Rates as Ratios
A ratio compares two amounts. A unit rate is a ratio with in the second slot, which makes scaling up easy: just multiply.
A ratio compares two quantities by division: red to blue, written . A rate is a ratio of two different units, like miles per hour, and a unit rate makes the second quantity . Equivalent ratios scale both numbers by the same factor — a ratio table is just multiplication in rows.
If grams of sugar go with one cup of flour, then cups need grams.
A bag has red marbles for every blue. Red to blue is . Each group of marbles has red, so red to all is : three fifths of the marbles are red.
Comparing ratios by subtracting instead of dividing. and are the same ratio even though the differences and differ.
A class has girls and boys. Write girls to boys in simplest form, and girls to the whole class.
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Girls to boys: . Both numbers share , so it is . The class has students, so girls to class is .
In Plain Terms & More Examples
A ratio compares two amounts. A unit rate rewrites it with in the second slot, which makes scaling as simple as multiplying.
If is earned in hours, the unit rate is per hour.
The ratio scaled up by becomes .
Writing the ratio backwards. " cups of juice for every cups of water" is juice to water . Writing describes cups of juice for of water, a much stronger drink. The order of the words is the order of the numbers.
You earn in hours. What is the unit rate, and how much would you earn in hours?
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per hour. In hours: .
Ratio Tables and Unit Rates
A ratio table multiplies both numbers by the same factor row by row. Dividing both by the second number gives the unit rate — "per one".
Every column of a ratio table is the first column multiplied by the same number, so every column describes the same mix. Dividing both entries by the second one gives the column whose second entry is , the unit rate, and from there any other column is one multiplication away.
pens cost . Per pen: . So pens cost .
Juice and water are mixed , and you want cups in all. One batch is cups, and batches. Juice: cups; water: cups. Check: .
Adding the same amount to both parts. and look alike, but and is adding, not scaling. Equivalent ratios multiply both parts: . In the blue is of the yellow instead of , a different colour.
Paint is mixed parts blue to parts yellow. How much blue for liters of paint?
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Each batch of liters has blue. batches, so liters of blue.
Word Problems
A rate compares two different kinds of quantity: kilometres per hour, dollars per kilogram, minutes per kilometre. Find the unit rate by dividing, then multiply to scale.
Dividing by the second quantity answers "how much for one?". Once that is known, any amount is one multiplication away, and two offers can be compared fairly by putting both in the same "per one" form.
A train goes km in hours: km/h. In hours it goes km.
Ana runs km in minutes. Her pace is minutes per kilometre. At that pace, km takes minutes.
Dividing the wrong way round. is kilometres per hour. is hours per kilometre, a different and tiny rate. Say the units out loud: "km per h" puts km on top.
Store A sells bagels for ; Store B sells for . Which is cheaper per bagel?
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A: . B: . Store B is cheaper.
Graphing Ratios
Plot each column of a ratio table as a point . Equivalent ratios fall on a straight line through , and the unit rate is how far the line rises for each step of across.
Every column is the column scaled up, so every point is the same "one across, rate up" step repeated from . Repeating one step traces a straight line. A point off that line is a different ratio.
A dough uses cups of flour for cups of water: . The points line up through , and for every cup of flour there are cups of water.
Continue the line to cups of flour: cups of water, the point . The table agrees: scaled by is .
Assuming any straight line means equivalent ratios. The points are in a line, but it does not pass through , and , and are all different. Equivalent ratios need a line through the origin.
Are , and in the same ratio? Is ?
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, so the first three are the same ratio, all on one line through . For , , so it is not.
Why It Works
Two ratios and are equivalent exactly when , i.e. when (multiply both sides by ). Scaling a ratio by keeps the fraction the same because — this is equivalent fractions again. A unit rate is the fraction itself: it answers "how much of the first for one of the second".
Do not read a proof the way you read a rule. Cover the last line, ask what the previous line forces, and only then uncover it. If a step feels like magic, that is the step to work by hand with a concrete number.