Linear models and rates — Digital SAT Math practice
Interpret slope and intercept in context; rate and mixture problems.
Key terms
- Slope: The constant rate of change per unit of x; always includes units like dollars per hour or miles per gallon.
- Y-intercept: The starting value when x equals zero; represents the initial amount before any changes occur.
- Rate: A ratio comparing two different quantities, showing how much one quantity changes for each unit of another.
- Concentration: The proportion of a specific ingredient within a mixture, usually expressed as a decimal or percentage.
- Piecewise Function: A model that switches to a different linear equation when the input crosses a specific threshold.
Common traps
- Confusing slope with the y-intercept by swapping their values when writing the equation from a word problem.
- Forgetting to include units when interpreting slope, which loses critical context about what is changing per unit.
- Adding concentrations directly in mixture problems instead of setting up separate equations for each ingredient amount.
Key ideas to know
Interpreting Slope in Context
A line model for a phone plan is y = 0.15x + 10, where x is minutes used and y is total cost in dollars. What does the slope represent?
The slope is 0.15. It represents the rate of change: $0.15 per additional minute used. Always state units as dollars per unit of x.
Watch out: Forgetting to include units or misidentifying the starting amount (b) as the rate. The intercept is the fixed cost when x = 0.
Total Cost Linear Model
A gym charges a one-time membership fee plus $8 per visit. Write the equation for total cost C after v visits, and identify fixed vs rate terms.
C = 8v + K (where K is the fixed fee). Fixed amount = starting value b; rate = slope m. Total = fixed amount + rate x quantity.
Watch out: Swapping the fixed fee and the per-visit rate in the equation. The fixed amount always stands alone as b, never multiplied by x.
Mixture Problem Setup Error
A chemist mixes a 20% acid solution with a 50% acid solution to get 10 liters of a 35% solution. The student wrote: 0.2x + 0.5y = 10. What is wrong?
That equation only tracks total volume, not acid amount. You need two equations: x + y = 10 (total volume) and 0.2x + 0.5y = 3.5 (acid amount).
Watch out: Writing only one equation for a two-variable mixture problem or forgetting to multiply concentration by the correct variable amount.
Sample questions
Question 1
A shirt costs $40. Its price is increased by 15%, and then a $5 coupon is applied. What is the final price?
- $46
- $41
- $35
- $41.75
Show answer and explanation
Answer: $41
40 x 1.15 = 46, then 46 - 5 = 41.
Question 2
Car A leaves traveling 60 mph. One hour later Car B leaves from the same point in the same direction at 80 mph. How many hours after Car B leaves does it catch up to Car A?
- 1
- 4
- 2
- 3
Show answer and explanation
Answer: 3
When B catches A, 60(t + 1) = 80t, so 60t + 60 = 80t and t = 3 hours.
Question 3
How many liters of a 20% acid solution must be mixed with 5 liters of a 50% acid solution to produce a 30% acid solution?
- 5
- 15
- 10
- 2.5
Show answer and explanation
Answer: 10
0.20x + 0.50(5) = 0.30(x + 5). Then 0.2x + 2.5 = 0.3x + 1.5, so 0.1x = 1 and x = 10.