The Central Idea
Suppose
\[ \frac{2}{3} = \frac{4}{6}. \]These ratios are equal because the second ratio was created by multiplying both parts of the first ratio by \(2\):
\[ \frac{2\cdot2}{3\cdot2} = \frac{4}{6}. \]The relationship between numerator and denominator has not changed.
This gives the central proportion principle:
If two quantities maintain the same multiplicative relationship, their ratios remain equal.
What Is a Proportion?
A proportion is an equation stating that two ratios are equal.
For example,
\[ \frac{3}{5} = \frac{12}{20}. \]Because both ratios simplify to
\[ \frac35, \]the equation is a true proportion.
Ratio, Rate, and Proportion
A ratio compares two quantities by division.
A rate is a ratio involving quantities with different units.
A proportion states that two ratios or rates are equal.
For example,
\[ \frac{60\text{ miles}}{1\text{ hour}} = \frac{180\text{ miles}}{3\text{ hours}} \]is a proportion comparing two equivalent rates.
Equivalent Ratios
Ratios are equivalent when they have the same value.
Starting with
\[ \frac{a}{b}, \qquad b\ne0, \]multiply numerator and denominator by the same nonzero number \(k\):
\[ \frac{ka}{kb}. \]Then
\[ \frac{ka}{kb} = \frac{k}{k}\cdot\frac{a}{b} = 1\cdot\frac{a}{b} = \frac{a}{b}. \]So common scaling preserves a ratio.
Scaling Up
Suppose the ratio is
\[ \frac{4}{7}. \]Multiply both quantities by \(3\):
\[ 4\cdot3=12, \] \[ 7\cdot3=21. \]Therefore,
\[ \frac47=\frac{12}{21}. \]The quantities changed, but the relationship between them did not.
Scaling Down
Consider
\[ \frac{18}{24}. \]Divide numerator and denominator by \(6\):
\[ \frac{18\div6}{24\div6} = \frac34. \]So
\[ \frac{18}{24} = \frac34. \]Reducing a ratio is scaling both quantities down by the same nonzero factor.
Proportions Preserve Multiplicative Structure
Suppose
\[ \frac{y}{x}=4. \]This means
\[ y=4x, \]provided \(x\ne0\).
If \(x\) doubles, then \(y\) doubles.
If \(x\) triples, then \(y\) triples.
The proportion captures a constant multiplicative relationship.
The Constant of Proportionality
If
\[ \frac{y}{x}=k, \]then \(k\) is the constant of proportionality.
Multiplying both sides by \(x\) gives
\[ y=kx. \]This is the standard equation of a direct proportional relationship.
Unit Rate as a Constant of Proportionality
Suppose a worker earns \$18 per hour.
Then
\[ \frac{E}{h}=18, \]where \(E\) is earnings and \(h\) is hours.
Therefore,
\[ E=18h. \]The unit rate
\[ 18\frac{\$}{\text{hour}} \]is also the constant of proportionality.
Building a Proportion from a Constant Rate
If a car travels at a constant rate of
\[ 60\text{ miles per hour}, \]then
\[ \frac{60\text{ mi}}{1\text{ hr}} = \frac{d}{t}. \]Any distance-time pair consistent with the constant rate forms an equivalent ratio.
For \(t=4\):
\[ \frac{60}{1} = \frac{d}{4}. \]The proportional relationship gives
\[ d=240. \]Solving by Scaling
Consider
\[ \frac{3}{5} = \frac{x}{20}. \]Ask:
What happened to the denominator?
Since
\[ 5\cdot4=20, \]multiply the numerator by the same factor:
\[ 3\cdot4=12. \]Therefore,
\[ \boxed{x=12}. \]Scaling Is Often the Fastest Method
For
\[ \frac{7}{9} = \frac{x}{36}, \]the denominator \(9\) was multiplied by \(4\).
