Understanding Continued Proportion

When three numbers are in a continued proportion, it means that the ratio of the first number to the second number is equal to the ratio of the second number to the third number. If the numbers are \( a, b, c \), they are in continued proportion if:

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

In this question, the three numbers in continued proportion are 22, x, and 88. Here, \( a = 22 \), \( b = x \), and \( c = 88 \).

Setting Up the Proportion

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

\( \frac{22}{x} = \frac{x}{88} \)

Solving for x in Continued Proportion

To find the value of \( x \), we can cross-multiply the terms in the proportion:

\( x \times x = 22 \times 88 \)

\( x^2 = 22 \times 88 \)

Now, we calculate the product on the right side:

\( 22 \times 88 = 1936 \)

So, the equation becomes:

\( x^2 = 1936 \)

To find \( x \), we need to take the square root of both sides of the equation:

\( x = \sqrt{1936} \)

Calculating the Value of x

We need to find the square root of 1936. Let's check some possibilities or use factorization:

• We know \( 40^2 = 1600 \) and \( 50^2 = 2500 \). So, the square root is between 40 and 50.
• The last digit of 1936 is 6. A number ending in 4 or 6 will have its square ending in 6. Let's check numbers ending in 4 or 6 in the 40s.
• \( 44 \times 44 \): \( 44 \times 40 = 1760 \), \( 44 \times 4 = 176 \). \( 1760 + 176 = 1936 \).

So, \( \sqrt{1936} = 44 \).

Therefore, the value of \( x \) is 44.

Comparing with Options

The calculated value of \( x \) is 44.

Let's look at the given options:

• A. 24
• B. 33
• C. 44
• D. 36

The value \( x = 44 \) matches option C.

Final Answer Derivation

The numbers 22, x, and 88 are in continued proportion. This leads to the equation \( \frac{22}{x} = \frac{x}{88} \). Solving this equation gives \( x^2 = 22 \times 88 = 1936 \). Taking the square root, \( x = \sqrt{1936} = 44 \).

Thus, the value of \( x \) is 44.

Revision Table: Continued Proportion

Additional Information: Mean Proportional and Continued Proportion

In a continued proportion \( a, b, c \), the middle term \( b \) is specifically known as the mean proportional between the first term \( a \) and the third term \( c \). The relationship \( a/b = b/c \) can be rearranged to \( b^2 = ac \), which means \( b = \sqrt{ac} \). This shows that the mean proportional is the geometric mean of the other two terms.

• If you are given two numbers, say \( p \) and \( q \), the mean proportional between them is \( \sqrt{pq} \).
• These three numbers, \( p, \sqrt{pq}, q \), are in continued proportion.

In this problem, \( x \) is the mean proportional between 22 and 88. We found \( x = \sqrt{22 \times 88} = \sqrt{1936} = 44 \). This confirms our result and the relationship between mean proportional and continued proportion.