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Chapter 4: Problem 102

Multiply, and write the answer in simplified form. \(\frac{4}{5} \cdot \frac{2}{7}\)

Short Answer

Expert verified
The simplified form is \(\frac{8}{35}\).

Step by step solution

01

- Multiply the numerators

To multiply fractions, start by multiplying the numerators (the top numbers) together. For the fractions \(\frac{4}{5}\) and \(\frac{2}{7}\), multiply 4 and 2: \[4 \times 2 = 8\]
02

- Multiply the denominators

Next, multiply the denominators (the bottom numbers) together. For the fractions \(\frac{4}{5}\) and \(\frac{2}{7}\), multiply 5 and 7: \[5 \times 7 = 35\]
03

- Write the fraction

Combine the results from the previous steps to form the new fraction: \[\frac{8}{35}\]
04

- Simplify if necessary

Check if the fraction can be simplified. The numerator 8 and the denominator 35 do not have any common factors other than 1, so the fraction \(\frac{8}{35}\) is already in its simplest form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Step-by-Step Multiplication
Understanding how to multiply fractions involves following a straightforward process. The key steps are as follows:
1. **Multiply the Numerators**: Begin by multiplying the numerators (top numbers) of both fractions. For instance, in multiplying \(\frac{4}{5}\) and \(\frac{2}{7}\), you multiply 4 and 2 to get 8.
2. **Multiply the Denominators**: Next, multiply the denominators (bottom numbers) of both fractions. So, you would multiply 5 and 7 to get 35.
3. **Form the New Fraction**: Combine these results to form the new fraction, resulting in \(\frac{8}{35}\).

This step-by-step multiplication approach helps simplify the process and ensures accurate results.
Fraction Simplification
After forming a new fraction from the multiplication, it's essential to check if the fraction can be simplified. Simplification of fractions involves dividing the numerator and the denominator by their greatest common factor (GCF).
For example, in our case, the fraction \(\frac{8}{35}\) does not have common factors other than 1.
This means the fraction is already in its simplest form.
To determine if a fraction can be simplified, always look for numbers that can evenly divide both the numerator and the denominator.
  • The GCF helps simplify the fraction further.
  • This makes the fraction easier to understand and work with.
Numerators and Denominators
In any fraction, the numerator and the denominator play specific roles. The **numerator** is the top number and represents how many parts of the whole are being considered.
The **denominator** is the bottom number and shows the number of equal parts the whole is divided into.
When multiplying fractions, both numerators and denominators are multiplied directly.

For instance:
  • In \(\frac{4}{5} \), 4 is the numerator and 5 is the denominator.
  • In \(\frac{2}{7} \), 2 is the numerator and 7 is the denominator.
Multiplying the numerators gives you a new top number while multiplying the denominators provides a new bottom number.

This is important because understanding the roles of these components helps in correctly forming and simplifying fractions.

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Most popular questions from this chapter

In the following exercises, simplify. $$ \frac{\frac{5}{14}+\frac{1}{8}}{\frac{9}{56}} $$

In the following exercises, solve the equation. $$ f+\left(-\frac{2}{3}\right)=\frac{5}{12} $$

In the following exercises, perform the indicated operation. $$ 2 \frac{3}{10}-1 \frac{9}{10} $$

In the following exercises, simplify. $$ \frac{5}{20} $$

In the following exercises, perform the indicated operation. $$ 3 \frac{5}{8}+3 \frac{7}{8} $$
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