Compare Fractions: Find the Symbol Between 3/7 and 1/6

Q: Why can't I just compare the numerators 3 and 1?

Because the denominators are different! $ \frac{3}{7} $ means 3 parts out of 7, while $ \frac{1}{6} $ means 1 part out of 6. You need equal-sized pieces to compare fairly.

Q: How do I find the common denominator?

Find the Least Common Multiple (LCM) of the denominators. For 7 and 6, since they share no common factors, the LCM is 7 × 6 = 42.

Q: Can I use cross-multiplication instead?

Yes! Cross-multiply: $ 3 \times 6 = 18 $ and $ 1 \times 7 = 7 $. Since 18 > 7, we get $ \frac{3}{7} > \frac{1}{6} $. This is actually faster!

Q: Do I always need to use the product of denominators?

No! Use the LCM (Least Common Multiple) instead. For example, comparing $ \frac{1}{4} $ and $ \frac{1}{8} $, use 8 (not 32) as the common denominator.

Q: How can I check my answer makes sense?

Convert to decimals as a quick check: $ \frac{3}{7} ≈ 0.43 $ and $ \frac{1}{6} ≈ 0.17 $. Since 0.43 > 0.17, our answer $ \frac{3}{7} > \frac{1}{6} $ is correct!

Fraction Comparison with Common Denominators

Fill in the missing sign:

$\frac{3}{7}☐\frac{1}{6}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the appropriate sign

00:03 We want to find a common denominator

00:06 Therefore we'll multiply each fraction by the denominator of the other fraction

00:09 Remember to multiply both numerator and denominator

00:19 Now we'll use the same method for the second fraction

00:24 Remember to multiply by the denominator of the second fraction

00:29 Remember to multiply both numerator and denominator

00:33 Now we have a common denominator between the fractions

00:37 When denominators are equal, the larger the numerator, the larger the fraction

00:41 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

Fill in the missing sign:

$\frac{3}{7}☐\frac{1}{6}$

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Determine a common denominator for the fractions.
Step 2: Convert each fraction to have the common denominator.
Step 3: Compare the numerators to find the correct inequality.

Now, let's work through each step:
Step 1: Determine a common denominator for $\frac{3}{7}$ and $\frac{1}{6}$ . The common denominator is the product of the denominators $7 \times 6 = 42$ .
Step 2: Convert each fraction to have this common denominator:
- Convert $\frac{3}{7}$ : Multiply the numerator and denominator by 6: $\frac{3 \times 6}{7 \times 6} = \frac{18}{42}$ .
- Convert $\frac{1}{6}$ : Multiply the numerator and denominator by 7: $\frac{1 \times 7}{6 \times 7} = \frac{7}{42}$ .
Step 3: Compare the numerators: $18$ and $7$ .
Since $18 > 7$ , we have $\frac{18}{42} > \frac{7}{42}$ . Thus, $\frac{3}{7} > \frac{1}{6}$ .

Therefore, the correct inequality sign is $>$ .

Final Answer

$>$

Key Points to Remember

Essential concepts to master this topic

Rule: Find common denominator to compare fractions accurately
Technique: Convert $\frac{3}{7} = \frac{18}{42}$ and $\frac{1}{6} = \frac{7}{42}$
Check: Compare numerators: 18 > 7, so $\frac{3}{7} > \frac{1}{6}$ ✓

Common Mistakes

Avoid these frequent errors

Comparing numerators without finding common denominators
Don't compare 3 and 1 directly = wrong conclusion that $\frac{3}{7} > \frac{1}{6}$ just because 3 > 1! Different denominators make direct comparison impossible. Always find a common denominator first to make accurate comparisons.

Practice Quiz

Test your knowledge with interactive questions

Fill in the missing sign:

$\frac{5}{25}☐\frac{1}{5}$

FAQ

Everything you need to know about this question

Why can't I just compare the numerators 3 and 1?

Because the denominators are different! $\frac{3}{7}$ means 3 parts out of 7, while $\frac{1}{6}$ means 1 part out of 6. You need equal-sized pieces to compare fairly.

How do I find the common denominator?

Find the Least Common Multiple (LCM) of the denominators. For 7 and 6, since they share no common factors, the LCM is 7 × 6 = 42.

Can I use cross-multiplication instead?

Yes! Cross-multiply: $3 \times 6 = 18$ and $1 \times 7 = 7$ . Since 18 > 7, we get $\frac{3}{7} > \frac{1}{6}$ . This is actually faster!

What if the fractions look very close in value?

That's when finding a common denominator is most important! Convert both fractions and compare the numerators carefully. Small differences become clear with the same denominator.

Do I always need to use the product of denominators?

No! Use the LCM (Least Common Multiple) instead. For example, comparing $\frac{1}{4}$ and $\frac{1}{8}$ , use 8 (not 32) as the common denominator.

How can I check my answer makes sense?

Convert to decimals as a quick check: $\frac{3}{7} ≈ 0.43$ and $\frac{1}{6} ≈ 0.17$ . Since 0.43 > 0.17, our answer $\frac{3}{7} > \frac{1}{6}$ is correct!

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