-slope option.
The Bottom Line:
This problem tests whether students understand that the y-intercept represents the theoretical inventory at month 0, not necessarily one of the given data points. The key insight is using the slope and any given point to work backwards to find what the initial inventory would have been.