QUESTION STEM:The table shows four values of t (in minutes) and the corresponding total volume \(\mathrm{V(t)}\) of water in a...
GMAT Algebra : (Alg) Questions
- The table shows four values of \(\mathrm{t}\) (in minutes) and the corresponding total volume \(\mathrm{V(t)}\) of water in a tank (in liters).
- The relationship between \(\mathrm{t}\) and \(\mathrm{V(t)}\) is linear and is defined by \(\mathrm{V(t) = rt - 20}\), where \(\mathrm{r}\) is a constant.
- What is the value of \(\mathrm{r}\)?
| \(\mathrm{t}\) | 8 | 12 | 16 | 20 |
|---|---|---|---|---|
| \(\mathrm{V(t)}\) | 84 | 136 | 188 | 240 |
Answer Format Instructions: Enter your answer as an integer.
Type: Fill-in-the-blank
1. TRANSLATE the problem information
- Given information:
- Table showing t values: 8, 12, 16, 20 minutes
- Corresponding V(t) values: 84, 136, 188, 240 liters
- Linear relationship: \(\mathrm{V(t) = rt - 20}\)
- Need to find: constant r
2. TRANSLATE a data point into an equation
- Choose any point from the table (let's use \(\mathrm{t = 8, V(t) = 84}\))
- Substitute these values into \(\mathrm{V(t) = rt - 20}\):
\(\mathrm{84 = r(8) - 20}\)
3. SIMPLIFY to solve for r
- Start with: \(\mathrm{84 = 8r - 20}\)
- Add 20 to both sides: \(\mathrm{84 + 20 = 8r}\)
- Combine: \(\mathrm{104 = 8r}\)
- Divide both sides by 8: \(\mathrm{r = 104/8 = 13}\)
4. Optional verification
- Check with another point (\(\mathrm{t = 12, V(t) = 136}\)):
\(\mathrm{136 = 12(13) - 20 = 156 - 20 = 136}\) ✓
Answer: 13
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Sign errors when isolating r
Students often write \(\mathrm{84 = 8r - 20}\), then incorrectly move the 20 by subtracting it from both sides instead of adding, getting \(\mathrm{64 = 8r}\), leading to \(\mathrm{r = 8}\). Or they make arithmetic errors like \(\mathrm{84 + 20 = 94}\) instead of 104.
This leads to confusion and incorrect numerical answers.
Second Most Common Error:
Poor TRANSLATE reasoning: Using the wrong relationship or misreading table values
Students might confuse which variable is which, or try to use multiple points simultaneously in a confusing way instead of systematically using one point at a time.
This causes them to get stuck and guess rather than following a clear solution path.
The Bottom Line:
This problem tests your ability to substitute known values into a linear equation and solve systematically. The key is being careful with arithmetic and maintaining clear organization when moving terms around.