Algebra & Linear Equations
Problem 01
Memory Key "TRAP: Read what they ask — value of expression, NOT variable alone"
The Hidden Expression Trick
A store sells two types of beverages. Coffee costs $3 more than twice the price of tea. If the total cost of 4 coffees and 4 teas is $60, what is the cost, in dollars, of one coffee and one tea?
What is the cost of one coffee + one tea? (Enter a number, no $ sign)
Problem 02
Memory Key "NO SOLUTION = parallel lines = same slope, different intercept → coefficients proportional, constants NOT"
Infinite vs. No Solution — The Classic Trap
Consider the system of equations:
4x − 6y = 10
kx − 9y = 15
For what value of k does the system have no solution?
kx − 9y = 15
What is the value of k?
Problem 03
Memory Key "PERCENT CHANGE: always divide by ORIGINAL (not new). Up then down ≠ back to start."
The Percent Up-Down Trap
A jacket is marked up 25% from its wholesale price. Then, during a sale, the marked-up price is reduced by 20%. The final sale price is what percent of the original wholesale price?
Final price = what percent (%) of wholesale? (Enter number only)
Problem 04
Memory Key "FLIP SIGN when multiplying/dividing by NEGATIVE. Constraint = find min/max by testing boundary."
The Constraint Boundary Trick
A driver charges a flat fee of $5 plus $2 per mile. A second driver charges $3 per mile with no flat fee. For what minimum integer number of miles is the first driver strictly cheaper?
What is the minimum integer miles where Driver 1 is cheaper?
Ratios, Rates & Proportions
Problem 05
Memory Key "MIXTURE: use (concentration × volume) = amount of substance. Keep track of what's ADDED vs. TOTAL."
The Mixture Concentration Trap
A scientist has 40 mL of a solution that is 30% acid. She wants to dilute it to a 20% acid solution by adding pure water. How many milliliters of water should she add?
How many mL of water to add?
Problem 06
Memory Key "WORK RATE = 1/time. Together: add rates (1/A + 1/B = 1/T). Never add times directly!"
Combined Work Rate — A Classic Mistake
Pump A can fill a tank in 6 hours. Pump B can fill the same tank in 4 hours. If both pumps work together, and Pump A breaks down after 1 hour, how many additional hours does Pump B need to finish filling the tank?
How many additional hours for Pump B alone? (Enter fraction or decimal)
Problem 07
Memory Key "UNIT CONVERSION: write units as fractions and CANCEL. Never rely on memory for direction."
Speed Unit Conversion Trap
A car travels at 90 kilometers per hour. A speed limit sign reads 55 miles per hour. Given that 1 mile = 1.6 km, is the car speeding? By approximately how many miles per hour is the car above or below the limit? (Round to nearest whole number.)
How many mph above (+) or below (−) the limit? (e.g. +5 or −3)
Functions & Graphs
Problem 08
Memory Key "f(x+k) = horizontal SHIFT left (k>0). f(x)+k = vertical shift up. Don't confuse them!"
The Function Input Swap Trick
The function f(x) = 3x² − 2x + 1. What is the value of f(x + 2) − f(x) when x = 1?
What is f(3) − f(1)?
Problem 09
Memory Key "TABLE → slope = Δy/Δx. Pick ANY two rows. If slope is constant → linear."
Reading a Table — Hidden Slope
The table below shows values of a linear function g(x):
What is g(10)?
| x | g(x) |
|---|---|
| 1 | 7 |
| 3 | 13 |
| 5 | 19 |
What is the value of g(10)?
Problem 10
Memory Key "EXPONENTIAL: y = a·(b)^t. Find 'a' (initial), 'b' (multiplier per period). Watch the TIME UNIT."
Doubling Time — The Time Unit Trap
A population of bacteria doubles every 3 hours. If there are 500 bacteria at time zero, how many bacteria are there after 12 hours?
How many bacteria after 12 hours?
Quadratics & Polynomials
Problem 11
Memory Key "VERTEX x = −b/(2a). Max/min VALUE → plug x back into equation. Area/profit → vertex = optimum."
