Understanding Proportions and Finding the Missing Term

A proportion is a statement that two ratios are equal. If we have four quantities, say $a, b, c,$ and $d$, they are said to be in proportion if the ratio of the first to the second is equal to the ratio of the third to the fourth. This is written as $a:b :: c:d$ or $\frac{a}{b} = \frac{c}{d}$.

In this proportion, $a$ is the 1st term, $b$ is the 2nd term, $c$ is the 3rd term, and $d$ is the 4th term.

The problem gives us the 1st, 2nd, and 4th terms of a proportion. We are given:

• 1st term ($a$) = 51
• 2nd term ($b$) = 68
• 4th term ($d$) = 108

We need to find the 3rd term ($c$).

Using the definition of a proportion, we can set up the equation:

$\frac{a}{b} = \frac{c}{d}$

Substitute the given values into the equation:

$\frac{51}{68} = \frac{c}{108}$

Solving for the 3rd Term in the Proportion

To find the value of $c$, we can use cross-multiplication. Cross-multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the products equal:

$51 \times 108 = 68 \times c$

Now, we need to calculate the product on the left side:

$51 \times 108 = 5508$

So, the equation becomes:

$5508 = 68c$

To isolate $c$, we need to divide both sides of the equation by 68:

$c = \frac{5508}{68}$

Performing the division:

$5508 \div 68 = 81$

Therefore, the 3rd term ($c$) is 81.

Let's verify this by plugging the value of $c=81$ back into the proportion equation:

$\frac{51}{68} = \frac{81}{108}$

We can simplify both fractions:

• For $\frac{51}{68}$: Both 51 and 68 are divisible by 17. $51 \div 17 = 3$ and $68 \div 17 = 4$. So, $\frac{51}{68} = \frac{3}{4}$.
• For $\frac{81}{108}$: Both 81 and 108 are divisible by 27. $81 \div 27 = 3$ and $108 \div 27 = 4$. So, $\frac{81}{108} = \frac{3}{4}$.

Since $\frac{3}{4} = \frac{3}{4}$, the proportion is valid, and our calculated 3rd term is correct.

The 3rd term in the proportion is 81.

Revision Table: Key Concepts in Proportion

Additional Information: Applications of Proportions

Proportions are widely used in various fields to solve problems involving relationships between quantities. Some common applications include:

• Scaling: Maps and architectural drawings use proportions to represent larger areas or structures on a smaller scale.
• Cooking: Recipes often require adjusting ingredient amounts proportionally when changing the number of servings.
• Chemistry: Stoichiometry uses proportions to determine the amounts of reactants and products in chemical reactions.
• Physics: Many physical laws involve proportional relationships, such as Ohm's Law ($V \propto I$ if $R$ is constant).
• Unit Conversion: Converting between different units of measurement (e.g., inches to centimeters) involves proportions.
• Similarity: Similar geometric figures have corresponding sides that are proportional.

Understanding how to set up and solve proportions is a fundamental skill in mathematics with practical relevance in everyday life and science.