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Chapter 5: Problem 23

Find the ratio of the first number to the second and simplify. $$ 4 \text { to } 6 $$

Short Answer

Expert verified
The simplified ratio is \( \frac{2}{3} \).

Step by step solution

01

- Write the Ratio

The ratio of 4 to 6 can be written as the fraction \( \frac{4}{6} \).
02

- Find the Greatest Common Divisor (GCD)

Determine the greatest common divisor of 4 and 6. The GCD of 4 and 6 is 2.
03

- Simplify the Fraction

Divide both the numerator and the denominator of \( \frac{4}{6} \) by the GCD, which is 2: \( \frac{4 \div 2}{6 \div 2} = \frac{2}{3} \).

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Key Concepts

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

greatest common divisor
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the largest positive integer that divides two or more integers without leaving a remainder. In other words, it is the highest number that can perfectly divide a set of given numbers.
To find the GCD of two numbers, you can use various methods:
  • Prime Factorization: Break down both numbers into their prime factors and then multiply the common prime factors.
  • Euclidean Algorithm: Repeatedly subtract the smaller number from the larger number until you reach zero. The last non-zero remainder is the GCD.
In our example of 4 and 6, the prime factors are 2. So, the GCD is 2.
simplifying fractions
Simplifying fractions means to reduce them to their simplest form. A fraction is simplified when the numerator and the denominator no longer have any common divisors other than 1.
Here are the steps to simplify a fraction:
  • Write down the fraction.
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
Using the ratio of 4 to 6 as an example, the fraction is written as \( \frac{4}{6} \). With a GCD of 2, we divide both parts by 2 to get \( \frac{2}{3} \). This is the simplified form.
finding ratios
Finding ratios is a way of comparing two quantities. A ratio can be expressed in three different ways: using a colon (4:6), as a fraction \( \frac{4}{6} \), or in words (4 to 6).
To find a ratio and express it in its simplest form, follow these steps:
  • Identify the two quantities you want to compare.
  • Write these quantities as a fraction (numerator over denominator).
  • Find the GCD of the numbers.
  • Divide both the numerator and the denominator by the GCD to simplify the fraction.
So, for the numbers 4 and 6, the initial ratio \( \frac{4}{6} \) simplifies to \( \frac{2}{3} \) using the GCD.
numerator and denominator
A fraction consists of two parts: the numerator and the denominator. Understanding these terms is key to simplifying and working with fractions.
  • **Numerator**: The top number in a fraction. It represents how many parts of the whole or group are being considered. In the fraction \( \frac{4}{6} \), 4 is the numerator.
  • **Denominator**: The bottom number in a fraction. It indicates the total number of equal parts the whole is divided into. In the fraction \( \frac{4}{6} \), 6 is the denominator.
By dividing both the numerator and the denominator by their GCD, we can simplify fractions. For example, dividing the numerator 4 and denominator 6 by the GCD 2 simplifies \( \frac{4}{6} \) to \( \frac{2}{3} \).

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Most popular questions from this chapter

Multiply. Simplify, if possible. $$ 3 \frac{4}{5} \cdot 2 \frac{1}{4} $$

Solve. $$ \frac{1.28}{3.76}=\frac{4.28}{y} $$

Find the unit price of each brand. Then, in each exercise, determine which brand is the better buy based on unit price alone. $$ \begin{array}{|c|c|c|} \hline \text { Brand } & \text { Size } & \text { Price } \\ \hline \mathrm{F} & 14 \mathrm{oz} & \$ 1.29 \\ \mathrm{G} & 19 \mathrm{oz} & \$ 1.99 \\ \mathrm{P} & 20 \mathrm{oz} & \$ 2.49 \\ \hline \end{array} $$

Solve. $$ \frac{\frac{5}{8}}{\frac{5}{4}}=\frac{y}{\frac{3}{2}} $$

Solve. $$ \frac{x}{3}=\frac{0}{9} $$
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