Solving for the Missing Fraction

The problem asks us to identify which fraction, when added to 5/7, equals 1. This involves basic fraction arithmetic and comparison.

Representing the Problem Mathematically

Let the fraction we need to find be represented by '$x$'. The problem statement can be translated into the following equation:

$$ \frac{5}{7} + x = 1 $$

Calculating the Unknown Fraction

To find the value of '$x$', we need to isolate it. We can achieve this by subtracting 5/7 from both sides of the equation:

$$ x = 1 - \frac{5}{7} $$

To subtract the fractions, we need a common denominator. We can write 1 as 7/7:

$$ x = \frac{7}{7} - \frac{5}{7} $$

Now, we subtract the numerators while keeping the denominator the same:

$$ x = \frac{7 - 5}{7} $$

$$ x = \frac{2}{7} $$

Therefore, the fraction needed is 2/7.

Evaluating the Given Options

We must now examine each provided option to see which one simplifies to 2/7.

• Option 1: 6/21
• Option 2: 4/2
• Option 3: 6/14
• Option 4: 5/3

Simplifying Each Fraction Option

We simplify each option:

• Option 1 (6/21): We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 3.
  $$ \frac{6 \div 3}{21 \div 3} = \frac{2}{7} $$
  This fraction is equal to 2/7.
• Option 2 (4/2): This fraction simplifies to a whole number.
  $$ \frac{4}{2} = 2 $$
  This is not equal to 2/7.
• Option 3 (6/14): We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2.
  $$ \frac{6 \div 2}{14 \div 2} = \frac{3}{7} $$
  This is not equal to 2/7.
• Option 4 (5/3): This fraction is already in its simplest form and is greater than 1.
  $$ \frac{5}{3} $$
  This is not equal to 2/7.

Final Answer Determination

By comparing the simplified options with our calculated required fraction (2/7), we find that only Option 1 (6/21) matches.

To confirm, let's add 5/7 and 6/21:

First, simplify 6/21 to 2/7.

$$ \frac{5}{7} + \frac{6}{21} = \frac{5}{7} + \frac{2}{7} $$

Since they have a common denominator, we add the numerators:

$$ \frac{5 + 2}{7} = \frac{7}{7} = 1 $$

Thus, the fraction 6/21, when added to 5/7, gives 1.