Understanding Ratio and Proportion

The question deals with the concept of ratio and proportion. A ratio is a comparison of two quantities, often expressed as \(a:b\) or \(\frac{a}{b}\). A proportion is an equation stating that two ratios are equal. The notation \(a : b :: c : d\) means that the ratio \(a:b\) is equal to the ratio \(c:d\).

Setting up the Proportion Equation

Given the proportion \(12 : 51 :: x : 17\), we can write this as an equation:

\(\frac{12}{51} = \frac{x}{17}\)

Our goal is to find the value of \(x\) that makes this equation true.

Solving for x in the Proportion

To find \(x\), we can isolate it on one side of the equation. We can do this by multiplying both sides of the equation by 17:

\(17 \times \frac{12}{51} = 17 \times \frac{x}{17}\)

\(17 \times \frac{12}{51} = x\)

Simplifying the Expression

Before multiplying, we can simplify the fraction \(\frac{12}{51}\). We can find a common factor for both 12 and 51. Both numbers are divisible by 3.

• \(12 \div 3 = 4\)
• \(51 \div 3 = 17\)

So, the fraction \(\frac{12}{51}\) simplifies to \(\frac{4}{17}\).

Substituting and Calculating x

Now substitute the simplified fraction back into the equation for \(x\):

\(x = 17 \times \frac{4}{17}\)

We can see that 17 in the numerator cancels out with 17 in the denominator:

\(x = \frac{17}{1} \times \frac{4}{17}\)

\(x = 4\)

Verification

We can check if the value \(x=4\) makes the original proportion true:

Is \(\frac{12}{51} = \frac{4}{17}\)?

We already simplified \(\frac{12}{51}\) to \(\frac{4}{17}\). Since \(\frac{4}{17} = \frac{4}{17}\), the equality holds true.

Therefore, the value of \(x\) is 4.

Revision Table: Key Concepts in Ratio and Proportion

Additional Information: Using the Property of Proportion

Another way to solve \(\frac{12}{51} = \frac{x}{17}\) is using the property that the product of the means equals the product of the extremes.

In the proportion \(12 : 51 :: x : 17\):

• Extremes are 12 and 17.
• Means are 51 and x.

According to the property:

Product of Extremes = Product of Means

\(12 \times 17 = 51 \times x\)

Now, solve for x:

\(51 \times x = 12 \times 17\)

\(51x = 204\)

Divide both sides by 51:

\(x = \frac{204}{51}\)

To simplify \(\frac{204}{51}\), we can perform division. Recognizing that \(51 = 3 \times 17\) and \(204 = 12 \times 17 = (3 \times 4) \times 17\), we get:

\(x = \frac{12 \times 17}{3 \times 17}\)

Cancel out the common factor of 17:

\(x = \frac{12}{3}\)

\(x = 4\)

Both methods yield the same result, confirming that the value of \(x\) is 4.