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Chapter 5: Problem 408

In the following exercises, write each ratio as a fraction. $$56 \text { to } 16$$

Short Answer

Expert verified
\( \frac{7}{2} \)

Step by step solution

01

- Write Ratio in Fraction Form

Express the given ratio '56 to 16' as a fraction. Therefore, the fraction is \ \( \frac{56}{16} \ \).
02

- Simplify the Fraction

Simplify the fraction \( \frac{56}{16} \) by finding the greatest common divisor (GCD) of 56 and 16. The GCD of 56 and 16 is 8.
03

- Divide Numerator and Denominator by GCD

Divide both the numerator (56) and the denominator (16) by their GCD (8): \ \ \( \frac{56 \div 8}{16 \div 8} = \frac{7}{2} \ \).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a crucial skill in mathematics. It makes working with fractions easier and often reveals the true 'size' or proportion of the fraction.
To simplify a fraction, follow these steps:
  • Find the greatest common divisor (GCD) of the numerator and denominator.
  • Divide both numerator and denominator by this GCD.
For example, the given fraction is \(\frac{56}{16}\).
The GCD of 56 and 16 is 8.
So, you divide both by 8 to get \(\frac{7}{2}\).
This process reduces the fraction to its simplest form.
Greatest Common Divisor
The Greatest Common Divisor (GCD) is the largest number that can divide both the numerator and the denominator without leaving a remainder.
Finding the GCD helps in simplifying fractions effectively.
You can use several methods to find the GCD:
  • Prime Factorization: Break both numbers into prime factors and multiply the common prime factors.
  • Euclidean Algorithm: Involves division and remainders to find the GCD. It's efficient for larger numbers.
For instance, to simplify \( \frac{56}{16} \), you find that their GCD is 8.
Dividing both by 8 simplifies the fraction to \(\frac{7}{2}\).
Numerator and Denominator
Understanding numerators and denominators is essential for working with fractions.
The numerator is the top part of a fraction, signaling how many parts we have.
The denominator is the bottom part, showing into how many equal parts the whole is divided.
For example, in \( \frac{56}{16} \):
  • Numerator (56): Tells us we have 56 parts.
  • Denominator (16): Indicates that these parts are divided into 16 equal pieces.
When you simplify the fraction to \( \frac{7}{2} \), just remember: the concept is still the same.
You're expressing the same quantity but in a more manageable form. This understanding of numerators and denominators helps in many areas of math such as ratios, proportions, and division.

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