A reliable method
Write conversion factors so unwanted units cancel. For a mixture, multiply each volume by its concentration, then divide total ingredient by total volume.
The common trap
A ratio between two parts is not a part-to-total fraction. A speed averaged over equal distances is generally not the arithmetic mean of the speeds.
Make the connection
Check whether a scale factor applies to length, area or volume. Put units beside every number before deciding which operations are sensible.
Worked example: choose the requested quantity
A map scale is 1 cm to 3 km. Two towns are 4 cm apart on the map. How many kilometers apart are they? Each map centimeter represents the stated real distance. Multiply map distance by kilometers per centimeter; centimeters cancel. The requested value is 12.
Contrast and diagnose an error
A 3-liter solution is 20% salt and a 3-liter solution is 40% salt. What percent salt is their combined solution? Find the amount of salt in each equal-volume solution. Divide total salt by total solution volume, then express the fraction as a percent. The requested value is 30. A tempting answer is 15: Halve the already calculated 30 percent mixture concentration again because there are two solutions.
Transfer practice and check
A cyclist rides 18 km at 6 km/h, then another 18 km at 9 km/h. What is the average speed for the whole trip in km/h? Try it before reading on. Check: 36/5. Compute the time for each leg using distance divided by speed. Divide total distance by total time, not by the average of the two speeds.
A map scale is 1 cm to 3 km. Two towns are 4 cm apart on the map. How many kilometers apart are they?
- Each map centimeter represents the stated real distance.
- Multiply map distance by kilometers per centimeter; centimeters cancel.
- The requested value is 12.
Compare the other answer choices
Report distance per centimeter rather than total distance.
Incorrectly multiply the kilometer result by 100 using a centimeters-per-meter factor.
Divide by the map conversion factor.
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.
ACT Mathematics content