The Same Relationship in Different Forms
A proportion is more than an equation containing fractions. It expresses the idea that two comparisons describe the same relationship.
Consider the ratios
\[ \frac{2}{3} \qquad\text{and}\qquad \frac{6}{9}. \]The numbers are different, but the second ratio is created by multiplying both parts of the first ratio by 3:
\[ \frac{2\times3}{3\times3} = \frac{6}{9}. \]Because the relationship has not changed, the ratios are equivalent:
\[ \frac{2}{3}=\frac{6}{9}. \]That equation is a proportion.
Equivalent Ratios
Two ratios are equivalent when they represent the same relative comparison.
For example,
\[ \frac{3}{5} = \frac{6}{10} = \frac{9}{15} = \frac{12}{20}. \]Each ratio has different numbers, but every ratio expresses the same relationship: 3 units of one quantity for every 5 units of another.
Equivalent ratios work much like equivalent fractions. Multiplying or dividing both parts of a ratio by the same nonzero number changes its appearance without changing its value.
\[ \frac{3}{5} = \frac{3\times4}{5\times4} = \frac{12}{20}. \]Scale Factors Preserve the Relationship
One useful way to recognize equivalent ratios is to look for a common scale factor.
Suppose we compare
\[ \frac{4}{7} \qquad\text{and}\qquad \frac{20}{35}. \]The numerator 4 became 20 by multiplying by 5:
\[ 4\times5=20. \]The denominator 7 changed by the same factor:
\[ 7\times5=35. \]Because both quantities were scaled by the same amount, the original relationship was preserved:
\[ \frac{4}{7}=\frac{20}{35}. \]This idea of preserving a relationship while quantities scale upward or downward is at the heart of proportional reasoning.
What Makes an Equation a Proportion?
A proportion has the general form
\[ \frac{a}{b}=\frac{c}{d}, \]where \(b\neq0\) and \(d\neq0\).
The denominators cannot be zero because division by zero is undefined.
The equation states that the ratio \(a:b\) has the same value as the ratio \(c:d\).
For example,
\[ \frac{5}{8}=\frac{15}{24} \]is a proportion because both ratios simplify to the same value.
But
\[ \frac{5}{8}\neq\frac{15}{20} \]is not a proportion because the two ratios are not equivalent.
Checking a Proportion by Simplifying
One way to determine whether two ratios form a proportion is to simplify them.
Consider
\[ \frac{8}{12} \qquad\text{and}\qquad \frac{14}{21}. \]Simplify the first ratio:
\[ \frac{8}{12}=\frac{2}{3}. \]Simplify the second:
\[ \frac{14}{21}=\frac{2}{3}. \]Since both ratios simplify to the same value,
\[ \frac{8}{12}=\frac{14}{21}. \]They form a proportion.
Cross-Products
There is another powerful way to test whether two ratios are equivalent. Consider a proportion written as
\[ \frac{a}{b}=\frac{c}{d}. \]The products formed diagonally across the equality are called cross-products.
\[ a\cdot d \qquad\text{and}\qquad b\cdot c. \]If the two ratios form a proportion, these products are equal:
\[ ad=bc. \]This relationship is commonly called the fundamental property of proportions.
Why Do Cross-Products Work?
Cross multiplication is not a mathematical trick. It follows directly from ordinary multiplication.
Begin with
\[ \frac{a}{b}=\frac{c}{d}. \]Multiply both sides by \(bd\):
\[ bd\left(\frac{a}{b}\right) = bd\left(\frac{c}{d}\right). \]On the left, the factor \(b\) cancels:
\[ ad. \]On the right, the factor \(d\) cancels:
\[ bc. \]Therefore,
\[ ad=bc. \]The cross-product property is simply a convenient consequence of the original equality of the two ratios.
Checking with Cross-Products
Consider
\[ \frac{4}{7}=\frac{12}{21}. \]Form the cross-products:
\[ 4(21)=84 \]and
\[ 7(12)=84. \]Since the cross-products are equal,
\[ 4(21)=7(12), \]the ratios form a proportion.
Now compare
\[ \frac{4}{7} \qquad\text{and}\qquad \frac{10}{20}. \]Their cross-products are
\[ 4(20)=80 \]and
\[ 7(10)=70. \]Since
\[ 80\neq70, \]the ratios are not equivalent and therefore do not form a proportion.
Three Ways to Recognize Equivalent Ratios
At this stage, you have several ways to decide whether two ratios form a proportion.
