To solve the problem [tex]\(\frac{2}{7} : \frac{5}{7}\)[/tex], we will follow these steps to understand and find the ratio between the two fractions:
### Step-by-Step Solution:
1. Understand the Colon Notation (:):
- The colon (:) in mathematics typically represents a ratio. So, [tex]\(\frac{2}{7} : \frac{5}{7}\)[/tex] is asking for the ratio of [tex]\(\frac{2}{7}\)[/tex] to [tex]\(\frac{5}{7}\)[/tex].
2. Convert the Ratio to Division:
- A ratio [tex]\(\frac{2}{7} : \frac{5}{7}\)[/tex] can be interpreted as [tex]\(\frac{\frac{2}{7}}{\frac{5}{7}}\)[/tex].
3. Simplify the Division of Fractions:
- When dividing fractions, we multiply by the reciprocal of the divisor.
- [tex]\(\frac{\frac{2}{7}}{\frac{5}{7}} = \frac{2}{7} \times \frac{7}{5}\)[/tex].
4. Multiply the Fractions:
- Multiply the numerators together and the denominators together:
[tex]\[
\frac{2}{7} \times \frac{7}{5} = \frac{2 \times 7}{7 \times 5} = \frac{14}{35}.
\][/tex]
5. Simplify the Resulting Fraction:
- Simplify [tex]\(\frac{14}{35}\)[/tex] by finding the greatest common divisor of 14 and 35. The GCD is 7.
- Divide both the numerator and the denominator by 7.
[tex]\[
\frac{14 \div 7}{35 \div 7} = \frac{2}{5}.
\][/tex]
Therefore, the ratio [tex]\(\frac{2}{7} : \frac{5}{7}\)[/tex] simplifies to [tex]\(\frac{2}{5}\)[/tex].
For exact values:
- The decimal representation of [tex]\(\frac{2}{7}\)[/tex] is approximately 0.2857142857142857.
- The decimal representation of [tex]\(\frac{5}{7}\)[/tex] is approximately 0.7142857142857143.
- When you divide 0.2857142857142857 by 0.7142857142857143, the result is approximately 0.4.
Thus, the ratio of [tex]\(\frac{2}{7} : \frac{5}{7}\)[/tex] is indeed [tex]\(\frac{2}{5}\)[/tex], which in decimal form is 0.4.
