Solving Proportion Problems and Finding the Mean Proportional

The problem involves a scenario where a number 'x' is subtracted from a series of numbers, and the resulting numbers are in proportion. We need to find the value of 'x' first, and then use it to calculate the mean proportional between two expressions involving 'x'.

Understanding Proportion

Four numbers, a, b, c, and d, are in proportion if the ratio of the first two is equal to the ratio of the last two. Mathematically, this is written as:

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

In this question, the numbers after subtracting 'x' are (22 - x), (39 - x), (56 - x), and (107 - x). Since these resulting numbers are in proportion in this order, we can write the equation:

$\frac{22 - x}{39 - x} = \frac{56 - x}{107 - x}$

Solving for x

To find the value of 'x', we need to solve the proportion equation. We can do this by cross-multiplication:

$(22 - x)(107 - x) = (56 - x)(39 - x)$

Expand both sides of the equation:

$(22 \times 107) - (22 \times x) - (x \times 107) + (x \times x) = (56 \times 39) - (56 \times x) - (x \times 39) + (x \times x)$

$2354 - 22x - 107x + x^2 = 2184 - 56x - 39x + x^2$

Combine like terms on each side:

$2354 - 129x + x^2 = 2184 - 95x + x^2$

Subtract $x^2$ from both sides of the equation:

$2354 - 129x = 2184 - 95x$

Now, rearrange the terms to isolate 'x'. Move the x terms to one side and the constant terms to the other side:

$2354 - 2184 = 129x - 95x$

$170 = 34x$

Divide both sides by 34 to find the value of 'x':

$x = \frac{170}{34}$

$x = 5$

So, the value of x is 5.

Calculating the Expressions (x + 3) and (3x - 7)

Now that we have the value of 'x', we can calculate the values of the two expressions:

• Expression 1: $(x + 3)$
• Substitute $x = 5$: $5 + 3 = 8$
• Expression 2: $(3x - 7)$
• Substitute $x = 5$: $(3 \times 5) - 7 = 15 - 7 = 8$

The two numbers are 8 and 8.

Finding the Mean Proportional

The mean proportional between two numbers, 'a' and 'b', is the square root of their product. The formula is:

Mean Proportional $= \sqrt{a \times b}$

In this case, the two numbers are 8 and 8. So, the mean proportional between (x + 3) and (3x - 7) is:

Mean Proportional $= \sqrt{(x + 3) \times (3x - 7)}$

Mean Proportional $= \sqrt{8 \times 8}$

Mean Proportional $= \sqrt{64}$

Mean Proportional $= 8$

The mean proportional between (x + 3) and (3x - 7) is 8.

Revision Table: Key Concepts

Additional Information: Types of Proportion

Beyond simple proportion (also known as direct proportion when dealing with variables, though here it's about a series of numbers), there are other related concepts:

• Direct Proportion: Two quantities are directly proportional if an increase in one quantity causes a proportional increase in the other, and vice versa. Example: Cost of pens is directly proportional to the number of pens.
• Inverse Proportion: Two quantities are inversely proportional if an increase in one quantity causes a proportional decrease in the other, and vice versa. Example: Time taken to complete a job is inversely proportional to the number of workers.
• Continued Proportion: Three numbers a, b, c are in continued proportion if a, b, b, c are in proportion, i.e., $\frac{a}{b} = \frac{b}{c}$. Here, b is the mean proportional between a and c. Our problem required finding a mean proportional between two specific calculated values, not necessarily a continued proportion involving the original or resulting numbers.

Understanding proportions is fundamental in various mathematical problems, including ratios, percentages, speed-time-distance, and more complex algebraic equations.