So multiply \(7\) by \(4\):
\[ x=28. \]No formal cross multiplication is necessary when the scale factor is obvious.
Solving by Unit Rate
Suppose
\[ \frac{15\text{ dollars}}{5\text{ items}} = \frac{x\text{ dollars}}{8\text{ items}}. \]First find the unit rate:
\[ \frac{15}{5}=3. \]So the price is
\[ \$3\text{ per item}. \]For 8 items:
\[ x=3\cdot8=24. \]Thus,
\[ \boxed{x=24}. \]Cross Multiplication
For a proportion
\[ \frac{a}{b} = \frac{c}{d}, \]with
\[ b\ne0,\qquad d\ne0, \]we may multiply both sides by \(bd\):
\[ bd\cdot\frac{a}{b} = bd\cdot\frac{c}{d}. \]Cancel:
\[ ad=bc. \]This relationship is often called cross multiplication.
Cross Multiplication Is Not a Separate Law
The rule
\[ \frac{a}{b} = \frac{c}{d} \quad\Longrightarrow\quad ad=bc \]comes from ordinary multiplication properties of equality.
It is not a mysterious shortcut.
We are simply multiplying both sides by the common nonzero quantity \(bd\).
Solving with Cross Multiplication
Solve
\[ \frac{4}{7} = \frac{x}{21}. \]Cross multiply:
\[ 4(21)=7x. \]So
\[ 84=7x. \]Divide by \(7\):
\[ \boxed{x=12}. \]Choose the Simplest Method
The proportion
\[ \frac{4}{7} = \frac{x}{21} \]can be solved by recognizing that \(7\) was multiplied by \(3\).
That immediately gives
\[ x=12. \]Cross multiplication gives the same result, but it is not always the most efficient route.
Solving When the Unknown Is in a Denominator
Solve
\[ \frac{5}{x} = \frac{10}{14}. \]The domain requires
\[ x\ne0. \]Cross multiply:
\[ 5(14)=10x. \]So
\[ 70=10x, \] \[ \boxed{x=7}. \]The solution satisfies the domain restriction.
Proportion Domain Restrictions
Every denominator in a proportion must be nonzero.
For
\[ \frac{x+1}{x-3} = \frac{2}{5}, \]we must record
\[ x\ne3. \]Any candidate equal to \(3\) must be rejected because the original ratio would be undefined.
Verification by Substitution
Suppose solving gives
\[ x=12 \]for
\[ \frac{3}{5} = \frac{x}{20}. \]Substitute:
\[ \frac35 = \frac{12}{20}. \]Simplify the right side:
\[ \frac{12}{20} = \frac35. \]The proportion is true, so the solution is verified.
Equivalent Rates with Units
Consider
\[ \frac{45\text{ miles}}{1\text{ hour}} = \frac{135\text{ miles}}{3\text{ hours}}. \]Both rates simplify to
\[ 45\frac{\text{miles}}{\text{hour}}. \]A proportion involving units is meaningful only when corresponding quantities are arranged consistently.
Keep Corresponding Quantities Aligned
Suppose 3 notebooks cost \$6 and 8 notebooks cost \(x\) dollars.
A consistent proportion is
\[ \frac{3\text{ notebooks}}{\$6} = \frac{8\text{ notebooks}}{x\text{ dollars}}. \]Another consistent form is
\[ \frac{\$6}{3\text{ notebooks}} = \frac{x\text{ dollars}}{8\text{ notebooks}}. \]Both work because corresponding units occupy corresponding positions.
An Inconsistent Setup
The expression
\[ \frac{3\text{ notebooks}}{\$6} = \frac{x\text{ dollars}}{8\text{ notebooks}} \]mixes notebooks-per-dollar on one side with dollars-per-notebook on the other.
These are reciprocal rates, not equal rates.
Correct setup matters before any algebra begins.
