Understanding Fourth Proportion

The question asks us to find the fourth proportion to the numbers 55, 60, and 33. When four numbers, say a, b, c, and d, are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. This relationship is written as \(a:b :: c:d\), which can also be expressed as a fraction: \(\frac{a}{b} = \frac{c}{d}\).

In this problem, we are given the first three numbers of a proportion (55, 60, and 33) and need to find the fourth number. Let's call the fourth proportion 'x'. So, the proportion is 55, 60, 33, and x. This means:

• The first term (a) is 55.
• The second term (b) is 60.
• The third term (c) is 33.
• The fourth term (d or x) is what we need to find.

Setting up the Proportion Equation

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

\(\frac{a}{b} = \frac{c}{x}\)

Substitute the given numbers into the equation:

\(\frac{55}{60} = \frac{33}{x}\)

Solving for the Fourth Proportion (x)

To find the value of x, we can cross-multiply the terms in the equation:

\(55 \times x = 60 \times 33\)

Now, we need to isolate x. We can do this by dividing both sides of the equation by 55:

\(x = \frac{60 \times 33}{55}\)

Let's simplify the expression. We can cancel common factors from the numerator and the denominator. Both 55 and 33 are divisible by 11:

Divide 55 by 11: \(55 \div 11 = 5\)

Divide 33 by 11: \(33 \div 11 = 3\)

So the equation becomes:

\(x = \frac{60 \times 3}{5}\)

Now, we can simplify further by dividing 60 by 5:

\(60 \div 5 = 12\)

Substitute this back into the equation:

\(x = 12 \times 3\)

Finally, calculate the value of x:

\(x = 36\)

Conclusion

The fourth proportion to 55, 60, and 33 is 36. This means that the numbers 55, 60, 33, and 36 are in proportion, satisfying the relationship \(\frac{55}{60} = \frac{33}{36}\). Let's check this: \(\frac{55}{60} = \frac{11 \times 5}{12 \times 5} = \frac{11}{12}\) and \(\frac{33}{36} = \frac{11 \times 3}{12 \times 3} = \frac{11}{12}\). The ratios are indeed equal.

Revision Table: Key Proportion Concepts

This table summarizes important terms related to proportion.

Additional Information: Types of Proportion Problems

Besides finding the fourth proportion, you might encounter other types of proportion problems:

• Finding the Third Proportion: This applies to three numbers \(a, b, c\) where \(a:b :: b:c\). Here, \(c\) is the third proportion to \(a\) and \(b\), and \(b\) is the mean proportion. The relationship is \(\frac{a}{b} = \frac{b}{c}\), so \(b^2 = ac\).
• Finding the Mean Proportion: This is the term \(b\) in the third proportion scenario \(a:b :: b:c\). The mean proportion \(b\) between two numbers \(a\) and \(c\) is given by \(b = \sqrt{ac}\).
• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and vice versa. (e.g., cost of items and number of items). Represented as \(y \propto x\) or \(y = kx\).
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and vice versa. (e.g., speed and time taken for a fixed distance). Represented as \(y \propto \frac{1}{x}\) or \(y = \frac{k}{x}\) or \(xy = k\).

Understanding these concepts helps in solving various problems involving ratios and proportions in mathematics.