Understanding the Problem: Proportion and Fourth Proportional

The question involves two main concepts: numbers in proportion and finding the fourth proportional. First, we are told that when a specific value, let's call it 'x', is subtracted from four given numbers (55, 50, 23, and 22), the resulting numbers are in proportion in that specific order. This means the ratio of the first resulting number to the second is equal to the ratio of the third resulting number to the fourth.

The second part asks us to find the fourth proportional of three given numbers: 3, 7, and the value of x that we found in the first part.

Finding the Value of x

Let the four numbers after subtracting x be:

• First number: \(55 - x\)
• Second number: \(50 - x\)
• Third number: \(23 - x\)
• Fourth number: \(22 - x\)

Since these numbers are in proportion, we can write the relationship as:

$$\frac{55 - x}{50 - x} = \frac{23 - x}{22 - x}$$

To solve for x, we can cross-multiply:

$$(55 - x)(22 - x) = (23 - x)(50 - x)$$

Now, let's expand both sides of the equation:

Left side: \(55 \times 22 - 55x - 22x + x^2 = 1210 - 77x + x^2\)

Right side: \(23 \times 50 - 23x - 50x + x^2 = 1150 - 73x + x^2\)

So, the equation becomes:

$$1210 - 77x + x^2 = 1150 - 73x + x^2$$

We can subtract \(x^2\) from both sides:

$$1210 - 77x = 1150 - 73x$$

Now, let's isolate the terms with x on one side and the constant terms on the other side:

$$1210 - 1150 = 77x - 73x$$

$$60 = 4x$$

Finally, solve for x:

$$x = \frac{60}{4}$$

$$x = 15$$

So, the value of x is 15.

Calculating the Fourth Proportional

The second part of the question asks for the fourth proportional of 3, 7, and x. We found that \(x = 15\). Let the fourth proportional be 'd'.

For four numbers a, b, c, and d to be in proportion, we have the relationship:

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

In our case, a = 3, b = 7, and c = x = 15. We need to find d.

So, we have:

$$\frac{3}{7} = \frac{15}{d}$$

To solve for d, we can cross-multiply:

$$3 \times d = 7 \times 15$$

$$3d = 105$$

Now, divide by 3:

$$d = \frac{105}{3}$$

$$d = 35$$

The fourth proportional of 3, 7, and 15 is 35.

Final Answer

The value of x is 15, and the fourth proportional of 3, 7, and x (which is 15) is 35.

Revision Table: Proportion Concepts

Additional Information: Solving Algebraic Equations

Solving equations like \((55 - x)(22 - x) = (23 - x)(50 - x)\) is a fundamental skill in algebra. It involves expanding binomials, combining like terms, and isolating the variable.

• Expanding binomials uses the distributive property (often remembered by FOIL: First, Outer, Inner, Last).
• Combining like terms means adding or subtracting coefficients of the same variable raised to the same power (e.g., -55x and -22x combine to -77x).
• Isolating the variable involves performing inverse operations (addition/subtraction, multiplication/division) to move terms around the equality sign. Remember to do the same operation on both sides to maintain equality.

In this problem, we encountered a quadratic term (\(x^2\)) which conveniently cancelled out, leaving a linear equation that was straightforward to solve.