Proportion as Matching Multipliers
Suppose
\[ \frac{a}{b} = \frac{c}{d}. \]If there is a scale factor \(k\) such that
\[ c=ka \]and
\[ d=kb, \]then the ratios are equal.
This gives a useful structural interpretation:
A true proportion often says that corresponding quantities were changed by the same multiplicative factor.
Proportions in Similar Figures
Suppose two similar rectangles have side lengths
\[ 4,\ 6 \]and
\[ 10,\ 15. \]Corresponding sides satisfy
\[ \frac{4}{6} = \frac{10}{15}. \]Both dimensions were multiplied by
\[ \frac{10}{4} = \frac{15}{6} = 2.5. \]Similarity preserves corresponding side ratios.
Scale Drawings
Suppose a map uses the scale
\[ 1\text{ cm}:5\text{ km}. \]A map distance of \(7\) cm corresponds to \(d\) kilometers.
Write
\[ \frac{1\text{ cm}}{5\text{ km}} = \frac{7\text{ cm}}{d\text{ km}}. \]Cross multiply:
\[ d=35. \]So the real distance is
\[ \boxed{35\text{ km}}. \]Recipes and Mixtures
Suppose a recipe uses 2 cups of flour for 12 cookies.
For 30 cookies, let \(f\) be the flour required.
\[ \frac{2\text{ cups}}{12\text{ cookies}} = \frac{f\text{ cups}}{30\text{ cookies}}. \]Cross multiply:
\[ 60=12f. \]Thus,
\[ f=5. \]The scaled recipe requires
\[ \boxed{5\text{ cups of flour}}. \]Percent as a Proportion
A percent compares a quantity with \(100\).
For example, \(35\%\) means
\[ \frac{35}{100}. \]If 35% of \(80\) is \(x\), write
\[ \frac{35}{100} = \frac{x}{80}. \]Then
\[ 100x=35(80), \]so
\[ x=28. \]Proportions and Probability
Suppose a sample contains 8 successes in 20 trials.
The observed success rate is
\[ \frac{8}{20} = 0.4. \]If that proportion remained constant over 50 comparable trials, the expected count \(x\) would satisfy
\[ \frac{8}{20} = \frac{x}{50}. \]This produces
\[ x=20. \]The mathematical scaling is exact; whether the real process actually maintains that proportion is a modeling assumption.
Not Every Pair of Quantities Is Proportional
Suppose a taxi charges a \$4 starting fee plus \$2 per mile.
The total cost is
\[ C=2m+4. \]The ratio
\[ \frac{C}{m} \]is not constant because of the fixed starting fee.
So cost and miles are related linearly, but not proportionally.
Proportional Relationships Pass Through the Origin
A direct proportional relationship has form
\[ y=kx. \]When
\[ x=0, \]we get
\[ y=0. \]Therefore, its graph passes through the origin.
A nonzero intercept indicates that the relationship is not directly proportional.
Tables of Proportional Values
Consider:
| \(x\) | \(y\) | \(y/x\) |
|---|---|---|
| 2 | 6 | 3 |
| 4 | 12 | 3 |
| 7 | 21 | 3 |
Because
\[ \frac{y}{x}=3 \]for every nonzero \(x\), the relationship is proportional:
\[ y=3x. \]A Nonproportional Table
| \(x\) | \(y\) | \(y/x\) |
|---|---|---|
| 2 | 7 | 3.5 |
| 4 | 11 | 2.75 |
| 6 | 15 | 2.5 |
The quotient is not constant.
Therefore, the relationship is not proportional.
Proportions and Fractions Are Closely Related
A proportion is an equation between fractions:
\[ \frac{a}{b} = \frac{c}{d}. \]So many tools learned for fractions apply directly:
- equivalent fractions;
- simplifying;
- common scaling;
- multiplication by reciprocals;
- and denominator restrictions.
