Number Comparison Problem: Determining Which Value is Larger

Q: Why can't I just look at the numerators and denominators separately?

Because fractions represent parts of different wholes! $ \frac{7}{11} $ might have a bigger numerator than $ \frac{2}{3} $, but the denominators make all the difference. Always compare the actual values.

Q: Is there a faster way than cross multiplication?

Yes! Convert to common denominators or decimal form. For example: $ \frac{2}{3} $ ≈ 0.667 and $ \frac{7}{11} $ ≈ 0.636, so $ \frac{2}{3} $ is larger.

Q: What if I get the same cross products?

Then the fractions are equal! For example, $ \frac{2}{3} $ and $ \frac{4}{6} $ give: 2×6=12 and 3×4=12, so they're equivalent fractions.

Q: How do I remember which cross product goes with which fraction?

Think of an X pattern: multiply the top-left with bottom-right, then top-right with bottom-left. The fraction with the larger cross product is the larger fraction!

Q: Should I simplify fractions before comparing?

It's helpful but not required! In this problem, $ \frac{6}{10} $ = $ \frac{3}{5} $ when simplified, but cross multiplication works either way.

Fraction Comparison with Cross Multiplication

Which is larger?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:02 Let's start by finding the largest fraction.

00:13 First, multiply the first fraction by the second fraction's denominator to get a common base.

00:24 Now, multiply the second fraction by the first fraction's denominator.

00:33 With equal denominators, compare the numerators. The bigger one shows the larger fraction.

00:51 You can reduce the fraction by dividing by 2 to find a common base.

00:58 Both fractions are equal here.

01:03 Again, multiply the first fraction by the other’s bottom number for a common base.

01:10 Multiply the other fraction by the first one's denominator.

01:18 Remember, with equal bottoms, a larger top number is the larger fraction.

01:23 And this is how we solve the problem. Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Which is larger?

Step-by-step solution

To solve this problem, we will compare the given fractions to determine which is largest:

Compare $\frac{2}{3}$ and $\frac{7}{11}$ :
- Cross multiply: $2 \times 11 = 22$ and $3 \times 7 = 21$ .
  Since $22 > 21$ , $\frac{2}{3}$ is larger than $\frac{7}{11}$ .
Compare $\frac{2}{3}$ and $\frac{6}{10}$ :
- Simplify $\frac{6}{10} = \frac{3}{5}$ .
  Cross multiply: $2 \times 5 = 10$ and $3 \times 3 = 9$ .
  Since $10 > 9$ , $\frac{2}{3}$ is larger than $\frac{3}{5}$ (same as $\frac{6}{10}$ ).
To confirm, compare $\frac{2}{3}$ and $\frac{3}{5}$ again directly:
- Cross multiply: $2 \times 5 = 10$ and $3 \times 3 = 9$ .
  Since $10 > 9$ , $\frac{2}{3}$ is also larger than $\frac{3}{5}$ .

Therefore, the largest fraction of all given choices is indeed $\frac{2}{3}$ .

Final Answer

$\frac{2}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Cross multiply to compare fractions with different denominators
Technique: For $\frac{2}{3}$ vs $\frac{7}{11}$ : 2×11=22 and 3×7=21
Check: Larger cross product means larger fraction: 22>21 so $\frac{2}{3}$ > $\frac{7}{11}$ ✓

Common Mistakes

Avoid these frequent errors

Converting all fractions to decimals incorrectly
Don't convert $\frac{2}{3}$ to 0.6 by doing 2÷3 carelessly = wrong comparison! Rounding too early leads to incorrect ordering. Always use cross multiplication or find exact decimal equivalents with sufficient precision.

Practice Quiz

Test your knowledge with interactive questions

Fill in the missing sign:

$\frac{5}{9}☐\frac{3}{9}$

FAQ

Everything you need to know about this question

Why can't I just look at the numerators and denominators separately?

Because fractions represent parts of different wholes! $\frac{7}{11}$ might have a bigger numerator than $\frac{2}{3}$ , but the denominators make all the difference. Always compare the actual values.

Is there a faster way than cross multiplication?

Yes! Convert to common denominators or decimal form. For example: $\frac{2}{3}$ ≈ 0.667 and $\frac{7}{11}$ ≈ 0.636, so $\frac{2}{3}$ is larger.

What if I get the same cross products?

Then the fractions are equal! For example, $\frac{2}{3}$ and $\frac{4}{6}$ give: 2×6=12 and 3×4=12, so they're equivalent fractions.

How do I remember which cross product goes with which fraction?

Think of an X pattern: multiply the top-left with bottom-right, then top-right with bottom-left. The fraction with the larger cross product is the larger fraction!

Should I simplify fractions before comparing?

It's helpful but not required! In this problem, $\frac{6}{10}$ = $\frac{3}{5}$ when simplified, but cross multiplication works either way.

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