7.5 Word Problems to Equations & Coordinate Geometry
Key Takeaways
- Assign x to the smallest or base quantity and express other quantities in terms of x; re-read the question before solving to confirm you answer what was asked.
- Translation traps: '5 less than x' is x - 5 (not 5 - x); 'of' means multiply; 'per' means divide; 'is' means equals.
- Average speed over a trip is total distance divided by total time, never the arithmetic mean of the two speeds.
- Coordinate geometry toolkit: slope m = (y2 - y1)/(x2 - x1), midpoint M = ((x1+x2)/2, (y1+y2)/2), distance d = sqrt((x2-x1)^2 + (y2-y1)^2).
- Parallel lines have equal slopes; perpendicular lines have slopes that multiply to -1 (m2 = -1/m1).
7.5 Word Problems to Equations & Coordinate Geometry
Quick Answer: Translate word problems by assigning a variable to the unknown and expressing each quantity in terms of it: rate problems use d = rt; mixtures track the substance of interest; work uses 1/t per worker; percents use part = percent × whole. In the coordinate plane: slope m = (y2 - y1)/(x2 - x1), midpoint M = ((x1+x2)/2, (y1+y2)/2), distance d = sqrt((x2-x1)^2 + (y2-y1)^2).
Why Translation and Coordinate Geometry Matter
Word problems are the GRE's favorite vehicle for testing algebra because they require you to set up the equation before solving. Coordinate geometry appears in 2-4 questions per Quant section, often as Numeric Entry or QC items requiring the slope, distance, or midpoint between two points.
Assigning Variables and the Translation Trap List
The single most important step is choosing what the variable represents. Three rules:
- Assign x to the smallest or base quantity, then express larger quantities in terms of x.
- Write the units next to the variable (x = miles, not just x).
- Re-read the question before solving to confirm you answer what was asked (the variable or a multiple of it).
Translation Traps
| Phrase | Math |
|---|---|
| "more than" | + (5 more than x is x + 5) |
| "less than" | - (5 less than x is x - 5) |
| "per" | ÷ (miles per hour = miles / hours) |
| "of" | × (half of 30 = (1/2) × 30) |
| "is" | = |
| "product of" | × |
| "quotient of" | ÷ |
| "twice" | × 2 |
| "consecutive integers" | n, n+1, n+2 |
The classic trap: "5 less than x" is x - 5, not 5 - x. The order flips because "less than" subtracts from the original quantity. Similarly, "three times the sum of x and 4" is 3(x + 4), not 3x + 4 — parentheses protect the sum.
Rate, Distance, Time
The fundamental relationship is d = r × t (distance = rate × time). Rearrange as r = d/t or t = d/r as needed.
Worked Example 1: A car travels 240 miles at 60 mph, then returns the same distance at 40 mph. What is the average speed for the whole trip?
- Outbound: t1 = 240/60 = 4 hours.
- Return: t2 = 240/40 = 6 hours.
- Total distance = 480 miles; total time = 10 hours.
- Average speed = 480/10 = 48 mph.
The trap: averaging 60 and 40 to get 50 is wrong because more time is spent at the slower speed. Average speed is total distance divided by total time, never the arithmetic mean of the two speeds.
Mixture Problems
Track the amount of the substance of interest (acid, salt, pure alloy), not the total volume.
Worked Example 2: How many liters of 30% acid must be added to 4 liters of 10% acid to make a 20% acid solution?
- Let x = liters of 30% solution.
- Acid in: 0.30x + 0.10(4) = 0.20(x + 4).
- 0.30x + 0.4 = 0.20x + 0.8, so 0.10x = 0.4 and x = 4.
- Answer: 4 liters.
Work Problems
If a worker completes a job in t hours, the work rate is 1/t per hour. Combined rates add.
Worked Example 3: Pipe A fills a tank in 6 hours; pipe B in 3 hours. How long together?
- Rates: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank per hour.
- Time = 1 / (1/2) = 2 hours.
Percent Change
Percent change = (new - old) / old × 100. Use the same formula for increase and decrease; the sign of the numerator tells the direction.
Worked Example 4: A price rises from $80 to $100. What is the percent increase?
