Square Area Calculation: Finding Area B Using 2/3 Ratio and 56-Unit Perimeter

Q: Why do I need to square the ratio 2/3?

The ratio $ \frac{2}{3} $ refers to side lengths, not areas! Since area = side × side, the area ratio becomes $ (\frac{2}{3})^2 = \frac{4}{9} $.

Q: Why is the answer 441 and not 294?

294 comes from using the wrong ratio! If you use $ \frac{196}{x} = \frac{2}{3} $, you get $ x = 294 $. But areas need the squared ratio: $ \frac{196}{x} = \frac{4}{9} $ gives the correct answer 441.

Area Ratios with Square Proportions

Square A is greater than square B by a ratio of $\frac{2}{3}$ .

If the perimeter of square A is known to be 56, what is the area of square B?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the area of square B

00:03 In a square all sides are equal, marked as A

00:11 The square perimeter equals the sum of its sides

00:15 This is the side length in square 1

00:24 Side ratio according to the given

00:32 We'll substitute square 1 side value to find Y

00:40 We'll multiply by the inverse to isolate Y

00:45 This is side length Y

00:52 Square area equals side squared

01:01 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Square A is greater than square B by a ratio of $\frac{2}{3}$ .

If the perimeter of square A is known to be 56, what is the area of square B?

Step-by-step solution

We will mark the side in square A as X

Therefore the perimeter will be:
$4x=56$

$x=14$

Now we can calculate the area of square A:

$14\times14=196$

As we are given the ratio between the areas:

$\frac{196}{S_2}=(\frac{2}{3})^2=\frac{4}{9}$

That is, the ratio will be:

$\frac{196}{X}=\frac{4}{9}$

The area of the square will be equal to:

$\frac{9\times196}{4}=\frac{1764}{4}=441$

Final Answer

441

Key Points to Remember

Essential concepts to master this topic

Perimeter Rule: Square perimeter equals 4 times side length
Area Ratio: Square area ratio $\frac{2}{3}$ becomes $\frac{4}{9}$ when squared
Check: Verify area B = 441 gives ratio $\frac{196}{441} = \frac{4}{9}$ ✓

Common Mistakes

Avoid these frequent errors

Using the original ratio instead of squaring it for areas
Don't use $\frac{2}{3}$ directly for area ratio = wrong answer 294! The ratio $\frac{2}{3}$ refers to side lengths, not areas. Always square the side ratio to get the area ratio: $(\frac{2}{3})^2 = \frac{4}{9}$ .

Practice Quiz

Test your knowledge with interactive questions

FAQ

Everything you need to know about this question

Why do I need to square the ratio 2/3?

The ratio $\frac{2}{3}$ refers to side lengths, not areas! Since area = side × side, the area ratio becomes $(\frac{2}{3})^2 = \frac{4}{9}$ .

How do I know which square is A and which is B?

The problem states "Square A is greater than square B by a ratio of $\frac{2}{3}$ ". This means A is smaller than B, so $\frac{\text{Area A}}{\text{Area B}} = \frac{2}{3}$ squared.

What if I calculated the side ratio backwards?

If you get a ratio > 1, check your setup! Since A is described as greater "by" the ratio $\frac{2}{3}$ , A is actually smaller than B. The word "by" indicates the proportion, not that A is larger.

Can I solve this without finding the area of square A first?

It's much clearer to find area A first! You need the perimeter (56) to find A's side length (14), then A's area (196). This gives you a concrete number to work with when setting up the ratio.

Why is the answer 441 and not 294?

294 comes from using the wrong ratio! If you use $\frac{196}{x} = \frac{2}{3}$ , you get $x = 294$ . But areas need the squared ratio: $\frac{196}{x} = \frac{4}{9}$ gives the correct answer 441.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Mathematics questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime