Mathematical Reasoning
The math the exam actually asks for: proportional reasoning, basic algebra, and reading data.
By the end of this lesson you can
- Solve problems involving ratios, rates, and percentages
- Set up and solve a one-variable equation from a word problem
- Read and interpret data from tables and graphs
- Use the formula sheet the exam provides
Lesson Notes
Read through the key concepts before you try the challenge.
Most of it is proportional reasoning wearing different clothes
You are looking at a practice math test.
The questions are about sale prices, fuel consumption, medication doses, and interest. They look like four different topics. They are mostly one topic — a comparison of two quantities — asked four ways.
Your task: Learn to recognize the same structure under different surfaces.
A large share of HSE math is proportional reasoning: percentages, ratios, rates, unit conversion, and scale. If you can set up and solve a proportion reliably, you have covered a great deal of the test. The rest is basic algebra, geometry formulas that are provided to you, and reading data.
Setting up a proportion from a word problem
A car uses 12 gallons of fuel to travel 384 miles. How far can it travel on 18 gallons?
- 1
Identify what is being compared, and keep the units in the same positions.
Miles to gallons, both sides. Writing 384/12 on one side and 18/x on the other inverts the relationship and produces a confidently wrong answer. The units are what stop this.
- 2
Write it as two equal fractions: 384/12 = x/18.
Miles on top on both sides, gallons underneath on both sides. Set it up this way and it is nearly impossible to get backwards.
- 3
Cross-multiply: 12x = 384 × 18, so 12x = 6,912.
Cross-multiplication turns a proportion into a one-step equation, which is the only algebra this problem needs.
- 4
Divide: x = 576 miles.
One operation. The setup was the whole problem, which is true of most proportion questions on this test.
- 5
Sanity-check: 18 gallons is 1.5 times 12, and 576 is 1.5 times 384.
This takes five seconds and catches an inverted setup immediately. If your answer had been smaller than 384, the setup was backwards.
Result: 576 miles, checked.
Keep units in matching positions, cross-multiply, and sanity-check the direction. Setup is the problem; the arithmetic is not.
| Question | Method | Example |
|---|---|---|
| What is 15% of 80? | Multiply by the decimal | 0.15 × 80 = 12 |
| 12 is what percent of 80? | Divide, then convert | 12 ÷ 80 = 0.15 = 15% |
| 12 is 15% of what? | Divide by the decimal | 12 ÷ 0.15 = 80 |
| A $80 item is 15% off | Find the discount, subtract | 80 − 12 = $68 |
| Increase 80 by 15% | Multiply by 1.15 | 80 × 1.15 = 92 |
| From 80 to 92 — what change? | Difference ÷ original | 12 ÷ 80 = 15% increase |
A jacket costs $60 and is marked 25% off. What is the sale price?
Challenge
Apply what you've learned in this lesson.
Proportional reasoning repays practice more than any other single topic here.
- Find the official formula sheet for your exam, print it, and use it for every practice problem.
- Do ten proportion problems, writing the units beside every number in the setup.
- Work all six percentage relationships from the table using the numbers from your own life — a bill, a paycheck, a sale.
- Take five word problems and, without solving them, write only the equation each one describes.
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