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This is the aptitude questions and answers section on "Ratios and Proportion " with detailed explanation for various interview, competitive examination and entrance test. Problems of three difficulty levels are given with detailed solution description, explanation, so that it becomes easy to grasp the fundamentals.

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**Ratio:**

A ratio is simply a fraction. The following notations all express the ratio of x to y

=> $x : y$ , $ x ÷ y$ , or $x/y$.

In the ratio x : y, we call a as the first term or ** antecedent** and b, the second term or

**.**

*consequent*Writing two numbers as a ratio provides a convenient way to compare their sizes. For example, since $3/π < 1$, we know that 3 is less than π.

A ratio compares two numbers. Just as you cannot compare apples and oranges, so the numbers you are comparing must have the same units.

For example, you cannot form the ratio of 2 feet to 4 meters because the two numbers are expressed in different units—feet vs. meters.

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**Example 1:** What is the ratio of 2 feet to 4 yards?

**(A)** 1 : 2 **(B)** 1 : 8 **(C)** 1 : 7 **(D)** 1 : 6 **(E)** 1 : 5

The ratio cannot be formed until the numbers are expressed in the same units. Let’s turn the yards into feet.

Since there are 3 feet in a yard, 4 yards = 4 * 3 feet = 12 feet .

Forming the ratio yields $\text"2 feet"/\text"12 feet" = 1/6 $ or $1 : 6$

The answer is** (D)**.

**Note.** Taking the reciprocal of a fraction usually changes its size. For example,

$3/4 ≠ 4/3$

So order is important in a ratio=> **3:4 ≠ 4:3.**

**Rule:** The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

**Ex.** 4 : 5 = 8 : 10 = 12 : 15.

Also, 4 : 6 = 2 : 3.

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**Proportion:**

The equality of two ratios (fractions) is called proportion. If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion.

Here *a* and *d* are called ** extremes**, while

*b*and

*c*are called

**terms.**

*mean* $$\text"Product of means = Product of extremes"$$

Thus, $$a : b :: c : d ⇔ (b * c)= (a * d)$$

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**Fourth Proportional:**

If a : b = c : d, then d is called the *fourth proportional *to a, b, c.

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**Third Proportional:**

a : b = c : d, then c is called the *third proportional* to a and b.

**Mean Proportional:**

*Mean proportional* between a and b is ab.

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**Comparison of Ratios:**

We say that $(a : b) > (c : d) ⇔ a/b > c/d$

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**Compounded Ratio:**

The *compounded ratio* of the ratios: $(a : b), (c : d), (e : f) is (ace : bdf)$

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**Duplicate Ratios:**

*Duplicate ratio* of $(a : b)$ is $(a^2 : b^2)$

** Sub-duplicate** ratio of $ (a : b)$ is $(a^{1/2} : b^{1/2})$

** Triplicate ratio **of $(a : b)$ is $(a^3 : b^3)$

** Sub-triplicate ratio** of $(a : b)$ is $(a^{1/3} : b^{1/3})$

If $a/b = c/d$ then, ${a + b}/{a - b} = {c + d}/{c – d}$ [**Componendo and Dividendo**]

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**Variations:**

We say that x is ** directly proportional **to y,

if x = ky for some constant k and we write, $x ∝ y$

We say that x is inversely proportional to y,

if xy = k for some constant k and we write, $x ∝ 1/y$

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