Problem 72 - Prealgebra

Question

Find the mistakes. Correct the right-hand side of each one. $$\frac{2}{7} \cdot \frac{3}{5}=\frac{5}{35}$$

Step-by-Step Solution

Answer

The mistake is in the product; it should be \( \frac{6}{35} \), not \( \frac{5}{35} \).

Step 1: Understand the Multiplication of Fractions

When multiplying fractions, multiply the numerators (top numbers) together and multiply the denominators (bottom numbers) together. The general rule is: \( \frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \).

Step 2: Identify the Numerators and Denominators

In the given fractions \( \frac{2}{7} \cdot \frac{3}{5} \), the numerators are 2 and 3, and the denominators are 7 and 5.

Step 3: Multiply the Numerators

Multiplying the numerators: \( 2 \cdot 3 = 6 \).

Step 4: Multiply the Denominators

Multiplying the denominators: \( 7 \cdot 5 = 35 \).

Step 5: Write the Product of the Fractions

After multiplying the numerators and denominators, the product of the fractions is: \( \frac{6}{35} \).

Step 6: Compare with the Given Equation

The given equation is \( \frac{2}{7} \cdot \frac{3}{5} = \frac{5}{35} \). Compare this with the correct product \( \frac{6}{35} \).

Step 7: Identify the Mistake and Correct

The mistake is that the right-hand side of the equation is \( \frac{5}{35} \), but the correct multiplication yields \( \frac{6}{35} \).

Key Concepts

Numerator and Denominator

Fractions consist of two parts: the numerator and the denominator. The numerator is the top number of the fraction and indicates how many parts of the whole are being considered. For instance, in the fraction \( \frac{2}{7} \), the numerator is 2. This tells us that we are considering 2 parts out of the entire set that is divided into 7 parts.
The denominator, on the other hand, is the bottom number of the fraction. It indicates the total number of equal parts into which the whole is divided. In the same example \( \frac{2}{7} \), the denominator is 7, showing that the whole is divided into 7 equal parts.
Understanding both components is crucial when performing operations with fractions, such as addition, subtraction, multiplication, or division.

Fraction Multiplication Rules

Multiplying fractions is a straightforward operation if the rules are followed carefully. The key to multiplying fractions is to multiply straight across the fraction. Specifically:

• Multiply the numerators (top numbers) to find the new numerator.
• Multiply the denominators (bottom numbers) to find the new denominator.

Let's use the exercise example: \( \frac{2}{7} \cdot \frac{3}{5} \).
First, focus on the numerators: \( 2 \times 3 = 6 \). Next, focus on the denominators: \( 7 \times 5 = 35 \).
This gives us the product \( \frac{6}{35} \), which is our final answer.
A common mistake is trying to multiply denominators with numerators, but remember to always keep numerators and denominators in their respective positions.

Common Mistakes in Fractions

Working with fractions can sometimes lead to common errors, especially in multiplication. Here are a few mistakes to watch out for:

• **Incorrect Numerator/Denominator Multiplication**: Mixing up numerators and denominators is a frequent error. Always keep the numerators and denominators separate in each step of multiplication.
• **Misinterpretation of Multiplication Rules**: Some may incorrectly add numerators and denominators separately instead of multiplying them. Ensure to follow the multiplication rule: numerator with numerator and denominator with denominator.
• **Simplification Error**: Sometimes, fractions are not simplified when they could be reduced. While not relevant in all cases, always check if the final answer can be simplified by dividing both parts by a common factor.

Correctly multiplying fractions relies heavily on understanding the distinction between numerators and denominators and consistently applying multiplication rules. Avoid these common pitfalls to achieve accurate results.