**1. Cross Product Property :**

The product of the extremes equals the product of the means.

If a/b = c/d, then ad = bc.

**2. Reciprocal Property :**

If two ratios are equal, then their reciprocals are also equal.

If a/b = c/d, then b/a = d/c.

**3. A****lternendo ****Property :**

If a/b = c/d, then a/c = b/d.

**4. Componendo**** ****Property :**

If a/b = c/d, then (a + b) / b = (c + d) / d.

**Example 1 : **

Say, whether the statement given below is true.

If x / 4 = y / 16, then x / y = 1 / 4

**Solution : **

Given :

x / 4 = y / 16

Using Alternendo Property,

x / y = 4 / 16

Simplify.

x / y = 1 / 4

Hence, the statement is true.

**Example 2 : **

Say, whether the statement given below is true.

If p / 3 = q / 5, then (p + 3) / 3 = (q + 3) / 5

**Solution : **

Given :

p / 3 = q / 5

Using Componendo Property,

(p + 3) / 3 = (q + 5) / 5

Because (p + 3) / 3 ≠ (q + 3) / 5, the conclusions are equivalent.

Hence, the statement is false.

**Example 3 : **

In the diagram shown below,

PQ / QS = PR / RT

Find the length of QS.

**Solution : **

Given :

PQ / QS = PR / RT

Substitute.

16 / x = (30 - 10) / 10

Simplify.

16 / x = 20 / 10

16 / x = 2 / 1

By reciprocal property, we have

x / 16 = 1 / 2

Multiply each side by 16.

16 ⋅ (x / 16) = (1 / 2) ⋅ 16

Simplify.

x = 8

Hence, the length of BD is 8 units.

The geometric mean of two positive numbers a and b is the positive number **x** such that

a / **x** = **x** / b

If we solve this proportion for **x**, we find that

**x** = √(a ⋅ b)

which is a positive number.

Fro example, the geometric mean of 8 and 18 is **12**.

Because 8 /**12** = **12** / 18 and also because

√(8 ⋅ 18) = √144 = **12**

**Example :**

International standard paper sizes are commonly used all over the world. The various sizes all have the same width-to-length ratios. Two sizes of paper are shown below, called A4 and A3. The distance labeled x is the geometric mean of 210 mm and 420 mm. Find the value of x.

Solution :

Using the given information, write proportion.

210 / x = x / 420

Using cross product property,

x^{2} = 210 ⋅ 420

Take radical on both sides.

√x^{2} = √(210 ⋅ 420)

Simplify.

x = √(210 ⋅ 210 ⋅ 2)

x = 210√2

The geometric mean of 210 and 420 is 210√2, or about 297.

Hence, the distance labeled x in the diagram is about 297 mm.

**Example : **

A scale model of the Titanic is 107.5 inches long and 11.25 inches wide. The Titanic itself was 882.75 feet long. How wide was it ?

**Solution :**

One way to solve this problem is to set up a proportion that compares the measurements of the Titanic to the measurements of the scale model

**Problem Solving Strategy :**

Multiply each side by 11.25

11.25 ⋅ (x / 11.25) = (882.75 / 107.5) ⋅ 11.25

Simplify.

x = (882.75 ⋅ 11.25) / 107.5

Using calculator, we have

x ≈ 92.4

Hence, the Titanic was about 92.4 feet wide.

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