Understanding the Problem: Numbers in Proportion

The question describes a scenario where a number, denoted by 'x', is subtracted from four different numbers: 19, 28, 55, and 91. The key information is that the resulting numbers, in that specific order, are in proportion. This means the ratio of the first resulting number to the second resulting number is equal to the ratio of the third resulting number to the fourth resulting number.

Once we find the value of 'x', we need to calculate the mean proportional between two other numbers derived from 'x': (x + 9) and x². The mean proportional between two numbers 'a' and 'b' is given by the square root of their product, i.e., $\sqrt{ab}$.

Setting up the Proportion Equation

When 'x' is subtracted from each number, we get:

• 19 - x
• 28 - x
• 55 - x
• 91 - x

Since these numbers are in proportion in the given order, we can write the equation:

$\frac{19 - x}{28 - x} = \frac{55 - x}{91 - x}$

Solving for x

To solve for 'x', we cross-multiply the equation:

$(19 - x)(91 - x) = (28 - x)(55 - x)$

Expand both sides of the equation:

$(19 \times 91) - (19 \times x) - (x \times 91) + (x \times x) = (28 \times 55) - (28 \times x) - (x \times 55) + (x \times x)$

$1729 - 19x - 91x + x^2 = 1540 - 28x - 55x + x^2$

Combine like terms on each side:

$1729 - 110x + x^2 = 1540 - 83x + x^2$

Subtract $x^2$ from both sides:

$1729 - 110x = 1540 - 83x$

Now, isolate the terms with 'x' on one side and constant terms on the other. Add 110x to both sides and subtract 1540 from both sides:

$1729 - 1540 = 110x - 83x$

$189 = 27x$

Divide by 27 to find the value of 'x':

$x = \frac{189}{27}$

$x = 7$

Calculating the Mean Proportional

We need to find the mean proportional between (x + 9) and x². Now that we know $x = 7$, we can find these two numbers:

• First number: $x + 9 = 7 + 9 = 16$
• Second number: $x^2 = 7^2 = 49$

The mean proportional between 16 and 49 is $\sqrt{16 \times 49}$.

$\sqrt{16 \times 49} = \sqrt{16} \times \sqrt{49}$

We know that $\sqrt{16} = 4$ and $\sqrt{49} = 7$.

So, the mean proportional is $4 \times 7 = 28$.

Conclusion

The value of x is 7. The two numbers are (x + 9) = 16 and x² = 49. The mean proportional between 16 and 49 is 28.

Revision Table: Key Concepts

Additional Information: Ratios and Proportions

A ratio is a comparison of two quantities by division. It is often written as a:b or a/b. A proportion is an equation stating that two ratios are equal. For example, a/b = c/d is a proportion. In a proportion a/b = c/d, 'a' and 'd' are called the extremes, and 'b' and 'c' are called the means. The property used to solve the equation in this problem, cross-multiplication, is derived from the fact that in a proportion, the product of the extremes equals the product of the means (ad = bc).

Understanding proportions is fundamental in various mathematical applications, including scaling, similarity in geometry, and solving problems involving rates.