Maximum Profit — Vertex Form
A company's daily profit P (in dollars) from selling x items is modeled by:
P(x) = −2x² + 80x − 300
What is the maximum daily profit, in dollars?
What is the maximum profit in dollars?
Problem 12
Memory Key "DISCRIMINANT b²−4ac: >0 two real roots, =0 one root (tangent), <0 no real roots. Tangent → EXACTLY ONE solution."
Line Tangent to Parabola — The One-Solution Trick
The line y = kx + 3 is tangent to the parabola y = x² + 1. What are the possible values of k?
Enter the TWO values of k separated by a comma (e.g. 2,−2)
Problem 13
Memory Key "ax²+bx+c: Sum of roots = −b/a · Product of roots = c/a. No need to solve!"
Sum of Roots Without Solving
The two solutions of 3x² − 12x + 7 = 0 are r and s. What is the value of (r + s)²?
What is the value of (r + s)²?
Statistics & Data Analysis
Problem 14
Memory Key "WEIGHTED AVERAGE: total sum = mean × count. Find missing value: new total − known total."
The Hidden Score — Weighted Average
A student's average on 5 tests is 82. After a 6th test, the average drops to 80. What did the student score on the 6th test?
What was the score on the 6th test?
Problem 15
Memory Key "LINE OF BEST FIT: slope = rate of change per 1 unit. Plug in x to predict y. Watch axis labels!"
Interpreting a Best-Fit Line
A study found that for every additional 1 hour of daily study, a student's test score increases by 4.5 points. The line of best fit has a y-intercept of 52. According to this model, what score is predicted for a student who studies 8 hours per day?
What is the predicted score?
Problem 16
Memory Key "TWO-WAY TABLE: P(A|B) = P(A and B)/P(B). Always divide by the CONDITION'S total, not grand total."
Conditional Probability — Two-Way Table
A survey of 200 students is summarized below:
Given that a randomly selected student likes math, what is the probability they are in Grade 11? (Enter as a fraction, e.g. 6/11)
| Likes Math | Dislikes Math | Total | |
|---|---|---|---|
| Grade 11 | 60 | 40 | 100 |
| Grade 12 | 50 | 50 | 100 |
| Total | 110 | 90 | 200 |
P(Grade 11 | Likes Math) = ?
Geometry & Word Contexts
Problem 17
Memory Key "SIMILAR TRIANGLES: ratio of sides is CONSTANT. Area ratio = (side ratio)². Don't mix up sides and area!"
Shadow Length — Similar Triangle Trap
A 6-foot person casts a 4-foot shadow. At the same time, a nearby building casts a 60-foot shadow. How tall, in feet, is the building?
What is the height of the building in feet?
Problem 18
Memory Key "ARC LENGTH = (central angle / 360) × 2πr. SECTOR AREA = (angle/360) × πr². Proportion is the key!"
Arc Length — The Proportion Trick
A circular pizza has a diameter of 16 inches. A slice is cut with a central angle of 45°. What is the arc length of the crust of this slice? (Leave answer in terms of π, e.g. write "2π")
Arc length of crust = ? (in terms of π)
Problem 19
Memory Key "VOLUME SCALE: if linear dimension × k, then volume × k³. Don't scale volume by k or k²!"
Scaling Volume — The Cube Trap
A cylindrical can has a radius of 3 cm and a height of 8 cm. A larger similar can has its radius doubled to 6 cm, with height also doubled to 16 cm. The larger can holds how many times the volume of the smaller can?
The larger can holds how many times the volume of the smaller?
Problem 20
Memory Key "MULTI-STEP: define variables → write equations → solve. The question asks for the EXPRESSION, not the variable!"
The Final Boss — Multi-Step Word Problem
A train leaves City A at 8:00 AM traveling toward City B at 60 mph. A second train leaves City B at 9:00 AM traveling toward City A at 90 mph. The cities are 300 miles apart. At what time do the trains meet? (Enter time in HH:MM AM format, e.g. 10:30 AM)
At what time do the trains meet?
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You completed all 20 problems!