- (100 - 80) / 80 × 100 = 20/80 × 100 = 25%.
The trap: a 25% increase is NOT undone by a 25% decrease. $100 × 0.75 = $75, not $80, because the base changed. Successive percent changes multiply, they do not add: a 25% increase followed by a 25% decrease nets to 1.25 × 0.75 = 0.9375, a 6.25% loss.
Ratio Problems
A ratio 3:2 with total 50 means 3x + 2x = 50, so 5x = 50 and x = 10. The parts are 30 and 20. Always introduce the multiplier x so the ratio is preserved while the total is matched.
Coordinate Geometry
Slope
m = (y2 - y1) / (x2 - x1). A line rising left-to-right has positive slope; falling has negative; horizontal has 0; vertical has undefined slope.
Midpoint Formula
M = ((x1 + x2)/2, (y1 + y2)/2).
Worked Example 5: Midpoint of (2, -3) and (8, 5) is ((2+8)/2, (-3+5)/2) = (5, 1).
Distance Formula
d = sqrt((x2 - x1)^2 + (y2 - y1)^2). This is the Pythagorean theorem in disguise.
Worked Example 6: Distance from (1, 2) to (4, 6) is sqrt((4-1)^2 + (6-2)^2) = sqrt(9 + 16) = sqrt(25) = 5.
Equations of Lines
| Form | Equation | Use |
|---|---|---|
| Slope-intercept | y = mx + b | given slope and y-intercept |
| Point-slope | y - y1 = m(x - x1) | given one point and slope |
| Standard | Ax + By = C | intercepts at (C/A, 0) and (0, C/B) |
Given a slope and a point, plug the point into y = mx + b and solve for b.
Parallel and Perpendicular Slopes
- Parallel: equal slopes.
- Perpendicular: slopes multiply to -1 (m2 = -1/m1, the negative reciprocal).
Worked Example 7: A line through (3, 4) perpendicular to y = 2x + 1 has slope -1/2. Equation: y - 4 = -(1/2)(x - 3). Rearranging: y = -(1/2)x + 11/2.
A horizontal line (slope 0) is perpendicular to a vertical line (undefined slope), and vice versa.
Intercepts and Reflections
- x-intercept: set y = 0 and solve for x.
- y-intercept: set x = 0 and solve for y.
- Reflection across x-axis: (x, y) -> (x, -y).
- Reflection across y-axis: (x, y) -> (-x, y).
- Reflection across origin: (x, y) -> (-x, -y).
- Reflection across y = x: (x, y) -> (y, x).
Reflections appear in QC and Multiple-Choice (single) items asking which transformed point satisfies a condition.
Tie to GRE Question Formats
- Numeric Entry: "What is the x-coordinate of the midpoint?" Enter a single value. The on-screen calculator handles the arithmetic, but reduce fractions before entering if the answer box accepts fractions.
- QC: Compare two slopes or two distances; compute both exactly rather than estimating, because coordinate grids are drawn to scale but the answer choices are not.
- Multiple-Choice (multiple answer): "Which of the following points lie on the line y = 2x - 3?" Test each candidate by substitution; select every point that satisfies the equation (no partial credit).
- Data Interpretation: coordinate graphs are drawn to scale, so visual estimation is allowed — but exact calculation eliminates rounding errors.
Strategic GRE Tips
- Always re-read the question before finalizing: word problems often ask for a quantity adjacent to your variable (e.g., you solved for x but the question asks for 2x).
- For rate problems with two legs, compute total distance and total time separately, then divide.
- For mixture problems, write the acid (or salt) balance, not the volume balance.
- For work problems, invert each worker's time to a rate, add rates, then invert the sum.
- For coordinate geometry, sketch the two points and the line; a quick sketch catches sign errors in slope and distance calculations.
- For perpendicular slopes, the product must be -1, not 1; check by multiplying the two slopes.
A car travels 240 miles at 60 mph and returns the same distance at 40 mph. What is the average speed for the entire trip?
How many liters of 30% acid solution must be added to 4 liters of 10% acid solution to produce a 20% acid solution?
What is the distance between the points (1, 2) and (4, 6)?
Which equation represents the line through (3, 4) perpendicular to y = 2x + 1?