Understanding the Fraction Problem

The question asks us to find a specific fraction. This fraction has a unique property: the difference between the fraction and 4/7 is the same as the difference between 5/6 and the fraction. In simpler terms, the fraction is exactly halfway between 4/7 and 5/6. It is equidistant from both 4/7 and 5/6.

Setting Up the Equation for the Equidistant Fraction

Let the unknown fraction be represented by \(x\). The condition "as much greater than 4/7 as it is less than 5/6" can be written as an equation:

The amount by which \(x\) is greater than \(4/7\) is \(x - \frac{4}{7}\).

The amount by which \(x\) is less than \(5/6\) is \(\frac{5}{6} - x\).

According to the question, these amounts are equal:

\[x - \frac{4}{7} = \frac{5}{6} - x\]

Solving for the Fraction using the Algebraic Method

We can solve this equation for \(x\). The goal is to isolate \(x\) on one side of the equation.

Step 1: Add \(x\) to both sides of the equation.

\[x - \frac{4}{7} + x = \frac{5}{6} - x + x\]

\[2x - \frac{4}{7} = \frac{5}{6}\]

Step 2: Add \(\frac{4}{7}\) to both sides of the equation.

\[2x = \frac{5}{6} + \frac{4}{7}\]

Step 3: Find a common denominator to add the fractions \(\frac{5}{6}\) and \(\frac{4}{7}\). The least common multiple (LCM) of 6 and 7 is 42.

• Convert \(\frac{5}{6}\): \(\frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}\)
• Convert \(\frac{4}{7}\): \(\frac{4}{7} = \frac{4 \times 6}{7 \times 6} = \frac{24}{42}\)

Now, substitute these equivalent fractions back into the equation:

\[2x = \frac{35}{42} + \frac{24}{42}\]

Step 4: Add the fractions.

\[2x = \frac{35 + 24}{42} = \frac{59}{42}\]

Step 5: Divide both sides by 2 to find the value of \(x\).

\[x = \frac{59}{42} \div 2\]

Dividing by 2 is the same as multiplying by \(\frac{1}{2}\).

\[x = \frac{59}{42} \times \frac{1}{2}\]

\[x = \frac{59 \times 1}{42 \times 2}\]

\[x = \frac{59}{84}\]

So, the fraction is \(\frac{59}{84}\).

Solving for the Fraction using the Midpoint Formula

As the fraction is equidistant from \(\frac{4}{7}\) and \(\frac{5}{6}\), it is essentially the midpoint of these two fractions. The midpoint of two numbers \(a\) and \(b\) is given by the formula \(\frac{a+b}{2}\).

Here, \(a = \frac{4}{7}\) and \(b = \frac{5}{6}\).

The unknown fraction \(x\) is:

\[x = \frac{\frac{4}{7} + \frac{5}{6}}{2}\]

Step 1: Add the two fractions in the numerator. Find a common denominator for \(\frac{4}{7}\) and \(\frac{5}{6}\), which is 42.

• \(\frac{4}{7} = \frac{4 \times 6}{7 \times 6} = \frac{24}{42}\)
• \(\frac{5}{6} = \frac{5 \times 7}{6 \times 7} = \frac{35}{42}\)

Add the fractions:

\[\frac{4}{7} + \frac{5}{6} = \frac{24}{42} + \frac{35}{42} = \frac{24 + 35}{42} = \frac{59}{42}\]

Step 2: Substitute the sum back into the midpoint formula.

\[x = \frac{\frac{59}{42}}{2}\]

Step 3: Divide the fraction \(\frac{59}{42}\) by 2.

\[x = \frac{59}{42} \div 2 = \frac{59}{42} \times \frac{1}{2} = \frac{59}{84}\]

Both methods yield the same result, \(\frac{59}{84}\).

Verifying the Answer

Let's check if \(\frac{59}{84}\) is indeed as much greater than \(\frac{4}{7}\) as it is less than \(\frac{5}{6}\).

Difference between \(\frac{59}{84}\) and \(\frac{4}{7}\):

\[\frac{59}{84} - \frac{4}{7}\]

Find a common denominator (84). \(\frac{4}{7} = \frac{4 \times 12}{7 \times 12} = \frac{48}{84}\).

\[\frac{59}{84} - \frac{48}{84} = \frac{59 - 48}{84} = \frac{11}{84}\]

Difference between \(\frac{5}{6}\) and \(\frac{59}{84}\):

\[\frac{5}{6} - \frac{59}{84}\]

Find a common denominator (84). \(\frac{5}{6} = \frac{5 \times 14}{6 \times 14} = \frac{70}{84}\).

\[\frac{70}{84} - \frac{59}{84} = \frac{70 - 59}{84} = \frac{11}{84}\]

The differences are equal (\(\frac{11}{84}\)), confirming that \(\frac{59}{84}\) is the correct fraction.

Final Answer

The fraction which is as much greater than 4/7 as it is less than 5/6 is \(\frac{59}{84}\).

Revision Table - Key Concepts

Additional Information - Working with Fractions

Fractions represent parts of a whole. They consist of a numerator (top number) and a denominator (bottom number).

• Adding/Subtracting Fractions: You must have a common denominator. Find the LCM of the denominators, convert the fractions, then add or subtract the numerators.
• Multiplying Fractions: Multiply the numerators together and multiply the denominators together. \(\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}\).
• Dividing Fractions: To divide by a fraction, multiply by its reciprocal. The reciprocal of \(\frac{c}{d}\) is \(\frac{d}{c}\). So, \(\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\).
• Comparing Fractions: To compare fractions, find a common denominator and compare the numerators.

Understanding these basic operations is crucial for solving fraction problems like the one discussed here.