Solving a Complex Fraction Word Problem

This problem involves finding the values of three fractions, A, B, and C, based on given relationships, and then calculating the difference between two of them. We are given three key pieces of information:

1. The sum of the three fractions A, B, and C is \(\frac{121}{60}\).
2. Fraction C divided by fraction B is \(\frac{9}{10}\).
3. The result of C divided by B (\(\frac{9}{10}\)) exceeds fraction A by \(\frac{3}{20}\).
4. We are also told that A > B > C.

Our goal is to find the difference between B and C, which is \(B - C\).

Step 1: Find the value of Fraction A

We are given that the result of C divided by B, which is \(\frac{9}{10}\), exceeds A by \(\frac{3}{20}\). This can be written as an equation:

\(\frac{9}{10} = A + \frac{3}{20}\)

To find A, we need to subtract \(\frac{3}{20}\) from \(\frac{9}{10}\):

\(A = \frac{9}{10} - \frac{3}{20}\)

To subtract these fractions, we find a common denominator, which is 20. We convert \(\frac{9}{10}\) to an equivalent fraction with a denominator of 20:

\(\frac{9}{10} = \frac{9 \times 2}{10 \times 2} = \frac{18}{20}\)

Now substitute this back into the equation for A:

\(A = \frac{18}{20} - \frac{3}{20}\)

\(A = \frac{18 - 3}{20}\)

\(A = \frac{15}{20}\)

We can simplify the fraction for A by dividing both the numerator and denominator by their greatest common divisor, which is 5:

\(A = \frac{15 \div 5}{20 \div 5} = \frac{3}{4}\)

So, the value of fraction A is \(\frac{3}{4}\).

Step 2: Set up equations for B and C

We know that the sum of A, B, and C is \(\frac{121}{60}\). Substitute the value of A we just found:

\(\frac{3}{4} + B + C = \frac{121}{60}\)

We also know that C divided by B is \(\frac{9}{10}\). This can be written as:

\(\frac{C}{B} = \frac{9}{10}\)

We can rearrange this equation to express C in terms of B:

\(C = \frac{9}{10} \times B\)

Step 3: Solve for Fraction B

Now substitute the expression for C (\(C = \frac{9}{10} B\)) into the sum equation:

\(\frac{3}{4} + B + \frac{9}{10} B = \frac{121}{60}\)

Combine the terms involving B. Note that \(B = 1 \times B = \frac{10}{10} B\):

\(\frac{3}{4} + \left(1 + \frac{9}{10}\right) B = \frac{121}{60}\)

\(\frac{3}{4} + \left(\frac{10}{10} + \frac{9}{10}\right) B = \frac{121}{60}\)

\(\frac{3}{4} + \frac{19}{10} B = \frac{121}{60}\)

Now, isolate the term with B by subtracting \(\frac{3}{4}\) from both sides:

\(\frac{19}{10} B = \frac{121}{60} - \frac{3}{4}\)

Find a common denominator for the right side, which is 60. Convert \(\frac{3}{4}\) to an equivalent fraction with a denominator of 60:

\(\frac{3}{4} = \frac{3 \times 15}{4 \times 15} = \frac{45}{60}\)

Substitute this back into the equation:

\(\frac{19}{10} B = \frac{121}{60} - \frac{45}{60}\)

\(\frac{19}{10} B = \frac{121 - 45}{60}\)

\(\frac{19}{10} B = \frac{76}{60}\)

To solve for B, multiply both sides by \(\frac{10}{19}\):

\(B = \frac{76}{60} \times \frac{10}{19}\)

We can simplify before multiplying. Note that 76 is \(19 \times 4\) and 60 is \(6 \times 10\).

\(B = \frac{4 \times 19}{6 \times 10} \times \frac{10}{19}\)

Cancel out the common factors 19 and 10:

\(B = \frac{4}{6}\)

Simplify the fraction for B:

\(B = \frac{4 \div 2}{6 \div 2} = \frac{2}{3}\)

So, the value of fraction B is \(\frac{2}{3}\).

Step 4: Find the value of Fraction C

We use the relationship \(C = \frac{9}{10} B\) and the value of B we just found:

\(C = \frac{9}{10} \times \frac{2}{3}\)

Multiply the numerators and the denominators:

\(C = \frac{9 \times 2}{10 \times 3}\)

\(C = \frac{18}{30}\)

Simplify the fraction for C by dividing both the numerator and denominator by their greatest common divisor, which is 6:

\(C = \frac{18 \div 6}{30 \div 6} = \frac{3}{5}\)

So, the value of fraction C is \(\frac{3}{5}\).

Step 5: Calculate the difference between B and C

The question asks for the difference between B and C, which is \(B - C\). We have the values for B and C:

\(B = \frac{2}{3}\) and \(C = \frac{3}{5}\)

Calculate the difference:

\(B - C = \frac{2}{3} - \frac{3}{5}\)

To subtract these fractions, find a common denominator, which is 15. Convert both fractions to equivalent fractions with a denominator of 15:

\(\frac{2}{3} = \frac{2 \times 5}{3 \times 5} = \frac{10}{15}\)

\(\frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15}\)

Now substitute these back into the difference calculation:

\(B - C = \frac{10}{15} - \frac{9}{15}\)

\(B - C = \frac{10 - 9}{15}\)

\(B - C = \frac{1}{15}\)

The difference between B and C is \(\frac{1}{15}\).

Verification

Let's quickly check if A > B > C holds: A=\(\frac{3}{4}\) (0.75), B=\(\frac{2}{3}\) (\(\approx\) 0.667), C=\(\frac{3}{5}\) (0.6). Yes, \(0.75 > 0.667 > 0.6\).

The difference between B and C is \(\frac{1}{15}\).

Revision Table: Solving Fraction Problems

Additional Information: Working with Fractions

Fractions represent parts of a whole. They consist of a numerator (the top number) and a denominator (the bottom number). The denominator indicates the total number of equal parts the whole is divided into, and the numerator indicates how many of those parts are being considered.

Key concepts when working with fractions:

• Equivalent Fractions: Fractions that represent the same value, even though they have different numerators and denominators (e.g., \(\frac{1}{2} = \frac{2}{4} = \frac{5}{10}\)). You create equivalent fractions by multiplying or dividing both the numerator and denominator by the same non-zero number. This is crucial for adding and subtracting fractions with different denominators.
• Simplifying Fractions: Reducing a fraction to its simplest form by dividing both the numerator and the denominator by their greatest common divisor (GCD). For example, \(\frac{15}{20}\) simplifies to \(\frac{3}{4}\) because the GCD of 15 and 20 is 5.
• Comparing Fractions: To compare fractions, it's easiest to convert them to equivalent fractions with a common denominator or convert them to decimals. For example, to compare \(\frac{2}{3}\) and \(\frac{3}{5}\), convert them to fifteenths: \(\frac{10}{15}\) and \(\frac{9}{15}\). Since \(10 > 9\), \(\frac{10}{15} > \frac{9}{15}\), so \(\frac{2}{3} > \frac{3}{5}\).
• Reciprocal: The reciprocal of a fraction is obtained by flipping the numerator and denominator. For example, the reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Reciprocals are used in fraction division.

Solving word problems involving fractions often requires translating the problem's sentences into mathematical equations and then using algebraic techniques to solve for the unknown values, as demonstrated in this problem.