2.1 Arithmetic, Fractions, and Ratios
Key Takeaways
- Percent change uses the original value as the denominator.
- Fractions require common denominators for addition and subtraction.
- Unit-rate problems become easier when units are written beside every number.
- Sign errors often start with subtracting negatives or distributing negatives.
Arithmetic Is a Gatekeeper
ALEKS can quickly expose weak arithmetic. A student who can set up a quadratic equation but loses signs, fractions, or units will still lose placement strength. Treat arithmetic as a speed-and-accuracy layer.
High-Yield Rules
| Topic | Rule | Common trap |
|---|---|---|
| Negative numbers | Subtracting negative means add | Keeping the minus sign |
| Fraction addition | Use a common denominator | Adding denominators |
| Percent change | Change divided by original | Dividing by the new value |
| Unit rates | Divide to one unit | Mixing miles and gallons |
| Scientific notation | 1 <= lead factor < 10 | Wrong exponent direction |
Work Method
Write units through the calculation. For a rate problem, decide whether you need miles per gallon, gallons per mile, cups per batch, or batches per cup. In percent problems, identify the original value before touching a calculator. In fraction problems, ask whether the operation is addition, subtraction, multiplication, or division; each uses a different workflow.
Small arithmetic misses have large downstream effects. One sign error can turn a correct inequality into the opposite interval. One denominator error can make a rational expression unsimplifiable. Drill these basics until they are automatic.
A price drops from 80 dollars to 60 dollars. What denominator belongs in the percent-decrease calculation?