Problem 41 - Prealgebra

Question

Simplify each expression as much as possible. $$10+\frac{11}{12} \div \frac{11}{24}$$

Step-by-Step Solution

Answer

The simplified expression is 12.

Step 1: Rewrite Division as Multiplication

When dividing fractions, we can rewrite the division of fractions as multiplication by the reciprocal of the divisor. So, we have: \[ \frac{11}{12} \div \frac{11}{24} = \frac{11}{12} \times \frac{24}{11} \]

Step 2: Calculate the Fraction Expression

We simplify the multiplication by canceling out common factors in the numerator and the denominator. First, multiply the numerators and the denominators:\[ \frac{11 \times 24}{12 \times 11} = \frac{264}{132} \]Now, cancel \(11\) from the numerator and denominator since they appear in both:\[ \frac{24}{12} = 2 \]

Step 3: Add the Whole Number

Now, we add the simplified result from Step 2 to the whole number from the original expression:\[ 10 + 2 = 12 \]

Key Concepts

Division of Fractions

Dividing fractions can seem tricky at first, but with a simple rule, it becomes quite manageable. When you need to divide one fraction by another, you don’t actually perform the division directly. Instead, you flip the second fraction (known as the divisor) upside down to find its reciprocal, and then multiply them.

For example, in the exercise, we have \( \frac{11}{12} \div \frac{11}{24} \). Here, the divisor \( \frac{11}{24} \) is flipped to become \( \frac{24}{11} \). Now, instead of dividing, we multiply:
\[ \frac{11}{12} \times \frac{24}{11} \]
This simple switch from division to multiplication makes solving the problem easier.

When dividing fractions, always remember:

• Change the division sign to multiplication
• Use the reciprocal of the divisor in your calculation

Multiplication by Reciprocal

Multiplying by the reciprocal is a fundamental step in division of fractions. A reciprocal is essentially what you get when you swap the numerator and the denominator of a fraction. So, for the fraction \( \frac{a}{b} \), its reciprocal is \( \frac{b}{a} \).

When applying this concept:

• Identify the divisor
• Find the reciprocal by flipping the numerator and denominator
• Multiply the fractions to simplify

In our problem, multiplying \( \frac{11}{12} \) by the reciprocal \( \frac{24}{11} \) simplifies the calculation, allowing us to further simplify by canceling common factors.

Simplifying Fractions

Once you have multiplied the fractions, simplifying is the next vital step. Simplifying fractions involves reducing them to their lowest terms by finding and canceling common factors.

Take our fraction from the exercise: \( \frac{11 \times 24}{12 \times 11} \). Here, you can see that \( 11 \) appears in both the numerator and the denominator, allowing it to be canceled out:
\[ \frac{24}{12} \]
Further simplification shows that \( 24 \) divided by \( 12 \) is \( 2 \).

Key steps to simplify:

• Identify common factors
• Cancel them to reduce the fraction
• Check if the result can be further simplified

Adding Whole Numbers to Fractions

After simplifying the fraction, you might need to add it to a whole number, as seen in our original problem. Adding whole numbers and fractions is straightforward.

Here is the sequence:

• Simplify the fraction first (which we did, resulting in \(2\))
• Add this simplified value to the whole number

In our example, the problem ended with \(10 + 2\), which equals \(12\).

Always ensure you've fully simplified the fraction before adding it to a whole number. This streamlines your work and minimizes errors, making calculations smoother and faster. By clear stepwise addition, you ensure every component of the problem is accurate and easily understandable.