LESSON 1 · 21 MIN
Applied mathematics
Name the requested quantity, translate the relationship with an explicit unit ledger, solve exactly, and reject results that fail scale, unit, or context checks.
ESSENTIAL QUESTIONWhat relationship is invariant—and what unit must remain after every valid factor cancels?
Build the equation first; require the requested unit to survive.
| Ledger step | Exact operation | Unit before cancellation | Result |
|---|---|---|---|
| Base rate | 12 ÷ 30 = 2/5 | miles/minute | 2/5 mile/minute |
| Conversion factor | × 60 minutes/hour | miles/minute × minutes/hour | minute units cancel |
| Requested rate | (2/5) × 60 = 24 | miles/hour | 24 miles/hour |
| Stage | Comparison base | Multiplier | Stage amount |
|---|---|---|---|
| Listed price | Original amount | 1 | $80.00 |
| After 15% discount | $80.00 | 17/20 | $68.00 |
| After 5% sales tax | $68.00 | 21/20 | $71.40 |
The conversion factor is a form of one, and each percentage acts on the amount present at that stage. Neither result depends on a surface keyword.
Preserve what adds, then test the result against unit and scale.
| Component | Amount | Concentration | Active amount |
|---|---|---|---|
| Solution A | 3 liters | 20% | 3/5 active liters |
| Solution B | 2 liters | 50% | 1 active liters |
| Combined mixture | 5 liters | 32% | 8/5 active liters |
| Quantity | Exact model | Value | Unit audit |
|---|---|---|---|
| Wall area | 50 feet × 12 feet | 600 | square feet |
| Crew A rate | stated constant rate | 120 | square feet/hour |
| Crew B rate | stated constant rate | 80 | square feet/hour |
| Combined rate | 120 + 80 | 200 | square feet/hour |
| Completion time | 600 ÷ 200 | 3 | hours |
| Context | Before calculating | Exact result | Acceptance check |
|---|---|---|---|
| Motion | More than 12 mph; less than 30 mph | 24 miles/hour | requested time denominator remains |
| Finance | Below $80; tax recovers part of the discount | $71.40 | tax uses the discounted subtotal |
| Mixture | Between 20% and 50%; closer to 20% | 32% | unequal volumes are weighted |
| Work + geometry | Faster than either crew alone | 3 hours | area divided by compatible combined rate |
Estimates are not replacement answers. They are acceptance tests that catch wrong bases, incompatible units, reversed ratios, and implausible calculator output.
Define before substituting
Name the target and each quantity before combining numbers, so the equation represents the stated relationship instead of a keyword guess.
- Distance equals rate times time; work equals rate times time.
- A percent multiplier acts on the amount that exists at that stage.
Carry the units
Treat conversion factors as forms of one and require unwanted units to cancel before arithmetic is accepted.
- Linear, square, and cubic measures transform differently.
- Mixture totals add amounts and active amounts separately.
Estimate, solve, audit
Predict direction and magnitude, calculate only after the model is stable, then interpret the exact value in the requested context.
- Combined positive work rates make completion faster than either rate alone.
- A precise display with the wrong unit or comparison base is not a valid answer.
Worked example
A rectangular wall is 50 feet long and 12 feet high. Two crews paint 120 and 80 square feet per hour. How long do they need together?
- 1
Find the required work: 50 feet × 12 feet = 600 square feet.
- 2
Add compatible work rates: 120 + 80 = 200 square feet per hour.
- 3
Divide work by combined rate: 600 square feet ÷ 200 square feet per hour = 3 hours.
ConclusionThe crews need 3 hours. Square feet cancel, leaving hours, and the result is faster than either crew working alone.
Calculator decision ledger
- Model mentally
- Name the target, quantities, relationship, restrictions, and unit conversion factors before entering any values.
- Estimate
- Predict whether the answer should grow or shrink and bracket its plausible magnitude from the context.
- Calculator
- Use the digital calculator for the final product or quotient only after the exact relationship and units are visible.
- Audit
- Confirm the surviving unit, sign, scale, feasible domain, requested quantity, and any necessary contextual rounding.
Close the notes first
Retrieve the relationship.
01What must happen to unwanted units in a valid conversion?
The remaining unit identifies the quantity the calculation actually produces.
02Why are successive percent changes multiplied instead of added?
A later tax, discount, or growth rate generally has a new comparison base.
03What two totals are preserved in a concentration mixture?
The final concentration is total active amount divided by total amount.