Understanding Proportionality and the Fourth Proportional

The question asks us to find the fourth proportional to three given numbers: 3.6, 6.9, and 11.4. 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 expressed as:

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

In this relationship, $d$ is called the fourth proportional to $a, b,$ and $c.$

Setting Up the Fourth Proportional Problem

We are given the first three numbers: $a = 3.6,$ $b = 6.9,$ and $c = 11.4.$ We need to find the fourth proportional, which we will call $x.$ Using the definition of proportion, we can set up the equation:

$$\frac{3.6}{6.9} = \frac{11.4}{x}$$

Solving for the Fourth Proportional

To find the value of $x,$ we can use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction and setting it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

Step 1: Perform cross-multiplication.

$$3.6 \times x = 6.9 \times 11.4$$

Step 2: Calculate the product on the right side of the equation.

$$6.9 \times 11.4$$

Let's calculate this value:

• Multiply 69 by 114 (ignoring decimal points for a moment).
• $69 \times 114 = 7866$
• Since there is one decimal place in 6.9 and one in 11.4, there will be a total of $1 + 1 = 2$ decimal places in the product.
• So, $6.9 \times 11.4 = 78.66$

Now, the equation becomes:

$$3.6 \times x = 78.66$$

Step 3: Isolate $x$ by dividing both sides of the equation by 3.6.

$$x = \frac{78.66}{3.6}$$

To perform this division, we can make the divisor a whole number by multiplying both the numerator and the denominator by 10:

$$x = \frac{78.66 \times 10}{3.6 \times 10} = \frac{786.6}{36}$$

Now, divide 786.6 by 36:

• Divide 786 by 36. $786 \div 36 \approx 21$. $36 \times 21 = 756$. Remainder is $786 - 756 = 30$.
• Bring down the next digit, which is 6, making it 306. Place the decimal point after 21.
• Divide 306 by 36. $306 \div 36$. We know $36 \times 8 = 288$ and $36 \times 9 = 324$. So it's 8 point something. Let's try $36 \times 0.85$ or $36 \times 8.5$.
• Let's do long division:

So, $786.6 \div 36 = 21.85.$

Therefore, the fourth proportional $x$ is 21.85.

Checking the Options for the Fourth Proportional

Let's look at the given options:

• A. 20.3
• B. 18.9
• C. 19.6
• D. 21.85

Our calculated value for the fourth proportional is 21.85, which matches option D.

Final Answer Determination

Based on our calculation, the fourth proportional to 3.6, 6.9, and 11.4 is 21.85.

Revision Table: Key Steps to Find Fourth Proportional

Additional Information: Concepts of Proportion

Proportion is a fundamental concept in mathematics that describes the equality of two ratios. When quantities are in proportion, they have a constant relationship.

• Ratio: A ratio compares two quantities. It can be written as $a:b$ or $\frac{a}{b}.$
• Proportion: An equation that states that two ratios are equal, e.g., $\frac{a}{b} = \frac{c}{d}.$
• Terms of a Proportion: In the proportion $a:b = c:d,$ the terms are $a, b, c,$ and $d.$ $a$ and $d$ are called the extremes, and $b$ and $c$ are called the means.
• Property of Proportion: The product of the extremes is equal to the product of the means. For $a:b = c:d,$ this means $a \times d = b \times c.$ This property is what we used in the cross-multiplication step to solve for the unknown fourth proportional.
• Types of Proportion:
  • Direct Proportion: If two quantities increase or decrease together at a constant rate.
  • Inverse Proportion: If one quantity increases as the other decreases, and vice versa, such that their product is constant.

Understanding proportion is crucial for solving various problems involving ratios, scale factors, percentages, and similar geometric figures.