Why Cross Products Are Equal
If
\[ \frac{a}{b} = \frac{c}{d}, \]then both ratios represent the same number, say \(k\):
\[ \frac{a}{b}=k, \qquad \frac{c}{d}=k. \]Thus,
\[ a=kb \]and
\[ c=kd. \]Then
\[ ad=(kb)d \]and
\[ bc=b(kd). \]Both equal
\[ kbd, \]so
\[ ad=bc. \]The Converse Also Holds
If
\[ ad=bc \]and
\[ b\ne0,\qquad d\ne0, \]then divide both sides by \(bd\):
\[ \frac{ad}{bd} = \frac{bc}{bd}. \]Simplify:
\[ \frac{a}{b} = \frac{c}{d}. \]Therefore, under nonzero denominator conditions,
\[ \frac{a}{b} = \frac{c}{d} \quad\Longleftrightarrow\quad ad=bc. \]Checking Whether Four Numbers Form a Proportion
Do
\[ 6,\ 9,\ 14,\ 21 \]form the proportion
\[ \frac69 = \frac{14}{21}? \]Compute cross products:
\[ 6(21)=126, \] \[ 9(14)=126. \]Since the cross products are equal, the proportion is true.
Order Matters
Suppose
\[ \frac{2}{3} = \frac{8}{12}. \]Reversing both ratios gives another true proportion:
\[ \frac32 = \frac{12}{8}. \]But randomly rearranging terms can destroy the correspondence.
Proportion setup should reflect the same quantity order on both sides.
Units Help Preserve Order
Suppose 4 meters correspond to 10 seconds and 14 meters correspond to \(t\) seconds.
Write
\[ \frac{4\text{ m}}{10\text{ s}} = \frac{14\text{ m}}{t\text{ s}}. \]The matching units make the intended correspondence obvious.
Keeping units visible is one of the best safeguards against reversed proportions.
Proportion Versus Difference
Proportional reasoning is multiplicative, not additive.
For example,
\[ 2:3 \]and
\[ 4:5 \]have the same difference:
\[ 3-2=1, \] \[ 5-4=1. \]But they are not proportional because
\[ \frac23\ne\frac45. \]Equal differences do not imply equal ratios.
Additive and Multiplicative Thinking
Compare:
\[ 3,\ 6,\ 9,\ 12 \]and
\[ 3,\ 6,\ 12,\ 24. \]The first pattern uses a constant additive difference of \(3\).
The second uses a constant multiplicative factor of \(2\).
Proportion belongs to the multiplicative family of relationships.
Worked Example: Scale Factor
Solve
\[ \frac{5}{8} = \frac{x}{40}. \]Since
\[ 8\cdot5=40, \]multiply the numerator by \(5\):
\[ x=25. \]Therefore,
\[ \boxed{x=25}. \]Worked Example: Cross Multiplication
Solve
\[ \frac{7}{12} = \frac{21}{x}. \]Domain:
\[ x\ne0. \]Cross multiply:
\[ 7x=12(21). \]So
\[ 7x=252, \] \[ \boxed{x=36}. \]Worked Example: Unit Rate
Five pounds of rice cost \$9.50. What will 8 pounds cost at the same rate?
First find the cost per pound:
\[ \frac{9.50}{5}=1.90. \]Then multiply:
\[ 1.90(8)=15.20. \]Therefore,
\[ \boxed{\$15.20}. \]Worked Example: Similar Figures
A small triangle has a side of \(6\) corresponding to a side of \(15\) in a larger similar triangle.
Another small side is \(10\). Let \(x\) be its corresponding large side.
\[ \frac{6}{15} = \frac{10}{x}. \]Cross multiply:
\[ 6x=150. \]Thus,
\[ \boxed{x=25}. \]Worked Example: Percent
What is \(18\%\) of \(250\)?
Write
\[ \frac{18}{100} = \frac{x}{250}. \]Cross multiply:
\[ 100x=18(250). \]Then
\[ x=45. \]Therefore,
\[ \boxed{45}. \]A Reliable Proportion Workflow
- Identify the two quantities being compared.
