Solving a Fraction Word Problem

This problem asks us to find an original fraction based on two conditions and then perform a division. Let's break it down step by step.

Understanding the Original Fraction

We are told that the numerator of the fraction is 3 more than the denominator. Let's represent the denominator with the variable \(d\).

• Denominator = \(d\)
• Numerator = \(d + 3\)
• The original fraction is \(\frac{\text{Numerator}}{\text{Denominator}} = \frac{d+3}{d}\)

Setting up the Equation from the Second Condition

The problem gives us a second condition: When 5 is added to the numerator and 2 is subtracted from the denominator, the new fraction becomes 8/3.

• New Numerator = (Original Numerator) + 5 = \((d+3) + 5 = d+8\)
• New Denominator = (Original Denominator) - 2 = \(d - 2\)
• The new fraction is \(\frac{d+8}{d-2}\)

According to the problem, this new fraction is equal to 8/3. So, we can write the equation:

\(\frac{d+8}{d-2} = \frac{8}{3}\)

Solving for the Denominator \(d\)

To solve for \(d\), we can cross-multiply:

\(3 \times (d+8) = 8 \times (d-2)\)

Now, distribute the numbers on both sides:

\(3d + 24 = 8d - 16\)

Next, we want to get all the terms with \(d\) on one side and the constant terms on the other. Subtract \(3d\) from both sides:

\(24 = 8d - 3d - 16\)

\(24 = 5d - 16\)

Now, add 16 to both sides:

\(24 + 16 = 5d\)

\(40 = 5d\)

Finally, divide by 5 to find \(d\):

\(d = \frac{40}{5}\)

\(d = 8\)

Finding the Original Fraction

Now that we know the denominator \(d=8\), we can find the numerator and the original fraction:

• Denominator \(d = 8\)
• Numerator \(d+3 = 8+3 = 11\)
• The original fraction is \(\frac{11}{8}\)

Performing the Final Division

The problem asks us to divide the original fraction \(\frac{11}{8}\) by \(5 \frac{1}{2}\). First, convert the mixed number \(5 \frac{1}{2}\) into an improper fraction.

\(5 \frac{1}{2} = 5 \times 2 + 1 = \frac{10+1}{2} = \frac{11}{2}\)

Now, divide \(\frac{11}{8}\) by \(\frac{11}{2}\). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of \(\frac{11}{2}\) is \(\frac{2}{11}\).

\(\frac{11}{8} \div \frac{11}{2} = \frac{11}{8} \times \frac{2}{11}\)

Multiply the numerators and the denominators:

\(\frac{11 \times 2}{8 \times 11}\)

We can cancel out the common factor of 11 from the numerator and the denominator:

\(\frac{\cancel{11} \times 2}{8 \times \cancel{11}} = \frac{2}{8}\)

Finally, simplify the resulting fraction \(\frac{2}{8}\) by dividing both numerator and denominator by their greatest common divisor, which is 2:

\(\frac{2 \div 2}{8 \div 2} = \frac{1}{4}\)

So, the fraction obtained after the division is \(\frac{1}{4}\).

Revision Table - Fraction Problems

Additional Information - Working with Fractions

Understanding fractions is fundamental in mathematics. Fractions represent parts of a whole. Here are some related points:

• A fraction has a numerator (top number) and a denominator (bottom number). The denominator cannot be zero.
• Proper fractions have a numerator smaller than the denominator (e.g., \(\frac{1}{4}\)). Improper fractions have a numerator greater than or equal to the denominator (e.g., \(\frac{11}{8}\)).
• Mixed numbers combine a whole number and a proper fraction (e.g., \(5 \frac{1}{2}\)). They can always be converted to improper fractions.
• Operations with fractions (addition, subtraction, multiplication, division) require specific rules. For example, adding/subtracting requires a common denominator, while multiplication/division is more direct.
• Reciprocal of a fraction \(\frac{a}{b}\) is \(\frac{b}{a}\) (if \(a \neq 0\)).

Solving word problems involving fractions often involves translating the given information into equations and then solving them using algebraic techniques.