What is Proportion ?
Proportion is a concept that is closely interlinked with ratios and fractions. A ratio is a comparison of quantities having the same unit. A ratio helps us determine how big or how small a quantity is when compared to another quantity. Proportion is an equation that states that two ratios or two fractions are equivalent. That is, two ratios are said to be proportional when they are equal. We learned that a ratio can be expressed in several forms by performing arithmetic operations on both sides of the ratio.
How do we represent a Proportion ?
3 : 5 and 6 : 10 are equivalent ratios. That means these ratios are proportional. We can represent this proportionality using fractions:
\(\frac{3}{5} = \frac{6}{10}\)
This conveys that the two ratios are proportional. To verify this proportionality, we can perform arithmetic operations on the left-hand side of the equation.
Multiplying both terms of the LHS by 2: \(\frac{3}{5}\times \frac{2}{2} = \frac{6}{10}\) = RHS
Multiplying the LHS by different terms gives us bigger ratios that are still equivalent.
\(\frac{3}{5}\times \frac{3}{3} = \frac{9}{15}\)
\(\frac{3}{5}\times \frac{4}{4} = \frac{12}{20}\)
There is an infinite number of proportional ratios for any given ratio.
How do we determine whether Ratios form a Proportion ?
The easiest way to check if two ratios form a proportion is by simplifying both ratios into their simplest forms. Consider the ratios 16 : 28 and 36 : 63.
The simplest form of 16:28 = \(\frac{16÷4}{28÷4} = \frac{4}{7}\)
The simplest form of 36:63 = \(\frac{36÷9}{63÷9} = \frac{4}{7}\)
Since 16 : 28 and 36 : 63 are essentially the same ratios, they are in proportion.
Cross product method
Alternatively, ratios can be tested for proportion using cross-product. To check whether ratios are in proportion, we can express them as fractions and move the denominators to the opposite sides.
\(\frac{16}{28} = \frac{36}{63}\)
⇒ \(16\times 63 = 36\times 28\)
⇒ 1008 = 1008
Since both sides of the equation have the same value, we can conclude that the ratios are in proportion. If they are not in proportion, the values on either side of the equation will be different.
Solved Proportional Relationship Examples
Example 1: Determine whether the ratios 24 : 36 and 8 : 12 are in proportion.
Solution:
The simplest form of 24 : 36
= \(\frac{24÷12}{36÷12} = \frac{2}{3}\)
The simplest form of 8 : 12
= \(\frac{8÷4}{12÷4}=\frac{2}{3}\)
The values of the ratios are equivalent.
Therefore, the ratios 24 : 36 and 8 : 12 are in proportion.
Example 2: Determine whether x and y are proportional.
Solution:
Let’s check whether the first two ratios are in proportion by using the cross-product property.
Let’s assume that
\(\frac{4}{1} = \frac{6}{2}\)
We have to check whether the values of ratios are equivalent.
⇒ \(4\times 2 = 6\times 1\) Find the cross products
⇒ 8 ≠ 6 The cross products are not equal.
Now, let’s check whether the next set of ratios is equivalent.
Assume that \(\frac{4}{1} = \frac{8}{3}\)
⇒ \(4\times 3 = 8\times 1\) Find the cross products.
⇒ 12 ≠ 8 The cross products are not equal.
Assume that \(\frac{4}{1} = \frac{10}{5}\)
⇒ \(4\times 5 = 10\times 1\) Find the cross products.
⇒ 20 ≠ 10 The cross products are not equal.
Hence, we can conclude that x and y are not in proportion.
Example 3: Suppose a new road is paved at a constant rate for 10 days. If the distance paved in 3 days is 14 miles, calculate the distance paved in 9 days.
Solution:
Since the road is paved at a constant rate, the ratio between the distance paved and the number of days is proportional. Hence, the value on either side of the equation after taking the cross product is the same.
Let us assume that the distance of the road paved in 9 days is \(x\).
Assume that \(\frac{3}{14} = \frac{9}{x}\)
⇒ \(3\times x = 9\times 14\) Find the cross products.
⇒ \(x = 126÷3\) Find the value of \(x\).
⇒ \(x = 42\)
Therefore, the distance paved in nine days is 42 miles.
Example 4: Determine whether the following triangles are proportional.
Solution:
If the two triangles are proportional, the lengths of their sides will also be proportional.
Let’s compare the side AB with PQ, AC with PR, and BC with QR.
\(\frac{AB}{PQ} = \frac{4}{12} = \frac{1}{3}\) Comparing AB and PQ
\(\frac{AC}{PR} = \frac{4}{12} = \frac{1}{3}\) Comparing AC and PR
\(\frac{BC}{QR} = \frac{5}{15} = \frac{1}{3}\) Comparing BC and QR
Since all three sides of the triangle are proportional, we can say that the triangles are proportional to each other.
Proportions and ratios are related concepts. Two ratios are said to be in proportion if they are equal. A proportion is an equation that relates two ratios.
To check whether two ratios are in proportion, we can either use the cross-product method or simplify the ratio into their simplest forms
Since ratios are the same as fractions, two fractions can be proportional as well. Two fractions are said to be proportional if they are equivalent, i.e. their simplest form is the same.