Turn relationships into equations
Solve linear, quadratic and proportional problems.
The idea
Write the relationship before calculating. Weighted means use group totals; similar shapes scale areas by the square of a length factor.
A method to try
- Define what each quantity represents.
- Calculate with units and check the result in the original relationship.
Follow the reasoning.
A workshop has 10 beginners averaging 60 points and 20 returning learners averaging 75. Find the combined mean.
- Recover each total: 10 × 60 = 600 and 20 × 75 = 1500. Together the groups earned 2100 points.
- Divide by the combined 30 learners: 2100 ÷ 30 = 70. The larger group pulls the overall mean closer to 75 than to 60.
- The unweighted mean of 60 and 75 is 67.5. That would be appropriate only if the group sizes were equal, so it fails this problem’s structure.
Think it through first.
Similar rectangles have corresponding lengths in a 2:3 ratio. What is their area ratio?
Show the explanation
4:9.
Each of two dimensions scales by 2/3, so the area scale is (2/3)².
Make it your own
After reading, explain the method in your own words and identify the condition that makes it valid. Then try the linked starter practice without referring back to the worked example. Review the reasoning behind an error before repeating the same item; remembering an answer is different from being able to solve a new problem.
This lesson is one part of a short starter pathway. It does not cover the full exam syllabus or establish a predicted score.
Official references
These provider and publisher links explain the exam or underlying concepts. Our lessons and questions are original, independent practice.
View 5 references and checked dates
- College Board: SAT structure (opens in a new tab)Checked 2026-09-20
- College Board: practice (opens in a new tab)Checked 2026-09-20
- College Board: dates and deadlines (opens in a new tab)Checked 2026-09-20
- College Board — SAT content domains (opens in a new tab)Checked 2026-09-20
Concept and scope reference. Our questions and explanations are original; no provider endorsement.
- Purdue OWL — Commas vs. Semicolons (opens in a new tab)Checked 2026-09-20
Concept and scope reference. Our questions and explanations are original; no provider endorsement.