If 162 &ratio; x &ratio;&ratio; x &ratio; 338, find the positive value of x.

Question

Prepp Team · Accepted Answer

Solving Ratio and Proportion Problems

The question asks us to find the positive value of x in the given relationship: 162 ∶ x ∶∶ x ∶ 338.

This notation represents a continuous proportion. A continuous proportion is one where the means (the middle terms) are the same. In the form a : b :: c : d, if b = c, then it is a continuous proportion: a : b :: b : d.

The given relationship 162 ∶ x ∶∶ x ∶ 338 means that the ratio of 162 to x is equal to the ratio of x to 338. This can be written as an equation:

$$\frac{162}{x} = \frac{x}{338}$$

Step-by-Step Solution to Find x

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

$$162 \times 338 = x \times x$$

$$162 \times 338 = x^2$$

Now, we need to calculate the product of 162 and 338:

$$162 \times 338 = 54756$$

So, the equation becomes:

$$x^2 = 54756$$

To find x, we need to take the square root of both sides of the equation:

$$x = \pm \sqrt{54756}$$

We need to find the square root of 54756. We can try to find factors or estimate the value. A number ending in 6 suggests the square root might end in 4 or 6. Let's estimate:

$200^2 = 40000$
$300^2 = 90000$

So, the square root is between 200 and 300. Since 54756 ends in 6, the square root could end in 4 or 6. Let's try squaring numbers ending in 4 or 6 near the middle of 200 and 300.

$230^2 = 52900$
$240^2 = 57600$

The number 54756 is between 52900 and 57600. Let's try squaring 234 or 236.

Let's calculate $234 \times 234$:

	2	3	4
200	40000	6000	800
30	6000	900	120
4	800	120	16

Summing the values: $40000 + 6000 + 800 + 6000 + 900 + 120 + 800 + 120 + 16 = 54756$.

So, $\sqrt{54756} = 234$.

Therefore, the possible values for x are $x = 234$ and $x = -234$.

The question asks for the positive value of x. Between 234 and -234, the positive value is 234.

Understanding Continuous Proportion

In a continuous proportion a : x :: x : b, the term 'x' is called the mean proportional between 'a' and 'b'. The relationship can be expressed as $x^2 = ab$. To find the mean proportional, you take the square root of the product of the two numbers.

In this problem, a = 162 and b = 338. So, the mean proportional x is found by $x = \sqrt{162 \times 338}$.

$$x = \sqrt{54756}$$

$$x = 234$$

The positive value of the mean proportional is 234.

Summary of Finding x

Given the proportion 162 ∶ x ∶∶ x ∶ 338:

Write the proportion as an equation: $\frac{162}{x} = \frac{x}{338}$.
Cross-multiply: $162 \times 338 = x^2$.
Calculate the product: $54756 = x^2$.
Take the square root: $x = \pm \sqrt{54756}$.
Find the square root: $\sqrt{54756} = 234$.
The values are $x = 234$ and $x = -234$.
Select the positive value: $x = 234$.

Revision Table - Key Concepts in Ratio and Proportion

Concept	Definition	Example
Ratio	Comparison of two quantities of the same kind by division (a:b or a/b)	3:5 or 3/5
Proportion	An equality of two ratios (a:b :: c:d or a/b = c/d)	2:3 :: 4:6 (since 2/3 = 4/6)
Extremes	The first and fourth terms in a proportion (a and d in a:b :: c:d)	In 2:3 :: 4:6, 2 and 6 are extremes
Means	The second and third terms in a proportion (b and c in a:b :: c:d)	In 2:3 :: 4:6, 3 and 4 are means
Property of Proportion	Product of extremes = Product of means (ad = bc)	In 2:3 :: 4:6, $2 \times 6 = 12$ and $3 \times 4 = 12$
Continuous Proportion	A proportion where the means are equal (a:b :: b:d)	4:6 :: 6:9 (since 4/6 = 6/9 = 2/3)
Mean Proportional	The middle term in a continuous proportion (b in a:b :: b:d). $b = \sqrt{ad}$	In 4:6 :: 6:9, 6 is the mean proportional between 4 and 9. $6 = \sqrt{4 \times 9} = \sqrt{36}$

Additional Information - Applications of Ratio and Proportion

Ratio and proportion are fundamental concepts with wide applications:

Scaling: Maps and blueprints use ratios to represent large distances or areas on a smaller scale.
Mixtures: Recipes, chemical solutions, and concrete mixes use ratios to specify the amounts of ingredients.
Finance: Ratios are used in financial analysis (e.g., debt-to-equity ratio) and currency exchange rates.
Physics: Many physical laws and relationships can be expressed using ratios and proportions (e.g., relationships between distance, speed, and time).
Geometry: Similar shapes have corresponding sides in proportion.
Data Analysis: Ratios and proportions help compare data points and understand relationships within datasets.

Understanding continuous proportion helps in finding the mean proportional, which is used in various geometric constructions and statistical calculations.