- Keep corresponding quantities in the same numerator/denominator positions.
- Write units when they help reveal the structure.
- Check denominator restrictions.
- Look for an obvious scale factor first.
- Use a unit rate when it makes the relationship clearer.
- Use cross multiplication when scaling is not obvious.
- Solve the resulting equation.
- Verify the candidate in the original proportion.
- Interpret the answer in context.
Common Errors
Treating Cross Multiplication as Magic
Cross multiplication comes from multiplying both sides of an equation by the denominators.
Mixing Corresponding Quantities
If one ratio is dollars per pound, the other must also be dollars per pound.
Using Additive Instead of Multiplicative Reasoning
Proportions preserve ratios, not differences.
Ignoring Zero Denominators
Every ratio denominator must be nonzero.
Assuming Any Linear Relationship Is Proportional
A relationship such as
\[ y=3x+2 \]is linear but not proportional because the ratio \(y/x\) is not constant and the graph does not pass through the origin.
Using a Proportion When the Rate Is Not Constant
Proportional predictions require a constant multiplicative relationship.
Practice
1. Scale Factor
Solve:
\[ \frac{4}{9} = \frac{x}{27}. \]Check Answer
Since \(9\cdot3=27\),
\[ x=4\cdot3=12. \]Answer:
\[ \boxed{x=12}. \]2. Cross Multiplication
Solve:
\[ \frac{5}{8} = \frac{15}{x}. \]Check Answer
\[ 5x=120, \] \[ \boxed{x=24}. \]3. Unit Rate
Seven notebooks cost \$17.50. What will 10 notebooks cost at the same rate?
Check Answer
Unit price:
\[ \frac{17.50}{7}=2.50. \]Then
\[ 2.50(10)=25. \]Answer:
\[ \boxed{\$25}. \]4. Check a Proportion
Is
\[ \frac{8}{12} = \frac{14}{21} \]a true proportion?
Check Answer
\[ 8(21)=168, \] \[ 12(14)=168. \]Yes.
5. Similar Figures
Solve:
\[ \frac{3}{5} = \frac{12}{x}. \]Check Answer
\[ 3x=60, \] \[ \boxed{x=20}. \]6. Percent
What is \(24\%\) of \(150\)?
Check Answer
\[ \frac{24}{100} = \frac{x}{150}. \] \[ 100x=3600, \] \[ \boxed{x=36}. \]7. Proportional or Not?
Is
\[ y=5x+1 \]a proportional relationship?
Check Answer
No. A direct proportional relationship has form
\[ y=kx. \]The nonzero intercept \(1\) means the graph does not pass through the origin.
Key Ideas
- A proportion is an equation stating that two ratios are equal.
- Equivalent ratios preserve the same multiplicative relationship.
- Scaling both parts of a ratio by the same nonzero factor leaves the ratio unchanged.
- A unit rate often serves as the constant of proportionality.
- Direct proportional relationships have the form \(y=kx\).
- Scaling and unit-rate reasoning are often more intuitive than cross multiplication.
- Cross multiplication follows from ordinary multiplication properties of equality.
- For nonzero denominators, \(\frac{a}{b}=\frac{c}{d}\) is equivalent to \(ad=bc\).
- Corresponding quantities and units must remain aligned.
- Proportional reasoning is multiplicative rather than additive.
- Not every linear relationship is proportional.
- Proportion methods apply only when the underlying multiplicative relationship is constant.
- Proportions connect rates, scaling, similarity, percent, measurement, and modeling.
Where This Leads
The proportion principle provides a common structure for many problems that appear unrelated on the surface.
Equivalent rates, scale drawings, similar figures, percent, conversion factors, direct variation, work and motion models, mixture problems, and many forms of mathematical modeling all depend on preserving or interpreting multiplicative relationships.