- Look for a common scale factor. If both parts of one ratio were multiplied or divided by the same nonzero number, the ratios are equivalent.
- Simplify both ratios. If they reduce to the same ratio, they are equivalent.
- Compare the cross-products. If the cross-products are equal, the ratios form a proportion.
These are not three unrelated techniques. They are three different ways of examining the same underlying fact: the ratios represent the same relationship.
Keep Corresponding Quantities in the Same Order
A proportion only makes sense when corresponding quantities are compared consistently.
Suppose 3 notebooks cost $12 and 5 notebooks cost $20. One correct proportion is
\[ \frac{3\text{ notebooks}}{\$12} = \frac{5\text{ notebooks}}{\$20}. \]Both ratios use \(\text{notebooks}/\text{dollars}\).
We could also reverse both ratios:
\[ \frac{\$12}{3\text{ notebooks}} = \frac{\$20}{5\text{ notebooks}}. \]Now both ratios use \(\text{dollars}/\text{notebooks}\).
What we should not do is reverse only one side:
\[ \frac{3\text{ notebooks}}{\$12} \neq \frac{\$20}{5\text{ notebooks}}. \]Writing units carefully is one of the best ways to keep a proportion organized correctly.
Proportions with Units
Proportions often compare measurements rather than plain numbers.
Suppose a vehicle travels 90 miles in 3 gallons of fuel and 150 miles in 5 gallons.
We can compare distance with fuel:
\[ \frac{90\text{ miles}}{3\text{ gallons}} = \frac{150\text{ miles}}{5\text{ gallons}}. \]Both sides have the same unit rate:
\[ 30\text{ miles per gallon}. \]The equality of the ratios tells us that the same distance-to-fuel relationship is being maintained.
Proportional or Not?
It is important to distinguish between quantities that merely change together and quantities that maintain a constant ratio.
Suppose 2 tickets cost $10 and 6 tickets cost $30.
\[ \frac{\$10}{2\text{ tickets}} = \$5\text{ per ticket} \]and
\[ \frac{\$30}{6\text{ tickets}} = \$5\text{ per ticket}. \]The unit rate remains the same, so the relationship is proportional.
Now suppose a service charges a $10 starting fee plus $5 for each hour. One hour costs $15, while two hours cost $20.
\[ \frac{\$15}{1\text{ hour}} \neq \frac{\$20}{2\text{ hours}}. \]The ratios are not equal. The quantities change together, but they do not form a proportional relationship.
Cross Multiplication Is Not the Definition
It is easy to begin thinking that a proportion is simply an equation where we cross multiply. That reverses the real logic.
A proportion exists because two ratios are equal.
Cross-products are useful because equal ratios have the property
\[ ad=bc. \]In other words, cross multiplication helps us analyze proportions; it is not what makes a relationship proportional in the first place.
Check Your Understanding
- Do \(\frac{4}{5}\) and \(\frac{12}{15}\) form a proportion?
- Do \(\frac{7}{9}\) and \(\frac{21}{30}\) form a proportion?
- What common scale factor connects \(\frac{5}{8}\) with \(\frac{20}{32}\)?
- What must be true of the cross-products if \[ \frac{a}{b}=\frac{c}{d}? \]
- Why is \[ \frac{4\text{ books}}{\$20} = \frac{\$35}{7\text{ books}} \] not written with corresponding quantities in consistent positions?
Show Answers
- Yes. \[ 4(15)=60 \] and \[ 5(12)=60. \]
- No. \[ 7(30)=210 \] while \[ 9(21)=189. \]
- The scale factor is \(4\): \[ 5(4)=20,\qquad 8(4)=32. \]
- The cross-products must be equal: \[ ad=bc. \]
- The first ratio compares books to dollars, while the second compares dollars to books. Both ratios must use the same ordering of corresponding quantities.
Key Ideas
- A proportion is an equation stating that two ratios are equal.
- Equivalent ratios represent the same relative relationship.
- Scaling both parts of a ratio by the same nonzero factor preserves the ratio.
- Equivalent ratios can often be recognized by simplifying them.
- If \(\frac{a}{b}=\frac{c}{d}\), then \(ad=bc\).
- The equality \(ad=bc\) is the fundamental property of proportions.
- Cross multiplication follows from the equality of ratios; it is not the definition of proportion.
- Corresponding quantities and units should remain in consistent positions.