A reliable method
Write a unit equation before calculating. Add amounts of the same substance in a mixture. Add work rates only when workers contribute independently to the same job at constant rates.
The common trap
Do not average two travel speeds unless the times are equal. For equal distances, the slower segment uses more time and therefore carries more weight.
Make the connection
Compare the answer with physical bounds: a mixture lies between source concentrations and a joint completion time should be shorter than either individual time.
Contrast the cases
Equal times at 30 and 60 km/h give average speed 45 km/h. Equal distances give 40 km/h: take 60 km on each leg, requiring two hours and one hour, so 120/3 = 40. Compare this with work: workers needing 6 and 3 hours have rates 1/6 and 1/3, totaling 1/2 job per hour and taking 2 hours together. Averaging times would not conserve distance or completed work.
Transfer and explain
Transfer: equal 80 km legs at 40 and 80 km/h take 3 hours total, so average speed is 160/3. State which quantity is equal before deciding whether a simple arithmetic average is justified.
A dye mixture uses red and blue in ratio 6:9. How many liters of red are in 25 liters of mixture?
- The ratio simplifies to 2:3, so red is 2/5 of the total.
- Multiply 25 by 2/5.
- The red amount is 10 liters.
Compare the other answer choices
Selects the blue share.
Uses the whole mixture as the red amount.
Uses blue alone as the denominator.
Splits the mixture equally despite unequal parts.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
GRE Quantitative Reasoning overviewSiyavula Grade 10 mathematics foundation reference