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Chapter 7: Problem 25

Find product. Write in simplest form. \(7\left(\frac{8}{21}\right)\)

Short Answer

Expert verified
The product in simplest form is \( \frac{8}{3} \).

Step by step solution

01

Understand the Problem

We are asked to find the product of a whole number, 7, and a fraction, \( \frac{8}{21} \). This involves multiplying the whole number by the numerator of the fraction and keeping the denominator the same.
02

Multiply the Whole Number by the Numerator

Multiply 7 by the numerator of the fraction, 8. \[ 7 \times 8 = 56 \] Thus, the new numerator will be 56, resulting in the fraction \( \frac{56}{21} \).
03

Simplify the Fraction

To simplify \( \frac{56}{21} \), find the greatest common divisor (GCD) of 56 and 21. The GCD of 56 and 21 is 7. Divide both the numerator and the denominator by 7. \[ \frac{56 \div 7}{21 \div 7} = \frac{8}{3} \] Thus, the simplified form of the fraction is \( \frac{8}{3} \).

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Key Concepts

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

Understanding Fractions
Fractions are a way to represent parts of a whole. Think of a pizza cut into slices. If a pizza is divided into 8 equal slices and you eat 3, you have eaten 3 out of 8 parts, which can be written as the fraction \( \frac{3}{8} \).
Fractions consist of two parts: the numerator and the denominator.
  • The numerator is the top number, indicating how many parts you have.
  • The denominator is the bottom number, showing how many equal parts the whole is divided into.
If a fraction like \( \frac{8}{21} \) is represented with 8 as the numerator, that means you have 8 parts. Meanwhile, the number 21 as the denominator shows that the whole is split into 21 parts.
Fractions can represent not only simple numbers but parts of any complete whole, and are critical in many areas of mathematics.
Multiplying Fractions and Whole Numbers
When you multiply a whole number by a fraction, you are essentially scaling the fraction up or down by that whole number. This is done by multiplying the numerator of the fraction by the whole number and keeping the denominator the same.
For example, take 7 multiplied by the fraction \( \frac{8}{21} \).
  • Start by multiplying 7 and 8 (the numerator). This gives you \( 7 \times 8 = 56 \).
  • Now, the fraction becomes \( \frac{56}{21} \).
It's important to remember that the product of a whole number and a fraction keeps the denominator of the fraction unchanged. So, even though the numerator becomes larger, the divisions remain consistent with the original whole it was a part of. This is an essential part of understanding how multiplication with fractions operates.
Simplifying Fractions
Simplifying a fraction means making it as simple as possible while representing the same value. This is achieved by finding a common factor that both the numerator and the denominator share, then dividing by that number.
Here's how it's done:
  • Identify the greatest common divisor (GCD) for the numerator and the denominator.
  • For \( \frac{56}{21} \), the GCD is 7.
  • Divide both parts of the fraction by their GCD: \( \frac{56 \div 7}{21 \div 7} = \frac{8}{3} \).
The fraction \( \frac{8}{3} \) is in its simplest form because 8 and 3 do not share any common divisors other than 1. Simplifying fractions helps to easily understand and compare them and is a practical skill in many mathematical operations.

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