What are Proportions?
Proportions are mathematical equations that are used to relate equivalent ratios. Two ratios having different antecedents and consequents can have the same value. This relation can be expressed with the help of proportions. Let us consider the ratios \( 2:5 \) and \( 8:20 \). When we simplify the ratio \( 8:20 \), we obtain \( 2:5 \). This implies that \( 2:5 \) and \( 8:20 \) are equivalent ratios.
In other words, these ratios are in proportion. If \( a:b \) and \( c:d \) are equivalent ratios, we can express the relation as \( a:b::c:d \), where the ‘\( ~::~ \)’ sign is used to express proportion. So, we can state the proportion in the example that was just observed as, \( 2:5 :: 8:20 \).
Writing Proportions as Fractions
We have learned that a ratio is a comparison of two quantities having the same unit. We also compare two quantities using fractions. Hence, a ratio can also be written as a fraction. For example, we can write the ratio \( a:b \) as \( \frac{a}{b} \) and \( c:d \) as \( \frac{c}{d} \).
Similarly, we can write a proportion as a fraction. Instead of writing \( 2:5::8:20 \) , we can write the proportion as \( \frac{2}{5} = \frac{8}{20} \) , and this makes it easier to perform operations on proportions to solve for unknown values.
Using Operations to Solve Proportions
Since a proportion is basically an equation, we can perform operations on them to find unknown values. We can use operations like addition, subtraction, multiplication, and division to solve a proportion. In most cases, we only need to use multiplication and division. Let’s consider a proportion in which one of the values is unknown.
For example, \( \frac{5}{8} = \frac{x}{40} \)
Use basic math operations to solve this equation. Begin by removing the denominator from both sides.
\( \frac{5 \times 8}{8} = \frac{x \times 8}{40} \) [Multiply both sides by \( 8 \)]
\( 5 = \frac{x}{5} \) [Simplify]
\( 5 \times 5 = \frac{x\times 5}{5} \) [Multiply both sides by \( 5 \)]
\( 25 = x \) [Simplify]
Hence, the value of \( x \) is \( 25 \).
Similarly, we can use a combination of mathematical operations to solve proportions.
Solved Examples
Example 1: Use math operations to find the value of \( x \) in the expression, \( \frac{3}{7} = \frac{x}{28} \).
Solution:
To find the value of \( x \), simplify the equation.
\( \frac{3}{7} = \frac{x}{28} \) [Write the equation]
\( \frac{3 \times 7}{7} = \frac{x \times 7}{28} \) [Multiply both sides by \( 7 \)]
\( 3 = \frac{x}{4} \) [Simplify]
\( 3 \times 4 = \frac{x \times 4}{4} \) [Multiply both sides by \( 4 \)]
\( 12 = x \) [Simplify]
So, the value of \( x \) is \( 12 \).
Example 2: Solve the proportion to find the unknown value: \( 15:y :: 25:55 \).
Solution:
The proportion is \( 15:y :: 25:55 \) and this expression can also be written as \( \frac{15}{y} = \frac{25}{55} \)
To find the value of \( y \), simplify the equation.
\( \frac{15}{y} = \frac{25}{55} \) [Write the proportion]
\( \frac{y}{15} = \frac{55}{25} \) [Taking reciprocal of both sides]
\( \frac{y \times 15}{15} = \frac{55 \times 15}{25} \) [Multiplying both sides by \( 15 \)]
\( y = \frac{11 \times 15}{55} \) [Simplify]
\( y = 11 \times 3 \) [Simplify]
\( y = 33 \) [Multiply]
Hence, the unknown value, \( y \) is \( 33 \).
Example 3: An athlete can run \( 100 \) meters in \( 11 \) seconds. If she runs at a constant pace, how long will she take to run \( 800 \) meters?
Solution:
Time taken by the athlete to cover \( 100 \) meters \( = 11 \) seconds
Let us assume the time taken by the athlete to cover \( 800 \) meters \( = x \) seconds
Since her speed is constant, the ratio of distance and time in both cases is in proportion.
Hence,
\( \frac{100}{11} = \frac{800}{x} \) [Write the above condition in proportion]
\( \frac{11}{100} = \frac{x}{800} \) [Taking reciprocal of both sides]
\( \frac{11 \times 100}{100} = \frac{x \times 100}{800} \) [Multiplying both sides by \( 100 \)]
\( 11 = \frac{x}{8} \) [Simplify]
\( x = 88 \) [Multiplying both sides by \( 8 \)]
Hence, the athlete will take \( 88 \) seconds to run \( 800 \) meters.
Yes, a proportion is an equation that states that two ratios are equivalent. So, we can say that proportions are directly related to ratios.
Since proportions are basically mathematical equations, we can perform all the mathematical operations on them, just like we do with a normal mathematical equation.
We can solve a proportion to find the unknown value by performing mathematical operations on them. The goal is to isolate the unknown value on one side of the equation. Thus, by solving the equation, we will get the value of the unknown on the other side of the equation.