A ratio compares quantities; a rate compares quantities with different units. In either case, order and labels matter. Two cups of concentrate for five liters of water is not the same relationship as five cups of concentrate for two liters of water. Write the units beside the numbers before simplifying or scaling.
MATH-103 · MODULE 07
Ratios, Rates, and Proportional Reasoning
Track how two quantities grow together, find a unit rate, and reason before reaching for a shortcut.
A rational number toolkit
Make the relationship visible before the procedure tries to hide it.
This course begins every module with a teaching conversation and treats fractions, decimals, percents, and ratios as connected ways to describe quantity. Work one decision at a time, choose a representation that fits, then check whether the result belongs in a sensible range.
Restoring this module…
Before you begin
Ratios and rates compare quantities with labels that must stay attached.
Proportional relationships grow or shrink by the same factor. Ratio tables and double number lines make that factor visible across several pairs of values. They are more informative than cross-multiplying at the start because they show what each number means and whether the relationship actually stays constant.
A unit rate is a useful comparison tool, not an automatic decision maker. A lower cost per kilogram may still require more money today or more storage. Use the calculation to clarify one part of a decision, then name the practical condition that still matters.
Start
Say what each number counts
Learning target
A ratio compares two labeled quantities; the order and units matter.
A ratio of 3 cups concentrate to 5 cups water is written 3:5. It says “for every 3 cups of concentrate, there are 5 cups of water.” It is a relationship, not automatically a piece of one whole.
Read or listen to the teaching conversation for this step
Module 7, Start
Transcript: Start by reading the relationship in a full sentence. Ratios need labels. Reversing the order changes the meaning, just as swapping miles per hour with hours per mile changes what is being described.
Guided practice
Make one decision, get a useful response, then continue.
A response counts as independently complete only when it is correct. Hints and recovery routes remain available whenever you need them.
Read a ratio in context
A mix uses 2 scoops of powder for every 3 cups of water. Which ratio names powder to water?